## 1 Introduction  
The continuous development of optical technologies has allowed for nano-scale measurements beyond the diffraction limitation [1]. This has led to nano-scale positioning measurements, giving rise to high-precision three-dimensional spatial positioning of fluorescent nanoparticles in cells. This technology has a major impact on the life sciences, drug design, and cancer treatment [2; 3; 4; 5].  
Through point spread function engineering, the spatial positioning of nanoparticles can be quickly and accurately realized [6]. The double-helix point spread function (DH-PSF) is a special point spread function which is the superposition of a group of Laguerre-Gaussian modes [7]. By designing a special phase modulation plate, the double-helix spots image of nanoparticles can be obtained. There is a functional relationship between the rotation angle of the double-helix spots and the defocus distance of the nanoparticles. In this manner, the nano-scale spatial positioning of fluorescent nanoparticles can be realized [8]. DH-PSF composed of Laguerre-Gaussian mode superposition is widely used in spatial nanoparticle positioning [1; 9; 10]. However, a problem is that the light intensity of double-helix spots is not fully concentrated in the main lobes but partially scattered to the side lobes, which results in poor imaging quality [11]. Moreover, during the imaging process,the optical system inevitably has aberrations that blur the double-helix spot image and affect the positioning accuracy [12].  
To solve the problem of low light intensity concentration in the main lobes, [PERSON] et al. added an axicon before the phase modulation plate to suppress the side lobes [13]. However, this method failed to solve the aberration problem of the imaging system. The best annular spiral phase plate was obtained by eliminating the side lobes of the vortex beam [14]. This was achieved by selecting an annular spiral phase plate with an appropriate width that used the different energy sources of the vortex beam, i.e., the main and side lobes [15; 16]. However, because this method involves amplitude modulation, it can significantly reduce the light intensity utilization in the optical path and impose significant restrictions under the condition of weak light imaging. The adaptive optical aberration correction method [17] can effectively improve the aberration problems of the imaging system. However, it requires complex optical devices and algorithms to measure the wavefront, which induces a high cost and increases the complexity of the system, hindering broader use [18; 19]. The aberration optimization algorithm based on deep learning [20] can solve this problem to a certain extent. However, training the neural network requires a large number of samples and data. For small-data research such as cell imaging, errors can be large. The uncertainty level can be increased, and the applicability is limited [21]. The aberration correction optimization algorithm, which is based on the particle swarm optimization algorithm, obtains the optimal solution by adjusting the weight of each item of Zernike polynomials and taking the image information entropy as the evaluation standard of image sharpness [22; 23]. However, this algorithm cannot ensure the double-helix characteristics of the modulation function, which means the functional relationship between the defocus distance and the double-helix spots rotation angle in the optimization process.  
Thus, adopting the Laguerre-Gaussian mode and Zernike polynomial as the core techniques, and based on an iterative algorithm, this paper proposes a DH-PSF optimization algorithm to obtain the optimized phase modulation plate. The imaging quality was compared before and after processing optimization by building a 4f positioning optical path system. Simulations and experimental results indicated that the optimization method proposed here could increase the main lobes' light intensity concentration by more than 45%, and could significantly improve the imaging quality.