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Abstract
The correlation Shack–Hartmann wavefront sensor (SHWFS) is widely used in many fields in addition to solar adaptive optics. The requirement for the SHWFS dynamic range increases with the diameter of the telescope, which means a larger detector array is needed. However, the size of the detector would be restricted by the high frame rate needed for the solar observation. To solve this problem, a new, to the best of our knowledge, method called the region-correlation algorithm (RCA) is proposed. In this method, the sub-image array is divided into several regions, and the slopes of sub-apertures are calculated by referring to a selected sub-image in each region. Note that the final slope over a sub-aperture is obtained by combining the relative slopes between the selected sub-image in different regions. The dynamic range in each region is similar to the conventional correlation algorithm, and the final dynamic range of the RCA would be accumulated from those of the regions. The reconstruction accuracy under large aberration would also be improved due to the extended dynamic range. Meanwhile, the RCA does not require any extra device and the increase in calculation time resulting from the RCA is acceptable. The results of numerical simulation and experiment, compared with conventional correlation algorithm, confirm the advantages in the performance of the RCA as well.
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Corrections
20 September 2022: A typographical correction was made to the author affiliations.
Solar adaptive optics is indispensable in ground-based solar observation, and the correlation Shack–Hartmann wavefront sensor (SHWFS) is adopted for wavefront sensing. A microlens array divides the telescope’s exit pupil into sub-apertures, and each microlens images a small region of the Sun [1,2]. The requirement of the SHWFS dynamic range increases with $D/{r_0}$ for a certain turbulence intensity. Meanwhile, the number of sub-apertures across the pupil also increases with the diameter of the telescope. However, a high frame rate is necessary due to the shorter atmospheric coherence times during the day for solar observation [3]. The size of the detector would be restricted by the frame rate of the camera. Considering the restricted size of the detector, using an optimization algorithm would be a good choice to extend the dynamic range compared with increasing the field of view (FoV).
The dynamic range of the SHWFS could be represented by the maximum slope that can be accurately measured. For point object and extended object wavefront sensing, the center of gravity (CoG) algorithm and correlation algorithms such as cross correlation (CC), square difference function, etc. [4], are separately employed to obtain the slopes. Figure 1 shows the different impacts of a large aberration on the point and extended object wavefront sensing. For the point object, the spots array would mismatch to sub-apertures. While for the extended object, because of the discrepancies among sub-images, the accuracy of the correlation algorithm would decrease.
Fig. 1. Sub-image array in the SHWFS for the (a) point object and (b) extended object under large aberration.
Some methods have been proposed to extend the dynamic range of the SHWFS. Generally, the method for the point object is to distinguish the spots of different sub-apertures by software or device-based approaches [5–7]. However, they are not suitable for the extended object due to the different impacts mentioned above. The method for the extended object is to improve the accuracy of the correlation algorithm. Sidick et al. proposed an adaptive cross correlation (ACC) algorithm that iteratively calculates the slope [8]. Wang et al. proposed a gradient cross correlation algorithm to solve the wraparound effect [9]. Phase diversity and machine learning also have been used to improve the accuracy of the slope calculation [10,11]. However, the ACC algorithm requires a larger FoV for slope calculation, and the gradient cross correlation algorithm is needed to solve the wraparound effect. All of these result in extra computation cost.
In this Letter, a new method called the region-correlation algorithm (RCA) is proposed with improved dynamic range and reconstruction accuracy for the extended object wavefront sensing. Unlike the conventional correlation algorithm that uses one reference sub-image to calculate the slope, in the RCA, the sub-image array is divided into several regions first, and the slopes of the sub-apertures are calculated by referring to a selected sub-image in each region. Note that the final slope over a sub-aperture is obtained by combining the relative slopes between the selected sub-image in different regions. The dynamic range in each region is similar to the conventional correlation algorithm, and the final dynamic range of the RCA would be accumulated from those of the regions. At the same time, the reconstruction accuracy under large aberration would also be improved with the extended dynamic range.
The detailed steps of the RCA are shown in Fig. 2. The sub-image array is divided into ${R_N} \times {R_N}$ regions according to
where N and n are the numbers of sub-apertures across the pupil and regions separately. Since the sub-image array may not be divided equally, the size of the edge region in one dimension isFig. 2. Process of the RCA for extended object wavefront sensing. (a) Sub-image array of the SHWFS is divided into different regions (shown by the yellow line), and the references of each region are indicated by the red block. (b) Slopes in each region refer to the reference of the region, and the relative slopes between regions. (c) Final slope of each sub-aperture. (d) Reconstructed wavefront phase.
The center sub-image in each region could be selected as the reference of the region. As shown in Fig. 2(b), for the region O, the slopes are first calculated by referring to the selected reference with the cross correlation algorithm:
where ${S_{i,O}}$ is a sub-aperture in the region O, ${S_{R,O}}$ is the reference in the region O, ${s_{xi,O}}$ and ${s_{yi,O}}$ are the slopes of sub-aperture i referring to the reference, and $C < \bullet > $ means cross correlation.Then, all references are considered as an image array, and the relative slopes between regions are calculated. Combining the results from the two steps, the final slope over each sub-aperture is obtained and the wavefront aberration could be restored as in the conventional methods.
To ensure the accuracy of the slope calculation, the relative slope referring to the center region could be given by accumulating the slopes between adjacent regions. Take region 1 in Fig. 2(a) as an example. The relative slopes referring to the center region (region 5) are calculated as
As shown in Fig. 3, for the slope calculation between adjacent regions, the idea of this relay calculation could be used in adjacent sub-apertures to further decrease the influence of the sub-image difference.
In the RCA, slope calculations are dispersed throughout regions. Based on the mechanism of the cross correlation algorithm, the maximum slope measured in each region is the same as the conventional correlation algorithm in theory. Meanwhile, the relative slope in each region could usually be smaller due to the graduality of the slope. As shown in Fig. 4, the dynamic range in each region is similar to the conventional correlation algorithm, and the final dynamic range of the RCA is accumulated from those of the regions. Furthermore, the reconstruction accuracy under large aberration would also be improved due to the extended dynamic range.
Fig. 4. Dynamic range of the conventional correlation algorithm and RCA. The red squares indicate the reference, and the yellow squares indicate the regions of the RCA.
The numerical simulation and experiments were performed to evaluate the performance of the RCA in improving the dynamic range and reconstruction accuracy of extended object wavefront sensing. Table 1 lists the parameters of the SHWFS used in experiments, which are also the input of the simulation.
Table 1. Key Parameters of SHWFS
In the simulation, first we compared the dynamic range of the RCA and CC by applying different magnitudes of defocus aberrations. The 5% difference between calculated and input slopes was adopted as the threshold for determining the dynamic range. The region sizes of 6, 4, 3, and 2 were settled to compare the impact of the region size. The results in Table 2 verified the capability of the RCA in expanding the dynamic range, and the performance of the RCA is better with smaller region sizes. Meanwhile, the optimized bound corresponds to the region size of 3, which is determined by the calculation method of the relative slope between adjacent regions, as shown in Fig. 3. The region size less than 3 would not bring more benefits in the dynamic range improvement but extra computation cost.
Table 2. Dynamic Range of Different Algorithms
As shown in Fig. 5, to demonstrate the reconstruction accuracy of the RCA, a simulated SHWFS image with given atmospheric aberration was generated with the following steps. First, the sub-image shifts in sub-apertures based on given aberration were obtained. Second, the known shifts were introduced to the granulation image by convolving the image with the point spread function (PSF) of the specified tilt phase. All sub-images obtained in this way were arranged as an SHWFS image. The sub-pixel image shifts could be simulated in this way.
Fig. 5. Steps of the generation of the SHWFS image with known aberration. (a) Granulation image. (b) PSF of tilt phase. (c) Convolved image with the known amount of shift. (d) SHWFS image with known aberrations.
A series of phase screens following the Kolmogorov turbulence model were generated using 3 to 25 Zernike polynomials [12]. The root mean square (rms) of the residual error (the difference between the input and restored aberration) is given as the reconstruction error. The averaged rms of 100 random phase screens was calculated based on the two algorithms to evaluate the reconstruction accuracy. The simulation results under various turbulence strengths are shown in Fig. 6. Because of the accumulation of calculation errors, CC has the better performance when the rms of the input phase aberration is less than 1$\lambda $. With the increases of the input wavefront aberration, the RCA has a better accuracy due to large dynamic range. At the same time, the reconstruction accuracy of the RCA is also better with smaller regions size.
Fig. 6. Simulation results of reconstruction accuracy of the two algorithms. The vertical bar reflects the standard deviation.
The performance of the RCA was also verified in the laboratory. As shown in Fig. 7(a), an extended object was generated by illuminating the target board with an LED source. A 241-element deformable mirror (DM) was used to generate a given aberration based on 3 to 25 Zernike polynomials. The linearity of the DM response was tested and calibrated. We first calibrated the system aberration and then generated specified aberration using the DM. The region size of the RCA was 3 and the reconstruction was also achieved with 3 to 25 Zernike modes.
Fig. 7. (a) Optical layout of the experiment. (b) SHWFS image when calibrating aberration. (c) SHWFS image with aberration generated by the DM.
The results are shown in Fig. 8. As we can see, when the rms of the input phase was larger than 1.1$\lambda $, the aberration exceeded the dynamic range of CC, and the RCA still had a good measurement accuracy, which is consistent with the simulation. However, the residual errors of the two algorithms were larger than that in simulation. As shown in Fig. 7(c), the extended object generated with the target board had fewer features than the granulation. The limited features might be shifted to the edge of the sub-image under large aberration and the calculation error would increase due to the impact of the wraparound effect.
Fig. 8. Experimental results of reconstruction accuracy of the two algorithms. The region size of the RCA is 3.
The extra computation cost is discussed. Taking the experiment with a region size of 3 as an example, the extra time caused by the RCA is approximately 17 microseconds using a processor with 18 cores and 3.5-GHz frequency. Therefore, we can say that the RCA is able to be used in real-time wavefront sensing, as it does not require any extra device and the increase in calculation time added by the RCA is acceptable. In applications of the RCA, the size and shape of the region can be selected flexibly based on the number of sub-apertures across the pupil, the microlens arrangement, the turbulence intensity, and the performance of the real-time controller, etc.
In conclusion, a new method of the RCA was proposed to extend the dynamic range of the extended object wavefront sensing. Meanwhile, the reconstruction accuracy under large aberration also improved with the extended dynamic range. Both numerical simulation and experiment verified the improvement of the RCA in the dynamic range and reconstruction accuracy. This algorithm enabled a conventional SHWFS to measure larger aberrations in high frequency. The remarkable advantage would make solar adaptive optics systems exhibit better performance under strong turbulence. At the same time, the RCA may provide a new idea for the parameter setting of the SHWFS in solar observation.
Funding
National Natural Science Foundation of China (11727805, 12073031, 12103057); Frontier Research Fund of Institute of Optics and Electronics, Chinese Academy of Sciences (C21K002); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021378).
Acknowledgments
The authors would like to thank the reviewers and Xiaosong Wu for the useful suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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