## 4 Results and Discussions
### High-Intensity Laguerre-Gaussian Beam  
The last type of incident beams are the Laguerre-Gaussian beams. The field patterns of focal spots for Laguerre-Gaussian beams are different from the former two types of beams. Figure 10a is the normalized light field distribution of the incident Laguerre-Gaussian beam of \(\mathrm{LG}_{1}^{2}\) mode with \(w_{0}=0.1\) m. Figure 10b-d show the field patterns of focal spots in the nominal focal plane corresponding to three different light-field intensities, \(E_{0}=1\times 10^{8}\) V/m, \(5\times 10^{8}\) V/m and \(1\times 10^{9}\) V/m, respectively. It is found that the pattern of focal spots for the Laguerre-Gaussian beam with \(E_{0}=1\times 10^{8}\) V/m is quite different to those of fundamental Gaussian beam and Hermite-Gaussian beam. Except the two annual rings with different light field intensities, there is an exceptional focus at the center of rings. In the case shown in Figure 10c with \(E_{0}=5\times 10^{8}\) V/m, the pattern of focal spots is similar to that of Figure 10b. However, the light-field intensities of the two rings become weakened and the light energy of focal spots further concentrates to the central focus. When the intensity of the incident laser beam rises to the level with \(E_{0}=1\times 10^{9}\) V/m, the pattern is left with the dominant central focus and several negligible halos. The results demonstrate that when the light intensity of an incident Laguerre-Gauss beam reaches a certain level, the laser beam changes from a double-loop structure to a single high-intensity focus due to the nonlinear Kerr effect. To further analysis the impact of the Kerr effect on the focusing performance of optical lens for different intensities of incident laser beams, the radial distributions of light field intensities of focal spots normalized by their corresponding levels of \(E_{0}\) are plotted in Figure 11. It can be clearly seen that as the field intensity of incident laser beam increases, the light energy of outer rings is continuously concentrated to the central focus. For example, the relative light-field intensity of central focus in the case with the highest intensity \(E_{0}=1\times 10^{9}\) V/m is approximately twice of that of the first case with the lowest intensity \(E_{0}=1\times 10^{8}\) V/m and \(1.7\) of that of the median case with the medium intensity \(E_{0}=5\times 10^{8}\) V/m. It is interesting to find that the incident LG beam can be focused into the central bright focus when the Kerr effect of lens is significant, which is quite different from the hollow structure of the incident LG beam with a node in the intensity at the center of beam. Figure 9: The different patterns of focal spots in the six planes the same as those specified in Figure 8 for the incident Hermite-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 \(\alpha=0\). In order to fully understand the influence of the Kerr effect on the focusing performance of lens for the case of the Laguerre-Gaussian beam (LG\({}_{1}^{2}\)) with \(E_{0}=5\times 10^{8}\) V/m, the field patterns of focal spots in the six planes adjacent to the nominal focal plane, which are of 0.03 mm, 0.02 mm, and 0.01 mm left to the nominal focal plane and 0 mm, 0.01 mm, and 0.02 mm right to the nominal focal plane, are depicted in Figure 12a\(-\)f, respectively. It can be clearly seen from Figure 12 that the central spot becomes brighter and larger and the field intensities of the surroundings rings become weakened when the plane is away from the lens. In Figure 13, we also show the field patterns of focal spots in the six planes without consideration of Kerr effect with the coefficient \(\alpha=0\). It also can be found that the field patterns of focal spots shown in Figure 13 without Kerr effect are opposite to those shown in Figure 12 with the Kerr effect when the plane of interest is away from the nominal focal plane.