## 5 Conclusions  
In this work, the impact of the nonlinear Kerr effect on the focusing performance of an optical lens is studied by comparing the field patterns of focal spots for three types of incident high-intensity laser beams with different levels of light field intensities. Based on the theory of nonlinear optics and Fresnel diffraction integral formula, the transfer function of optical lens that takes the Kerr effect into consideration is derived for the incidence of high-intensity laser beams. The 2D-FFT algorithm was employed to reduce the computation resource for the focus of large incident laser beams, and the Kerr effect is considered by adding phase term proportional to the Kerr effect in order to get the light field distributions of focal spots in the nominal focal planes. The angular spectrum algorithm is utilized to calculate the field patterns of focal spots in the several transversal planes adjacent to the nominal focal plane for the three different types of incident laser beams. The obtained results show that the Kerr effect really affects the focusing performance of lens when the field intensity of incident laser lasers is above a threshold value. We find that the actual position of focal plane may be different from the  
Figure 14: The influence of different focal lengths and refractive indices of lens on the patterns of focal spots in the nominal focal plane for the three different incident laser beams of field intensity \(E_{0}=5\times 10^{8}\) V/m, the fundamental Gaussian beam, \(\text{HG}_{11}\) beam, and \(\text{LG}_{1}^{2}\) beam from left to right, where the lens parameters are (**a–c**) \(f_{0}=1\) m, \(n_{0}=1.5\); (**d–f**) \(f_{0}=2\) m, \(n_{0}=1.5\); (**g–i**) \(f_{0}=2\) m, \(n_{0}=1.7\), respectively.  
position of nominal focal plane, depending on the type and light intensity of incident laser beams. The focal spots become larger and more divergent for the cases of fundamental Gaussian beams and Hermite-Gaussian beams due to the Kerr effect of the lens. On the contrary, unlike the other two types of laser beams, the field distribution of focal spots for the case of Laguerre-Gaussian beam becomes more concentrated with the increment of light intensity. Meanwhile, the parameters of lens, such as the focal length and refractive index, also can influence the field patterns of focal spots. The theory, algorithms and results presented in this work have potential applications in research fields such as the high-power laser physics and the design of focusing lens in the systems of large ICF laser facility.  
Conceptualization, Z.L.; methodology, B.Y., Z.L., and J.P.; software, B.Y. and Z.L.; validation, X.C. and W.B.; formal analysis, B.Y. and J.P.; investigation, B.Y. and X.C.; data curation, B.Y. and W.Q.; writing--original draft preparation, B.Y.; writing--review and editing, Z.L.; visualization, B.Y. and W.Q.; supervision, Z.L.; project administration, Z.L.; funding acquisition, Z.L., X.C., W.Q. and J.P. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Science Foundation of China (Nos. 61101007, 61575070, 61605049, and 11774103), the Program for New Century Excellent Talents of Fujian Provincial Universities (No. MJK2015-54), the Key Program of Natural Science Fund for Young Scholars of Fujian Provincial Universities (No. JZ160408), the Natural Science Foundation of Fujian Province of China (No. 2018)01003), the Research Fund of Huaqiao University (No. 13B8408), and the Subsidized Project for Postgraduates' Innovative Fund in Scientific Research of Huaqiao University (No. 17013082013).  
The authors declare no conflict of interest.