Due to the influence of the processing error of optical elements, system assembly error, and ambient temperature, the optical system usually has static aberrations, which reduce the imaging quality and significantly affects the optical performance. The stochastic parallel gradient descent (SPGD) algorithm is widely used to correct static aberrations. For the SPGD algorithm, the performance metrics, usually including the Strehl ratio (SR), encircled energy (EE), and mean radius (MR), have a great influence on the correction accuracy. In practical applications, SR is seldom chosen as it is difficult to achieve. In terms of EE, the convergence speed and correction accuracy depend closely on the selected encircled area, and hence, EE can only represent the total energy distribution of the surrounding area but cannot effectively reflect the entire intensity distribution. In contrast, the performance metric MR takes the whole intensity distribution into consideration, and higher correction accuracy is thus obtained. However, it is sensitive to small disturbance voltage, which makes the correction unstable. Therefore, to achieve high-precision and stable correction of static aberrations, we propose a combination method of performance metrics, which can not only concentrate the spot energy but also make the energy distribution uniform.

The proposed method combines the performance metrics EE and MR to correct the static aberrations. EE is first chosen as the performance metric and is computed with the acquired image. The control voltage is calculated and applied on the deformable mirror to correct the distorted wavefront. When most of the energy is concentrated in the encircled area, the performance metric is switched from EE to MR. Afterward, the energy distribution is further unified, which can also reduce the MR of the light spot. Thus, with the combination method, the energy can be better concentrated, and the intensity distribution can be more uniform. Meanwhile, the static aberrations can be corrected with higher accuracy.

First, to concentrate the energy at the center of the image plane, we use EE for correction. After the energy is concentrated, it is then corrected with MR so that the energy distribution can be more uniform (Fig. 3). EE, MR, and the combination method of performance metrics are simulated and compared to verify the effectiveness of the proposed method (Fig. 12), and the root-mean-square values of residual aberrations are 0.22λ, 0.43λ, and 0.01λ, respectively. Compared with the EE and MR methods, the combination method can achieve better spot image quality and dramatically increased peak intensity. In addition, the corresponding SRs are computed to be 0.53, 0.78, and 1.00, respectively. Moreover, the simulations of the correction results based on the three methods under different noises (Fig. 14), different encircled diameters (Fig. 15), and multiple random static aberrations (Fig. 16) are compared and analyzed. For the combination method, SR remains stable at 1.00 after correction. Finally, an experiment is performed to further validate the proposed method. As a result, the corrected resolutions are improved to be 2.15, 1.40, and 1.05 times the diffraction limit for the three methods, respectively, and in particular, the diffraction limit of the optical system is almost achieved with the combination method (Fig. 18). The research reveals that the proposed method can realize higher correction accuracy and stability.

The combination method of performance metrics proposed in this paper can effectively improve the correction accuracy and stability of static aberrations. Simulations show that the SR corrected by the combination method can keep stable at 1.00 under different encircled diameters, different noises, and multiple static aberrations. As further demonstrated by experiments, with the combination method, the light spot is the most focused, and the resolution is improved to be 1.05 times the diffraction limit. Both the experimental results and simulations confirm the effectiveness of the proposed method in static aberration correction. The method provides a facile and effective way for the correction and elimination of static aberrations in optical systems with optical performance close to the diffraction limit.