## 2 Theoretical Background  
According to the nonlinear optics, the Kerr effect is a nonlinear optical effect occurring when the highly intense light is interacting with crystals, glasses, gases or other media. Under the light field strength \(\mathbf{E}(\omega)\) of a high-power laser with angular frequency \(\omega\), the total polarization of a nonlinear optical medium up to the third order is described by [14]  
\[\mathbf{P}(\omega)=\mathbf{P}^{(1)}(\omega)+\mathbf{P}^{(3)}(\omega)=\varepsilon_{0} \chi^{(1)}\mathbf{E}(\omega)+3\varepsilon_{0}\chi^{(3)}\big{|}\mathbf{E}(\omega)\big{|} ^{2}\mathbf{E}(\omega), \tag{1}\]  
where \(\mathbf{P}^{(1)}\) and \(\chi^{(1)}\) are the first-order linear polarization and susceptibility, respectively; \(\mathbf{P}^{(3)}\) and \(\chi^{(3)}\) are the third-order nonlinear polarization and susceptibility, respectively, which also represent the Kerr effect; and \(\varepsilon_{0}\) is the permittivity of free space. By comparing the constitutive equation \(\mathbf{D}(\omega)=\varepsilon(\omega)\mathbf{E}(\omega)\) and the polarization relationship \(\mathbf{D}(\omega)=\varepsilon_{0}\mathbf{E}(\omega)+\mathbf{P}(\omega)\), we have the dielectric constant of a nonlinear optical material described by  
\[\varepsilon(\omega)=\varepsilon_{0}\Big{[}1+\chi^{(1)}+\frac{3}{4}\chi^{(3)} \big{|}\mathbf{E}(\omega)\big{|}^{2}\Big{]}, \tag{2}\]  
Based on the formula of refractive index for nonmagnetic materials, \(n=\sqrt{\varepsilon_{r}}=\sqrt{\varepsilon/\varepsilon_{0}}\), we have the complete expression for the refractive index \(n\) of a Kerr medium  
\[n=n_{0}\sqrt{1+\frac{3\chi^{(3)}}{4n_{0}}\big{|}\mathbf{E}(\omega)\big{|}^{2}}, \tag{3}\]  
where the linear portion of refractive index \(n_{0}\) is given by \(n_{0}=\sqrt{1+\chi^{(1)}}\). when the nonlinear item \(3\chi^{(3)}\big{|}\mathbf{E}(\omega)\big{|}^{2}/(4n_{0})\) in the square root is much less than unit one, we obtain the approximate but simplified expression of refractive index  
\[n\approx n_{0}+\alpha\big{|}\mathbf{E}(\omega)\big{|}^{2}, \tag{4}\]  
where \(\alpha{=3\chi^{(3)}/(8n_{0})}\) is the nonlinear coefficient. It can be seen that the refractive index of a Kerr medium is depending on the square of field magnitude \(\big{|}\mathbf{E}(\omega)\big{|}^{2}\), which is proportional to the light intensity \(I(\omega)\) of the incident laser beam. Suppose that the optical lens is located on the transverse plane with coordinates \((x,y)\) that is perpendicular to the optical axis. Then the phase transfer function of lens is  
\[\varphi(x,\ y)=kn(x,\ y)\Delta(x,\ y)+k[\Delta_{0}-\Delta(x,\ y)], \tag{5}\]  
where \(\Delta_{0}\) is the maximum thickness of lens, \(\Delta(x,\ y)\) is the thickness function of lens and \(n(x,\ y)\) is the space-dependent refraction index of lens at position \((x,\ y)\). If the lens is considered as the superposition of three parts, as shown in Figure 1, the total thickness function can be written as  
\[\Delta(x,\ y)=\Delta_{1}(x,\ y)+\Delta_{2}(x,\ y)+\Delta_{3}(x,\ y), \tag{6}\]which is the sum of three individual thickness functions. According to the geometric relationship illustrated in Figure 1, the thickness function \(\Delta_{1}(x,\ y)\) of the left part is given by  
\[\Delta_{1}(x,\ y)=\Delta_{01}-(R_{1}-\sqrt{{R_{1}}^{2}-x^{2}-y^{2}})=\Delta_{01} -R_{1}(1-\sqrt{1-\frac{x^{2}+y^{2}}{{R_{1}}^{2}}}). \tag{7}\]  
where \(R_{1}\) is the radius of the left spherical surface of optical lens and \(R_{1}=\infty\) for a planar lens surface. Similarly, the right part has a thickness function given by  
\[\Delta_{2}(x,\ y)=\Delta_{02}-(R_{2}-\sqrt{{R_{2}}^{2}-x^{2}-y^{2}})=\Delta_{0 2}+R_{2}(1-\sqrt{1-\frac{x^{2}+y^{2}}{{R_{2}}^{2}}}). \tag{8}\]  
where \(R_{2}\) is the radius of the right spherical surface of optical lens and \(R_{2}=\infty\) for a planar lens surface.