## 4 Results and Discussions  
### High-Intensity Fundamental Gaussian Beam  
The normalized cross-sectional light field distribution of a fundamental Gaussian beam, \(E(x,y)/E_{0}\), on the waist plane with waist radius \(w_{0}=100\) mm is depicted in Figure 2a. It can be seen from Figure 2a that the radius of the circle where the magnitude of light field drops to 1/e of the central maximum value is 99.61 mm. Three different levels of light field intensities of incident laser beams with \(E_{0}=1\times 10^{8}\) V/m, \(5\times 10^{8}\) V/m and \(1\times 10^{9}\) V/m are applied, which are below, near and above the maximum damage threshold of fused silica, \(40\) GW/cm\({}^{2}\)[22]. The different field patterns of focal spots in the nominal focal plane of lens are shown in Figure 2c,d corresponding to the above three cases, respectively. It can be seen from Equation (3) that the different changes in refractive index for different levels of light intensities result in the different patterns of focal spots. For the case of incident laser beam with \(E_{0}=1\times 10^{8}\) V/m as shown in Figure 2b, the radius of focal spot on which the electric field magnitude drop to 1/e of the central maximum intensity is 2.4 \(\upmu\)m. Further comparative study shows that the shape and radius of focal spot nearly do not change when \(E_{0}<1\times 10^{8}\) V/m, in which situation the impact of Kerr effect can be ignored. For the case with \(E_{0}=5\times 10^{8}\) V/m, the shape of focal spot is different from that of the former case, but the Gaussian-type distribution still holds, and the radius of focal spot with 1/e of field magnitude increases to 2.9 \(\upmu\)m. However, as for the case with \(E_{0}=1\times 10^{9}\) V/m, it can be seen that the Kerr effect has significant influence on the focusing performance of optical lens. It is noted that there are two distinct annular light rings appearing around the center focus. By measuring the obtained result, the radii of the two annular light rings are 3.9 \(\upmu\)m and 11.9 \(\upmu\)m, respectively.  
In order to further evaluate the impact of Kerr effect on the focusing performance of lens for the Gaussian beams with three different intensities, the light field intensities of focal spots are normalized by \(E_{0}\). The radial distributions of light field intensities of focal spots are plotted in Figure 3. It can be seen that as the intensity of the incident laser beams increases, the relative light field intensity of the central focus becomes lower. The light energy becomes divergent and annular light rings gradually formed. 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 one-third of that of the first case with the lowest intensity \(E_{0}=1\times 10^{8}\) V/m. The obtained results imply that, in order to avoid the influence of the Kerr effect on the focusing performance of lens, the intensity of the incident laser beam must be less than a certain value, which is relevant to the type of material used for lens fabrication. For the fused silica, the threshold of field strength is approximately \(E_{0}=1\times 10^{8}\) V/m.  
focal plane as that shown in Figure 2c for the convenience of comparison. We find that the focal spots gradually diverge with the increment of interval distance \(\Delta z\). In Figure 5, we also present the patterns of focal spots in the six planes the same as those specified in Figure 4 when the Kerr effect is not taken into consideration by setting the coefficient \(\alpha=0\). It can be seen that the sizes and patterns of the focal spots are nearly the same in the six planes. Thus, the Kerr effect of optical lens enhances the diffraction of focal spots.