## 1 Introduction  
With the rapid development of high-energy and high-power laser technology, laser irradiation intensity can reach \(10^{22}\) W/cm\({}^{2}\) (equivalent to field strength \(E_{0}=2.7\times 10^{14}\) V/m, and peak power can be increased to the order of a petawatt (\(10^{15}\) W). In the near future, laser power will reach 10 PW or even 100 PW, and laser intensity will exceed the order of \(10^{23}\) W/cm\({}^{2}\). This technology provides a powerful experimental means for the autonomous and controllable research of extreme physical state in the laboratory and in the research field of inertial confinement fusion (ICF) [1; 2], which is a process where nuclear fusion reactions are initiated by high-power lasers and compressing a fuel target, typically in the form of a pellet that most often contains a mixture of deuterium and tritium. Meanwhile, the impact of the nonlinear optical effect of materials on the working performance of optical components has become one of the non-negligible considerations of scientists and engineers. When a high-intensity laser beam with non-uniform cross-sectional field distribution is incident on an optical component, it can cause the space-dependent change of refractive index of optical material due to the Kerr effect according to the theory of nonlinear optics. Consequently, the working performance of optical components with extremely high intensity laser incidence is different from that with relatively low intensity laser incidence [3]. The optical lens is an important optical component that consists of a single piece of transparent material with curved sides that can focus or diverge an incident light beam by means of light refraction. Here, we focus on the numerical and quantitative analysis of the influence of Kerr effect on the focusing performance of a convex optical lens with extremely intense laser incidence. In the literature, some traditional methods have been proposed to study the effect of optical nonlinearity on the focusing performance of optical components. As early as 1977, [PERSON] presented a general method for describing self-focusing phenomena in various media by means of a geometrical-optics approach [4]. In 2004, [PERSON] discussed the paraxial lens approximation and self-focusing theory [5]. In the same year, [PERSON] and [PERSON] proposed a new ray-tracing scheme to simulate the nonlinear processes of ultra-short pulse propagation, including the self-focusing and self-guiding [6]. They divided the propagation media into multiple thin layers and traced the evenly spaced light rays in the transverse plane to simulate the Kerr effect. In 2014, based on the work of [PERSON] and [PERSON] [6], [PERSON] and [PERSON] proposed a multilayer thin-lens self-focusing model to understand the formation and propagation characteristics of self-focusing beam spot inside nonlinear samples based on Fermat principle, which was utilized to calculate the light field distributions of nanoscale spots on the focal plane [7]. However, the ray-tracing approaches based on geometric optics have inherent shortcomings since the light field distributions are obtained indirectly according to the relationship between the number density of light rays and light intensity, which is with limited simulation accuracy. By using the finite-difference time-domain (FDTD) method, [PERSON] et al. analyzed the self-focusing effects in a nonlinear Kerr film and showed the formation of multiple filamentations. This method can discretely calculate the full-wave spatio-temporal distribution of optical fields, but its consumption of computer resource is also great and only micron-scale problems can be modeled and simulated [8]. In 2011, [PERSON] et al. presented an alternative wide-angle beam propagation method (BPM) to model nonlinear Kerr-type optical devices by using a reformulated Fourier-based complex Jacobi iterative technique [9].