Atmospheric turbulence can affect the quality of light waves, thereby impacting the clarity and resolution of astronomical observations and the transmission of free-space optical communications^[1-2]. Adaptive optics technology was therefore proposed to correct phase aberrations caused by atmospheric turbulence using efficient wavefront sensing^[3-4].

The traditional Shack–Hartmann wavefront sensor and the shearing interferometer have problems that prevent them from restoring phase distortion very well. The Shack–Hartmann wavefront sensor can only measure the phase aberration with limited spatial resolution because of the sub-aperture constriction^[5-6]. The shearing interferometer splits the beam into two wavefronts, lowering the light energy utilization efficiency and subsequently reducing the accuracy of wavefront sensing^[7-8]. In recent years, image-based wavefront sensing has gained attention. With the help of deep learning (DL), wavefront reconstruction is now much more efficient and accurate^[9-12]. Paine et al^[13] predicted Zernike polynomials (ZPs) coefficients from a computational simulated point spread function (PSF) using a convolutional neural network (CNN) for wavefront reconstruction in 2018. Nishikazi et al^[14] experimentally verified the effectiveness of CNN in predicting coefficients of ZPs and estimating wavefront aberrations in 2019. Ge et al^[15] further used a DL network to achieve high-precision mapping of phase features to wavefront aberrations in phase reconstruction in 2024.

Most image-based wavefront sensing methods use ZPs, a classic way to represent phase aberrations, as mentioned above. The higher the term of ZPs used, the more high-frequency components captured, and the more accurate the restoration of phase aberrations^[16]. However, using more ZPs makes the computer work harder, making predictions by the CNN model less accurate^[17], while using fewer ZPs reduces the model’s generalization ability to deal with complex environments such as strong turbulence.

The statistical Principal Component Analysis (PCA) method is becoming popular. PCA can identify the most dominant features from a large dataset. Over the past few years, researchers have successfully applied PCA to reduce speckle noise^[18], denoise meteorological echoes^[19], and combine with neural networks to correct non-common path aberrations^[20].

Inspired by the successful implementation of PCA, we performed PCA in the representation of phase aberrations caused by atmospheric turbulence and proved its validity. This paper discusses the factors affecting the restoration accuracy by principal components (PCs), mainly including the size of sample space and sampling interval of $D/{r_0}$ on the basis of characterizing phase aberrations by PCs. Section 2 provides a basic overview of the method used in the paper, and Section 3 introduces the simulation process. Section 4 verifies the representation and restoration performance of the PCA method and compares with the traditional ZPs method. Our analysis demonstrates that the PCA method outperforms traditional ZPs across varying atmospheric turbulence strengths, especially in challenging situations such as strong turbulence, providing a statistical reference for data acquisition for PCs model deployment in real applications.

To simulate atmospheric turbulence accurately, we created a phase screen data set that satisfies the modified Von Karman power spectrum using the Fast Fourier Transform^[21], and the inner and outer scales of atmospheric turbulence were set as 0.005 m and 10 m. According to Noll^[22], the phase aberration $W(\rho ,\theta )$ can be represented as a combination of ZP ${Z_{{j}}}$$(\rho ,\theta )$ with coefficients ${a_j}$. This paper ignores the first three terms of ZPs, which do not change the morphology of the aberration, and focuses on the aberration above the 4^th term: defocus. As ZP patterns are generally grouped in terms of spherical aberration, it is generally accepted that aberration patterns before the tertiary spherical aberration carry more weight and that using more terms increases the computational burden. Still, the improvement in aberration restoration is not significant. So, the 4^th to 37^th ZPs (tertiary spherical aberrations) were used to fit the original phase aberration and set the piston, x tilt, and y tilt terms to 0 to generate a new phase aberration dataset, as shown in the following expression:

$ {W_1}(\rho ,\theta ) = \sum\limits_{j = 4}^{37} {{a_j}{Z_j}(\rho ,\theta )} \quad, $ (1)

where $\rho $ and $\theta $ are the radial and azimuthal variables in a polar coordinate. ZPs have a specific pattern for each term, with more information in higher terms. More terms allow the phase aberration to be restored more finely, but it is slower and less efficient. A few terms are therefore often used, but higher frequencies are often missed. We propose the PCA method, a useful statistical tool for reducing multiple complex variables, to represent and restore the phase aberrations. The old variables are combined to form new variables as:

$ {{\boldsymbol{y}}_{i}} = {\boldsymbol{V}}'{{\boldsymbol{x}}_i}\quad, $ (2)

where V is the transformation matrix and the symbol ' means transpose of the vector or the matrix. where ${{\boldsymbol{y}}_i} = ({y_{1i}},{y_{2i}}, \cdots ,{y_{ni}})'$ is the new variable, and ${{\boldsymbol{x}}_i} = ({x_{1i}},{x_{2i}}, \cdots ,{x_{ni}})'$ is the original variable. This approach keeps the information in lower- and higher-term components in each new single mode. The first $m$ new variables with the highest variance, i.e., the PCs, are selected to distill the essence of the original dataset. This summary shows the main features of the original data and reduces the number of dimensions. To be consistent with Eq.(1) above, the representation of the phase aberration fitted using the first 34 terms of PCs is shown below:

$ {W_2}(r ,q ) = \sum\limits_{j = 1}^{34} (b_jPC_j) \quad, $ (3)

where $PC_j$ is the $j^{th}$ PC term with coefficient $b_j $. To analyze the validity of PCA in different conditions, we need to create datasets that show different turbulence levels. Based on Lane’s derivation^[23], $D/{r_0}$ is used to characterize the strength with ${r_0}$ being the atmospheric coherence length and D being the pupil diameter. The numerical interval is $D/{r_0}$ from 1 to 30. 1−10 is relatively weak turbulence, 10−20 is medium, and 20−30 is relatively strong. Further, the PSF’s Strehl ratio (SR) is used to evaluate the effectiveness of phase restoration by performing a fast Fourier transform on the residual phase aberration.

This paper focuses on analyzing factors influencing the accuracy of PCs restoration of phase aberration: the original phase data’s sample space size and the sampling interval of $D/{r_0}$. In order to conduct a more comprehensive analysis of PCs’ restoration effectiveness in different propagation conditions, generating phase datasets that can encompass different turbulence strengths is necessary.

The simulation was performed according to the following steps:

(1) The original sample space was randomly generated containing N distorted phases, then a PCA was performed on the base data to generate the PCs. N = 5000, 10000, and 30000 were chosen.

(2) For the equivalent sample space size, two sample spaces were generated with different $D/{r_0}$ sampling intervals. Space A had an average distribution between 1−30, while space B had $D/{r_0}$=5, 15, and 25 as representative values of weak, medium and strong turbulence, respectively. A PCA was performed on both datasets A and B to extract corresponding PCs.

(3) The phase aberrations of the test set were restored under different turbulence strengths using 8, 19, 34 terms of ZPs and PCs obtained from different sample spaces. The terms employed were based on the Zernike primary, secondary, and tertiary spherical aberrations (11^th, 22^nd, and 37^th). That is, using the ZPs starting from the 4^th term of defocus and ending with the spherical aberration term, a total of 8, 19, and 34 terms of modes were included. The restoration effects of the two methods were compared.

Fig. 1 (color online) shows the restoration effects of equivalent terms of ZPs and PCs obtained from A-5000, A-10000, and A-30000 datasets. In order to provide a comprehensive demonstration of the restoration effects, we took test sets of $D/{r_0}$= 8, 16, and 24, and restoration was performed using the first 8, 19, and 34 terms PCs and ZPs, respectively. The terms employed start with 4^th ZP: defocus, and end with the Zernike primary, secondary, and tertiary spherical aberrations (11^th, 22^nd, and 37^th). Fig. 1 shows the positive correlation between the original data amount and the PCs’ restoration effect. When 5000 sets of original phase data are used, the advantage of PCs may not be immediately apparent. The corrected SR obtained using 8 terms of PCs obtained from 5000 sets of original data in Fig. 1(a) is only slightly higher than SR obtained by ZPs; as the amount of original data increases, the accuracy of PCs improves, and the gap with ZPs gradually widens. When using the PCs generated from 30000 sets of original data, the corrected spot is clear and bright as shown in Fig. 1(c), where the turbulence is stronger and the SR can reach 0.2183. At this point, the SR after restoration using PCs obtained from 5000 sets is 0.1561, while the SR after restoration using equivalent terms of ZPs is only 0.1171. It can be seen that the advantage of PCs is more significant under strong turbulence, showing that PCs are stable and consistent and can adapt to different environments.

Furthermore, Fig. 2 (color online) shows the effect of using different amounts of terms of PCs to restore the same phase aberration under the same turbulence strength. The PCs used were extracted from 30000 sets of original data. Combined with Fig. 1, it shows that under weak turbulence, the phase aberration can be effectively restored using only 8 terms of PCs, and the corrected SR reaches 0.4131, further indicating that the PCs can effectively extract the main features of the phase aberrations; under medium turbulence, the aberration can be effectively restored using 19 terms of PCs, and under strong turbulence, the phase aberration can still be stably restored using 34 terms of PCs, and the corrected SR is about 0.2. This result again shows that PCs have the robustness to manage different levels of environmental turbulence and may be more suitable than ZPs for working under challenging conditions.

Fig. 3 (color online) further explores the mean SR after restoring phase aberrations at different turbulence strengths using equivalent terms of ZPs and PCs that were extracted from different sample spaces. It shows that the PCs obtained from the 30000 sets of original phase data are the most effective in restoring the phase aberrations. This is because the larger the data volume, the broader the model. Notably, PCs from 10000 sets (blue triangular scatter lines) and 30000 sets (orange circular scatter lines) almost overlap in Fig. 3, suggesting sample space size has limited impact. If the original data already covers enough phase information, adding more sample data won’t significantly improve the accuracy of PCs, but will just add redundant data, which will hinder the rapid deployment of models in real-world applications. This provides an important basis for model optimization of PCs in practical applications.

Although increasing the size of the sample space can improve the accuracy of PCs, it can also increase the computational and time costs. In the actual model deployment, it’s desirable to obtain the most accurate PCs of the phase aberrations caused by the local atmospheric turbulence with the shortest possible sampling time. The $D/{r_0}$ sampling interval was increased from 1 in A-space to 10 in B-space to analyze the impact of the sampling interval on the accuracy of the PCs. Fig. 4 compares the restoration for the same phase aberrations by PCs obtained from A-5000 and B-5000. It demonstrates that PCs obtained from the B-5000 with larger sampling intervals are significantly better for restoration than those from the A-5000.

Tab. 1 compares the restoration by PCs from B-5000 and A-30000, demonstrating that when the sampling interval is increased, the accuracy of the PCs obtained from only 5000 sets of sampled data can be comparable to that obtained from 30000 sets of sampled data.

This is because when the sampling interval is increased, the number of sampling points is reduced, and the size of the sample space corresponding to a single sampling point is increased, enabling the dataset to contain more varied information in a wider range. Therefore, when it is necessary to deploy the PCs model quickly for restoration, the sample space size can be reduced and the sampling interval can be increased to ensure the model’s generalization ability and robustness while minimizing the data’s redundancy.

Fig. 5 (color online) shows some examples of restoration by PCs generated from the B-5000 and ZPs of equivalent terms. Fig. 5(c) indicates that PCs are more effective than ZPs for restoring phase aberration in challenging environments, such as strong turbulence. Using the first 34 terms, the SR of the PSF after restoration by PCs reached 0.1585, which meets the requirements of engineering applications. At this time, the SR of the PSF after restoration by ZPs is only 0.02, and compared with the SR of the original spot PSF, which is 0.007, the effect of the ZP for restoration is almost negligible. This shows that PCs perform better, further supporting the stability of PCs demonstrated in Fig. 1 and 3.

In this paper, the factors affecting the restoration accuracy of PCs, mainly sample space size and the $D/{r_0}$ sampling interval, are discussed in depth on the basis of characterizing phase aberrations by PCs. The results show that the more sample data, the higher the accuracy of the PCs and the better the adaptability in restoring phase aberrations in different environments. However, when the sample space contains enough phase information, a further increase in sample data no longer improves the accuracy of the PCs but results in data redundancy. When the model needs to be deployed quickly, the generalizability and robustness of PCs can be ensured by appropriately reducing the sample space size and increasing the sampling interval.

In general, our work demonstrates that the PCA method outperforms traditional ZPs across varying atmospheric turbulence strengths, especially in challenging situations such as strong turbulence, indicating that the PCA method can serve as a better alternative in restoring the phase aberrations induced by atmospheric turbulence. These findings may help to reduce data dimensionality, i.e., using PCs of phase aberrations with fewer terms than traditional ZPs, which can help improve model and deep learning based adaptive optics correction.