## 4 Results and Discussions
### High-Intensity Hermite-Gaussian Beam  
As the second example, the incident beam is assumed to be the Hermite-Gaussian beam of HG\({}_{11}\) mode. The normalized light field distributions of the incident HG\({}_{11}\) laser beam in the lens plane is shown in Figure 6a with waist size \(w_{0}=0.1\) m. Figure 6b-d show the different field patterns of focal spots in the nominal focal plane corresponding to three incident laser beams of different  
Figure 4: The different field patterns of focal spots in the six planes, which are of (**a**) 0.03 mm, (**b**) 0.02 mm, and (**c**) 0.01 mm left to the nominal focal plane, respectively, and (**d**) 0 mm, (**e**) 0.01 mm, and (**f**) 0.02 mm right to the nominal focal plane for the incident fundamental Gaussian beam with field intensity \(E_{0}=5\times 10^{8}\) V/m. Figure 5: The different patterns of focal spots in the six planes the same as those specified in Figure 4 for the incident fundamental Gaussian beam with field intensity \(E_{0}=5\times 10^{8}\) V/m, where the Kerr effect of optical lens is not taken into consideration by setting the nonlinear coefficient \(\alpha=0\). intensities with \(E_{0}=1\times 10^{8}\) V/m, \(5\times 10^{8}\) V/m and \(1\times 10^{9}\) V/m, respectively. Similarly, for the case with \(E_{0}=1\times 10^{8}\) V/m, the pattern of focal spot shown in Figure 6b is similar to that of incident light field shown in Figure 6a, which indicates that the Kerr effect has little influence on the lens's focusing performance when the light field intensity is smaller than \(E_{0}=1\times 10^{8}\) V/m. However, for the case with \(E_{0}=5\times 10^{8}\) V/m, it can be seen that the pattern of focal spot is significantly different from the former case. The pattern of focal spots changes from four spots into an array of spots. It is also noted that the closer to the periphery, the lower the intensity of spots. If we further increase the intensity of incident laser beam to \(E_{0}=1\times 10^{9}\) V/m, the divergence of light energy become more apparent. Except for the central four focused spots, many light speckles are distributed on the entire calculation domain. Further calculation verifies that the increment of spatial sampling density and the increased number of discrete grids do not change the pattern of focal spots. Therefore, the speckles are correct results, but not due to under-sampling and aliasing phenomena in the numerical calculation implemented by the 2D-FFT algorithm. In order to fairly compare the field patterns of focal spots shown in Figure 6b-d for the three different intensities of incident laser beams, we first normalize the field intensities of focal spots by \(E_{0}\) and the normalized light field distributions along the diagonal lines of computational domains are plotted in Figure 7. It can be seen that as the intensity of incident laser beam increases, the degree of divergence of focal spots is higher, and the intensity of the central focal spot becomes lower. In particular, for the case with \(E_{0}=5\times 10^{8}\) V/m, the pattern of the four central focal spots are more complicated than the other two cases. It is found that the normalized light field intensity of the third case with \(E_{0}=1\times 10^{9}\) V/m is about 0.4 of that of the first case with \(E_{0}=1\times 10^{8}\) V/m. Figure 6: (**a**) The normalized cross-sectional light field distribution of incident Hermite-Gaussian beam (HG\({}_{11}\)) in the lens plane. The three field patterns of focal spots corresponding to the incident Hermite-Gaussian beams with three different light field intensities: (**b**) \(E_{0}=1\times 10^{8}\) V/m, (**c**) \(E_{0}=5\times 10^{8}\) V/m, and (**d**) \(E_{0}=1\times 10^{9}\) V/m.