## 2 Optimization of the Positioning Method of the Nano-Scale Double-Helix Point Spread Function
### Algorithm Simulation and Optimization Results  
At the end of each iteration, the algorithm computes the average efficiency of the main lobes, i.e., the ratio of the main lobe energy of the spot over the total energy of the light intensity distribution, and compares it with the previous iteration. If the efficiency converges, the iteration ends. As shown in Figure 3, before the iterative optimization, the weight of each Laguerre-Gaussian mode was 1, and the average efficiency of the main lobe was 35.44%. After the first iterative optimization, the average efficiency of the main lobes was increased to 38.77% and the variation range was obvious compared to the initial average efficiency. After the 19 th iteration, the average efficiency saturates to a stable value ranging from 41.7 to 42.2%. The maximum value is 42.02%, which is 18.57% higher than the one prior to the optimization. After the algorithm implementation, the new weight value  
Figure 2: Iterative algorithm flow. of each Laguerre-Gaussian mode is obtained. Due to the double-helix characteristic of the point spread function, which is determined by its superimposed Laguerre-Gaussian mode, the contribution of each part to the sidelobe energy is different [24]. Therefore, according to the order of magnitude of the new weight value, the initial weight is adjusted before the optimization, keeping its double-helix characteristics unchanged, and, subsequently, the iterative algorithm is executed again. The variation trend observed is similar. At the 19 th iteration, the average efficiency of the main lobe converges, and the algorithm is terminated. By this time, the average efficiency of the main lobe of the spot remains stable between 51.3 and 51.8%, with a maximum of 51.57%. Compared to the average efficiency of the main lobes before the optimization process, the efficiency was improved by 45.51%. The CPU is an Intel Core [EMAIL_ADDRESS] GHz (Intel, Santa Clara, CA, USA) and the memory is 16.0 GB. Two rounds of iterative optimization are required according to the iterative optimization algorithm proposed in this paper. After the 19 th iteration, the average efficiency of the main lobes converged, and the algorithm ended with a time of 1355.072190 s. The two rounds took a total of 2710.14438 s.  
After the iterative optimization algorithm, the phase modulation function is shown in Figure 4. Figure 4a depicts the phase modulation function before the optimization, and Figure 4b shows the phase modulation function after the optimization. The overall structure of the optimized phase modulation function is the same as the one before the optimization, indicating that the double-helix property of the modulation function does not change significantly. Note that the ripple pattern in the center is significantly reduced and the phase variation of each part is gentler. Figure 3: Iterative optimization results. In the iterative process, the double-helix characteristics of the modulation function remain unchanged. Figure 5a-c illustrate the double spot patterns for defocus distances of \(-2\), \(0\) and \(2\) mm before optimization. Similarly, Figure 5d-f illustrate the double spot patterns at the same defocus distances after optimization. Figure 5 also shows that the double spot rotation angle at the corresponding defocus distance does not change before and after the optimization, indicating that the iterative optimization process has no impact on the double-helix characteristics of DH-PSF. Figure 4: (**a**) Phase modulation function before optimization; (**b**) phase modulation function after optimization. Figure 5: Optimize the double spot pattern at the defocus distance of \(-2\), \(0\), and \(+2\) mm before and after: (**a-c**) are double spot patterns before optimization; (**d–f**) are double spot patterns after optimization. Figure 6 shows the double spot pattern and the corresponding light intensity distribution curve at a defocus distance of 0 mm.