## 3 Calculation Method and Parameter Selection
In most cases, a laser emits light in the form of a laser beam often close to a fundamental Gaussian beam, where the transverse profile of the optical intensity of the beam can be described with a Gaussian function and the variation of beam size can be very small for beams with large width. Thus, the first type is the fundamental Gaussian beam, whose electric field is given by  
\[E(x,y,z)=E_{0}\frac{w_{0}}{w(z)}\exp\left(-\frac{x^{2}+y^{2}}{w(z)^{2}}\right) \exp\left[-\frac{\mathrm{i}k(x^{2}+y^{2})}{2R(z)}\ -\mathrm{i}kz+\mathrm{i}\varphi(z)\right], \tag{19}\]  
where \(E_{0}\) is the maximum field amplitude at the center of beam waist, \(w_{0}\) is the radius of beam waist, \(z_{\mathrm{R}}=\pi w_{0}^{2}/\lambda\) is the Rayleigh length, \(w(z)=w_{0}\sqrt{1+\left(z/z_{\mathrm{R}}\right)^{2}}\) is the beam radius at z, \(R(z)=z+z_{\mathrm{R}}^{2}/z\) is the radius of curvature of the beam's wavefront at \(z\), and \(\varphi(z)=\arctan(z/z_{\mathrm{R}})\) is the Gouy phase shift. Therefore, we can substitute Equation (19) into Equation (17) to calculate the light field distribution of focal spots in the nominal focal plane of lens. The second type are the Hermite-Gaussian beams. A Hermite-Gaussian beam is one of the higher-order solutions of the paraxial Helmholtz equation in Cartesian coordinate system. The expression of light field for the \((m,\ n)\)th order Hermite-Gaussian laser beam is written as  
\[ \begin{split} E(x,\ y,\ z)&=E_{0}\frac{w_{0}}{w(z) }H_{m}\left(\frac{\sqrt{2}\,x}{w(z)}\right)H_{n}\left(\frac{\sqrt{2}\,y}{w(z)} \right)\exp\left(-\frac{x^{2}+y^{2}}{w^{2}(z)}\right)\times\\ &\exp\left(-\mathrm{i}\frac{k(x^{2}+y^{2})}{2R(z)}\right)\exp \left(\mathrm{i}\varphi_{mn}(z)\right)\end{split}  \tag{20}\]  
where \(\varphi_{mn}(z)=(m+n+1)\tan^{-1}(z/z_{\mathrm{R}})\) is the Gouy phase shift for the Hermite-Gaussian beams, \(H_{m}(\cdot)\) and \(H_{n}(\cdot)\) are the \(m\)th and \(n\)th Hermite polynomials [16]. It should be noticed that the Hermite-Gaussian beams degenerate to the fundamental Gaussian beam when \(m=n=0\). The third type are the Laguerre-Gaussian beams. A Laguerre-Gaussian beam is one of the high-order solutions of the paraxial Helmholtz equation in cylindrical coordinate system. The light field of the \((p,\ l)\)th order Laguerre-Gaussian beam is given by \[E_{lp}(r,\,\phi,\,z)=E_{0}\frac{w_{0}}{w(z)}\Bigg{(}\frac{\sqrt{2}r}{w(z)}\Bigg{)} ^{\parallel l}L_{p}^{\parallel l}\Bigg{(}\frac{2r^{2}}{w^{2}(z)}\Bigg{)}\exp \Bigg{[}-\mathrm{i}\Bigg{(}\frac{kr^{2}}{2q(z)}+kz+l\phi-\varphi_{pl}(z)\Bigg{)} \Bigg{]}, \tag{21}\]  
where \(\varphi_{pl}(z)=(2p+\parallel l+1)tan^{-1}(z/z_{R})\) is the Gouy phase shift, and \(L_{p}^{l}(\cdot)\) is the associated Laguerre polynomial. A Laguerre-Gaussian beam of LG\({}_{p}^{l}\) mode is specified by two mode indices, the angular mode index \(l\) (\(l=0,\,\pm 1,\,\pm 2,\,\ldots\)) and the transverse radial mode index \(p\) (\(p=0,\,\,1,\,2,\,\ldots\)). When \(p=l=0\), the Laguerre-Gaussian beams degenerate to the fundamental Gaussian beam. When the angular mode index \(l\) is greater than zero, the light field has an azimuthal phase change of \(2\pi l\), which results in a phase singularity in the field and a node in the intensity at the center of the beam. Thus, the laser beams propagating in Laguerre-Gaussian modes may have orbital angular momentum and multiply connected topology.