## 2 Optimization of the Positioning Method of the Nano-Scale Double-Helix Point Spread Function  
### Optimization of Double Spot Main Lobes Efficiency  
As shown in Figure 1, the light intensity distribution of the double-helix spots consists of the main lobes and several side lobes. The main lobes refer to the two spots in the middle where the light intensity is at its maximum (as shown in Part a). The side lobes refer to the corrugated secondary light spots with lower intensity (as shown in Part b). The side lobes' intensity is gradually weakened outside of the main lobes, and the larger the topological charge of the Laguerre-Gaussian mode is, the larger the side lobe becomes [24; 25; 26]. Thus, based on the optical characteristics analysis of the image and the definition of the main lobe efficiency formula, this work proposes an iterative scheme to optimize the Laguerre-Gaussian mode. The core idea is to optimize the weight of each Laguerre-Gaussian mode through iteration. The purpose is to reduce the influence of the side lobes and the attendant aberrations on the energy concentration of the main lobe without affecting the characteristics of the defocus double helix. Analyzing the spiral double-spot shows that the main lobe distribution can be approximately considered as a Gaussian spot formed by the point light source which passes through the imaging system. Therefore, two Gaussian spots centered on the central point of the main lobe are selected as the final optimization goal. The energy efficiency of the main lobes refers to the proportion of the light intensity of the main lobes to the overall light intensity. Its calculation formula is  
\[\eta_{\rm main}=\frac{I_{main}}{I_{total}} \tag{1}\]  
where \(I_{\rm main}\) represents the light intensity of the main lobes and \(I_{\rm total}\) represents the overall light intensity. In the iterative algorithm, a group of defocus planes with a defocus distance of \(-2\) to \(+2\) mm were selected as optimization objects to ensure that the defocus rotation characteristics of the double spot did not change. The average main lobe efficiency refers to the average of the main lobe efficiency corresponding to these defocus distances. Its calculation formula is  
\[\overline{\eta}=\frac{\sum\limits_{j=1}^{K}I_{mainj}}{K} \tag{2}\]  
where I\({}_{\rm mainj}\) represents the main lobe efficiency at the j defocusing distance, and K represents the total K main lobe efficiency. To facilitate calculation, the main lobes were regarded as the same point light source to form two Gaussian light spots with the maximum light intensity of the main lobes as the center. The rest were calculated as side lobe intensity. The derivation process is as follows: according to [27], the expression of different Laguerre-Gaussian modes after linear superposition is  
\[\mathrm{S}_{k}(r)=\sum\limits_{j=1}^{Q}A_{j_{k}}u_{m_{j}n_{j}}(r) \tag{3}\]  
where \(k\) represents the \(k^{\rm th}\) iteration, \(A\) represents the weight coefficient of the Laguerre-Gaussian mode, and \(Q\) is the number of Laguerre-Gaussian modes. Then, the light field distribution of the point light source on the image plane can be expressed as  
\[U_{i}(x_{i},y_{i})=\frac{1}{i\lambda f}\iint U_{spp}(x_{spp},y_{spp})S(x_{spp},y_{ spp})\times\exp\biggl{[}-i\frac{2\pi}{\lambda f}(x_{i}x_{spp}+y_{i}y_{spp}) \biggr{]}dx_{spp}dy_{spp} \tag{4}\]  
The respective strength distribution can be simplified, leading to  
\[I_{1k}(x,y)=|U_{k}|^{2} \tag{5}\]  
Figure 1: The simulation image of double-helix spots (**a**) the main lobes; (**b**) the side lobes. The aberrations of the system are expressed as the sum of Zernike polynomials, and their expression is  
\[\phi(u,v)=\sum_{j=1}^{K}C_{j}Z_{j}(u,v) \tag{6}\]  
The final light intensity distribution at the image plane is  
\[I_{2k}(x,y)=I_{1k}\otimes\phi(u,v) \tag{7}\]  
The aforementioned analysis indicates that the ideal light intensity distribution is the Gaussian spot formed by the point light source that focuses on the focal point, and its size range is fixed.