## 4 Conclusions  
In conclusion, an automatic optimized method for a DOPC system based on the MPGA has been presented to optimize the compensation quality of DOPC. This method improves the compensation quality of DOPC by compensating the overall phase introduced by the problems of pixel alignment, optical aberration, and mechanical error. For comprehensively solving the above problems, we introduced the concept of global optimization by Zernike polynomials (Zernike modes) to characterize overall imperfections and used the MPGA to search for the optimal Zernike coefficients in the DOPC system. In addition, to clearly characterize the advantages of Zernike modes, we introduced the \(\Delta\)-xyz modes as optimization parameters for comparison. Experimentally, the PBR of the OPC focus with Zernike modes increased 174 times to that of the original focus, which achieves about 3.3 times the enhancement of compensation effect compared with the \(\Delta\)-xyz modes. The correlation coefficient increased from the original 0.41 to 0.78, which is better than the 0.73 score of the \(\Delta\)-xyz modes. Furthermore, we evaluated the optimization results of the proposed method with the fitness functions of intensity fitness and correlation coefficient fitness in the MPGA. The results show that the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitness Function** & **Normalized** & **Correlation** & **PBR** & **FWHM** \\  & **Intensity** & **Coefficient (\(\gamma\))** & & \\ \hline Intensity fitness & 0.55 & 0.68 & 20,400 & 23.3 \(\upmu\)m \\ Correlation & 0.77 & 0.78 & 54,000 & 19.7 \(\upmu\)m \\  & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of optimization abilities of two fitness functions.  
Figure 5: Analysis of experimental results in intensity fitness and correlation coefficient fitness in the MPGA. (**a**) Typical normalized intensity in intensity fitness and correlation coefficient fitness; (**b**) correlation coefficient in intensity fitness and correlation coefficient fitness; (**c**) line profiles of the central rows of focal points in intensity fitness and correlation coefficient fitness.  
optimized capability is excellent and the correlation coefficient is more suitable as a fitness function in Zernike modes. Moreover, it should be mentioned that this method only demands a rough initial pixel alignment. In the future, further optimization of the devices is expected to further shorten the optimization time of the algorithm and improve experimental efficiency [48]. For different grit sizes of diffuser media, the smaller grit size introduces larger speckles and output angles [49]. The larger output angle exceeds the collection capacity of the microscope objective. It will reduce the measurement accuracy of scattering wavefronts in the scattering medium and affect the final PBR of DOPC. Therefore, for different grit sizes of diffuser media, it is necessary to use the appropriate microscope objective to collect maximum scattering information. Finally, we hope that this work contributes to the practical uses of the DOPC system.  
[PERSON] and [PERSON] came up with the idea; [PERSON] and [PERSON] designed the experiment and the software program; [PERSON], [PERSON], and [PERSON] performed the experiment; [PERSON] analyzed the data and wrote the original manuscript; [PERSON], [PERSON], and [PERSON] did the review and editing. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Shanghai Sailing Program (18 YF1425900), the National Natural Science Foundation of China (11774364), the Bureau of International Cooperation, the Chinese Academy of Sciences (I81231 KYSB20170022), and the Strategic Priority Research Program of Chinese Academy of Sciences Grant No. (XDA25020302).  
The authors declare no conflict of interest.