## 3 Calculation Method and Parameter Selection  
The Fresnel diffraction integral formula given by Equation (14) is usually too difficult to be analytically calculated since the integrand may be complicated. Thus, the numerical methods are often applied. We can rewrite Equation (14) into the form of two-dimensional Fourier transform,  
\[E(u,\;v)=\frac{\exp[\mathrm{i}\frac{k}{2f_{0}}(u^{2}+v^{2})]}{\mathrm{j} \lambda f_{0}}\exp[\mathrm{j}kn_{0}\Delta_{0}]\times F\Big{\{}E(x,\;y)\exp \left[\mathrm{j}k\beta(x,\;y)\left|E(x,\;y)\right|^{2}\right]\Big{\}}\Big{|}_{ f_{x}=\frac{\alpha}{\lambda f_{0}},\;f_{y}=\frac{v}{\lambda f_{0}}} \tag{17}\]  
To numerically calculate the double integral of two-dimensional Fourier transform in Equation (17), the efficient algorithm based on the two-dimensional fast Fourier transform (2D-FFT) is utilized. For this purpose, we first spatially discretize the front plane \((x,y)\) of optical lens, make the 2D-FFT manipulation, and then apply the variable substitutions \(f_{x}=u/(\lambda f_{0})\) and \(f_{y}=v/(\lambda f_{0})\) to obtain the light field distribution in the focal plane \((u,v)\). Assume that the sizes of computational domain in the front lens plane are \(L_{x}\times L_{y}\), and the gird numbers of the two-dimensional discrete mesh are \(N_{x}\times N_{y}\) with discrete grid sizes \(\Delta x=L_{x}/N_{x}\) and \(\Delta y=L_{y}/N_{y}\). Then, the discrete parameters in the spatial domain and the frequency domain satisfy the relationship  
\[ \begin{array}{ll}\Delta f_{x}=\frac{1}{\lambda xN_{x}}=\frac{1}{L_{x}},\; \Delta u=\lambda f_{0}\Delta f_{x}=\frac{\lambda f_{0}}{L_{x}},\\ \Delta f_{y}=\frac{1}{\lambda yN_{y}}=\frac{1}{L_{y}},\;\Delta v=\lambda f_{0} \Delta f_{y}=\frac{\lambda f_{0}}{L_{y}}.\end{array} \]It can be seen that the discrete grid sizes in the focal plane, \(\Delta u\) and \(\Delta v\), are determined by the sizes of computational domain, \(L_{x}\) and \(L_{y}\), rather than the grid numbers, \(N_{x}\) and \(N_{y}\). In practice, the of computational sizes \(L_{x}\) and \(L_{y}\) are determined by two factors--the waist size of incident laser beam and the grid fineness of focal plane. In order to avoid the aliasing errors in the 2D-FFT manipulation, it is necessary to ensure that the spectral components of the original function satisfy the Nyquist sampling conditions [18; 19]. That is, the highest spectral component of  
\[U(x,\ y)=E(x,\ y)\exp\left[\mathrm{i}k\beta(x,\ y)\big{|}E(x,\ y)\big{|}^{2}\right] \tag{18}\]  
should satisfy the conditions, \(f_{x\mathrm{max}}\lesssim 1/(2\Delta x)\) and \(f_{y\mathrm{max}}\lesssim 1/(2\Delta y)\). The requirements can be utilized to choose the proper grid sizes, \(\Delta x\) and \(\Delta y\). Then the grid numbers, \(N_{x}\) and \(N_{y}\), can be determined. For the incidence of an axisymmetric laser beam, the square computational domain with square discrete grids is set, so that \(L_{x}=L_{y}=L\) and \(N_{x}=N_{y}=N\). In the following calculations, the nominal focal length and thickness of the focusing lens made by fused silica glass are assumed to be \(f_{0}=2.2\) m and \(\Delta_{0}=0.01\) m, respectively. The wavelength of incident laser beams is \(\lambda=351\) nm in vacuum, which are usually applied in large laser facilities for realizing ICF. At this wavelength, the linear portion of refractive index of fused silica glass is \(n_{0}=1.4767\)[20], the nonlinear refractive-index coefficient of light intensity is \(\gamma=3.0\times 10^{-16}\) cm\({}^{2}\)/W [21], and the nonlinear refractive-index coefficient of squared electric-field magnitude is \(\alpha=\varepsilon_{0}cn_{0}\gamma/2\approx 5.3\times 10^{-23}\) m\({}^{2}\)/V\({}^{2}\). Moreover, the loss coefficient of fused silica is commonly between \(8\times 10^{-6}\) and \(2\times 10^{-5}\), depending on the purity of material. Thus, the absorption effect and loss of optical lens is ignored in this work. In this work, three classical types of incident laser beams are considered.