## Abstract

Speckle image reconstruction, in which the speckle transfer function (STF) is modeled as annular distribution according to the angular dependence of adaptive optics (AO) compensation and the individual STF in each annulus is obtained by the corresponding Fried parameter calculated from the traditional spectral ratio method, is used to restore the solar images corrected by AO system in this paper. The reconstructions of the solar images acquired by a 37-element AO system validate this method and the image quality is improved evidently. Moreover, we found the photometric accuracy of the reconstruction is field dependent due to the influence of AO correction. With the increase of angular separation of the object from the AO lockpoint, the relative improvement becomes approximately more and more effective and tends to identical in the regions far away the central field of view. The simulation results show this phenomenon is mainly due to the disparity of the calculated STF from the real AO STF with the angular dependence.

© 2014 Optical Society of America

## 1. Introduction

Atmospheric turbulence degrades the angular resolution of the images captured by the ground-based telescopes. For large telescopes without compensation, the resolution is approximately equal to that of a telescope with the diameter equates to the atmospheric Fried parameter. Many techniques for overcoming the effects of atmospheric turbulence have been proposed. Adaptive optics (AO) can compensate the aberrations induced by the atmospheric turbulence in real time. However, traditional AO correction is never perfect and limited to the isoplanatic patch. To enlarge the FOV, Multi-Conjugate Adaptive Optics (MCAO) which uses two or more DMs conjugated to the several heights of the main turbulence layers can provide real-time diffraction limited imaging over an extended FOV. However, the implementation has some fundamental issues. Trade-offs will have to be performed in order to optimize the MCAO performance toward specific science experiments. The FOV over which diffraction limited observations have to be performed to achieve scientific goals as well as the desired Strehl will drive the complexity of MCAO and the zenith distance for which the observations are possible. Post-facto image processing techniques also can remove the aberrations introduced by the atmospheric turbulence and obtain the diffraction limited images in large FOV. However, the accuracy of the reconstruction is mainly determined by the signal to noise ratio of the degraded images and is difficult to get the real-time diffraction limited imaging. Speckle image reconstruction [1, 2] which utilizes the statistical information of the atmospheric turbulence to recover the high resolution images is commonly used. The reconstructions of the object’s power spectrum [1, 3] and Fourier phase [4–6]are needed in this method.

Although the detection and correction of traditional AO system only limited within the isoplanatic lock patch, the AO images have two main advantages compared to the seeing-limited images without compensation in speckle image reconstruction. Firstly, the static aberrations in the system, which influence both the reconstruction of the Fourier amplitude and Fourier phase, can be removed. Secondly, the AO correction provides the higher signal to noise ratio (SNR) of the images and a higher accuracy of the Fourier phase reconstruction can be acquired [7]. Thus, the combination of the traditional AO correction and post processing are commonly used to obtain the high resolution images over a large FOV.

However, after the correction of the AO system, the STFs used to recover the power spectrum of the object are anisotropic and depend on the information of the AO system. Therefore, the traditional speckle image reconstruction method [2] using only one Fried parameter${r}_{0}$and the corresponding modeled STF to recover all the isoplanatic patches is not suitable for the AO images. To compensate the influences of the AO correction on the speckle image reconstruction, the method mentioned in Ref [8]. which includes the effect of an arbitrary correction as well as anisoplanatism has good photometric accuracy, whereas the numerical calculations performed for the spectral ratios [3] and the STFs are too complicated and difficult to implement in real time. Two simple methods, namely the method B and the method C in reference [7] were also used for AO supported speckle image reconstruction. The method C gets the STFs by the simulated atmospheric turbulence and real-time AO correction data, the quality of the reconstructed image depends on the consistency of the simulation with the real atmospheric condition and AO parameters. The method B uses the calculated Fried parameters for individual annulus from the traditional spectral ratio method and the related STFs to compensate the influence of AO correction and ignores the disparity from the Kolmogorov spectrum.

In this paper, the solar images corrected by a 37-element AO system [9] are reconstructed using the annular STF similar with the method B proposed in Ref [7]. Due to the angular dependence of AO compensation, the STF is modeled as annular distribution and then the individual annular Fried parameter of the atmospheric turbulence and the corresponding STF are obtained by the traditional spectral ratio method. The angular dependence of the AO correction ability is compensated by the annular distribution of the calculated Fried parameter and the corresponding theoretical STF. The speckle image reconstruction method is introduced and the results for the real photometric data observed by AO are presented in this paper. Specially, the image improvement with dependence of the angular separation of the object departure from the AO lockpoint in the whole field of view (FOV) is investigated. Moreover, the simulations of the anisoplanatism imaging with AO correction are done to explain the experimental results.

## 2. Power spectrum estimation

In extended object’s speckle image reconstruction, the Labeyrie method [1] and the spectral ratio method [3] are combined to recover the power spectrum of the object. The spectral ratio method calculates the Fried parameter of the atmospheric turbulence and the corresponding theoretical STFs are used to recover the Fourier amplitudes of the object in the Labeyrie method.

The spectral ratio is defined as

As the STF is known, the power spectrum of the object can be calculated following the Labeyrie method:

## 3. Image reconstruction

The raw images were obtained at the 1-m New Vacuum Solar Telescope of Full-shine Lake (also called Fuxian Lake) Solar Observatory with a 37-element solar AO system [11]. For solar AO system, the correlating Hartmann–Shack wavefront sensor (WFS) is generally used. In this system, the absolute difference algorithm with the parabolic subpixel interpolation is used to detect the subpixel shifts in subapertures. When AO system is closed-up, the anisoplanatism noise of the WFS is mainly related to the size of the reference image FOV. The correlating Hartmann–Shack WFS is averaging all the information that arrives from angles within the reference image FOV. When The FOV of the reference image is larger than the isoplanatic patch size, the anisoplanatism noise must be considered. The accuracy of the slope measurement is also determined by the WFS noise. It should also be noted that the wavefront sensor noise is object dependent in the sense that high signal to noise (S/N) ratio can be achieved for high contrast objects, such as sunspots, whereas the S/N ratio for tracking the low contrast granulation is much lower. Such the FOV of the WFS must consider the object size, the atmospheric condition including the seeing, the isoplanatic angle and so on.

In this system, the subaperture size is about 14cm. The FOV of the WFS is$1{6}^{\u2033}\times 1{3}^{\u2033}$. The FOV of the reference subimage is$7.{9}^{\u2033}\times 6.{6}^{\u2033}$. In the following two sets of solar images, the sunspots with high contrast are used as the beacons of the AO WFS. The isoplanatic angle is about${8}^{\u2033}$which is approximately equal to the FOV of the reference subimage. Meanwhile the anisoplanatism imaging of the correlating Hartmann-Shack WFS contributes to about 10nm Root-Mean-Square (RMS) wavefront error, which has a little effect on the AO system, in the central FOV according to simulation results of the Ref [12]. All of the raw images are captured at 705.5nm with a 10 nm bandwidth. Since the speckle image restoration method requires the isoplanatic condition, the raw images are divided into isoplanatic patches with 50% overlap and reconstructed separately. Also considering the seeing condition and the speed of the Fast Fourier transform, the size of the patches is chosen as $4.60{8}^{\u2033}\times 4.60{8}^{\u2033}$($64\times 64$pixels).

In traditional speckle image reconstruction [2], all the isoplanatic patches are used to calculate the spectral ratio and only one${r}_{0}$and the related STF is estimated in the FOV because the homogeneous of the atmospheric turbulence and the wavefronts. The compensation of AO changes the statistical information of the wavefronts and leads the STFs to unusual structure. Thus the traditional method cannot be used to reconstruct the AO images. To model the anisoplanatism of STFs, the image FOV is divided into several annuluses across AO lockpoint in this paper. The Fig. 1 shows the segmentations of the raw image and the calculated $D/{r}_{0}$ by the traditional spectral ratio method in each annulus. The blue contours are the boundaries of the annuluses and the red points superimposed on the images denote the center points of the patches. The lock patch marked with the smallest circle is estimated as the patch with the lowest variance of the residual aberrations,

The power spectrum of the object is reconstructed by the Labeyrie method and the patches with their center points at the same annulus share the same STF which related with the estimated ${r}_{0}$in this area. In Fig. 1, the values of $D/{r}_{0}$ in each annulus are calculated by the averaged spectral ratio of the patches which have their center points in the annulus. From the much larger calculated ${r}_{0}$around the lock patch of the AO in Fig. 1, it is obviously that the correction is mainly effective near the isoplanatic patch. Because the volumic distribution of the atmospheric turbulence, with the patches are away from the lockpoint, the correlations of the aberrations degraded those patches and the detected phases from the WFS decrease and the AO correction is less effective, thus the values of $D/{r}_{0}$ increase with increasing distance from the lockpoint. The static error of the whole system is only corrected as the patches are far away from the isoplanatic patch.

As mentioned in the former, the anisoplanatism noise of the WFS is mainly related to the size of the reference image FOV. Although the FOV of WFS is larger than the isoplanatic angle, the FOV of the reference image with$7.{9}^{\u2033}\times 6.{6}^{\u2033}$in the slope calculation process is approximately equal to the isoplanatic angle, thus the AO system in this condition didn’t suffer the severe anisoplanatism noise. In the FOV of the isoplanatic angle, the AO correction is very effective. With the increase of angular separation of the object from the central FOV, the anisoplanatism effect of the atmospheric turbulence gradually decrease the AO correction ability until AO correction has no effect for the atmospheric turbulence.

The Knox-Thompson method [2] is used to reconstruct the Fourier phases. The recovered patch is obtained by inverse transform the Fourier amplitude and the Fourier phase. Then all the recovered patches are combined to form an image of the object in FOV. The original image and the reconstruction are shown in Fig. 2. Compared with the raw AO images in Figs. 2(a) and 2(c), the recovered images in Figs. 2(b) and 2(d) have better image quality. Closely inspect the raw image and the reconstruction in Fig. 3, it is found that the quality of the reconstruction is field dependent.

To evaluate the quality of the reconstruction in the FOV, the contrast is taken as the criterion. The size of the evaluated patches is $32\times 32$pixels across the center point of the recovered patches to alleviate the apodization effect. The contrast is defined as

Where the symbols${()}_{std}$and${()}_{mean}$signify the standard deviation and mean value of the matrix enclosed in braces, respectively. The relative improvement of the contrast isWhere ${c}_{r}$ and ${c}_{o}$are the contrast of the reconstruction and the raw image, respectively.The mean contrast in each annulus is displayed at Figs. 4 (a) and4 (b). From Fig. 4, it is seen that the contrasts of the reconstruction images show little dependence on the field angle as the patches away from the isoplanatic angle and the relative improvement of the contrast is approximately more effective than the lock patch and tends to identical in those regions.

## 4. Simulation

To investigate how the AO correction ability influences the accuracy of the reconstruction, the numerical simulation of anisoplanatism imaging and AO correction is used. The wavefront in different angle are generated by adding the aberrations in the path of the light. In this paper, three random phase screens with different $D/{r}_{0}$are set in different height to simulate the anisoplanatism imaging. The simulated AO effect is that the wavefront aberration at the lock patch was fully corrected with the first several Zernike modes and the wavefront aberration from other field angle away from the lock patch are compensated by the same correction.

It is known that the accuracy of the recovered Fourier amplitude depends on the consistency of the STFs after the AO correction ($sit{f}_{o}$) and the theoretical STFs calculated by the spectral ratio method ($sit{f}_{c}$). To show how the field angle away from the lockpoint of the AO system influence the consistency of $sit{f}_{c}$ and $sit{f}_{o}$at different field angle, the disparity of the calculated STF from the AO STF is defined as

For our AO system, the matching arrangement of the actuators of DM and the subapertures of the WFS is shown in Fig. 5 (a), in which the circle characters denote the actuators, and the subapertures are illustrated in the square grids. Figure 5 (b) shows the relative RMS error defined as the ratio of the residual RMS error to the original RMS error for the individual Zernike aberrations with this AO system. It can be seen from Fig. 5 that the 37-element low-order AO system can be efficiently used to correct the first 5 order Zernike aberrations (the first 20 modes).

According to the seeing condition of the observation state and compensation capability of the AO system, the seeing condition is chosen as$D/{r}_{0}=10$and the correction numbers of the Zernike modes are 15, 25 and 35 in the simulations. The behaviors of the calculated$D/{r}_{0}$with the angle from the lockpoint shown in Fig. 6 (a) are similar with the results shown in Fig. 1. When the displacement angle is within the isoplanatic angle, the calculated$D/{r}_{0}$ is much smaller than the original$D/{r}_{0}$. As the displacement angle is away from the isoplanatic angle, the calculated $D/{r}_{0}$slightly increases and becomes closed to the original one. The disparities of the calculated STF from the AO STF at different field angles are displayed in Figs. 6(b)-6(d). From these figures, it can be seen that the STF disparity within the isoplanatic angle is large. With the displacement angle is far from the isoplanatic angle, the consistencies of the calculated STFs and the AO STFs become good and independent with the field angle.

This is the result of considering the residual wavefront aberration after AO correction following the Kolmogorov spectrum. The bigger calculated Fried parameter can only partially compensate the influence of the AO correction. The AO correction which introduced more effective correction at the locked isoplanatic patch leaves larger disparity of the aberrations from the Kolmogorov spectrum. Thus lead to the less consistency between the STFs. The rapidly decrease of the similarities between the wavefront from the lockpoint and the wavefront from the angle away from the isoplanatic angle which result with a little effective correction from the AO system and less influence on the statistical information of the wavefronts. Therefore, speckle image reconstruction method which considered the residual wavefront followed the Kolmogorov spectrum is more effective to restore those patches.

The simulations above show that the FOV within the isoplanatic angle suffers the maximum impact of the AO correction and the photometric accuracy of the reconstruction using the theoretical STF calculated by the spectral ratio method is low. As the patches away from the lockpoint received less effective correction, thus the reconstruction of those areas has better photometry accuracy than the locked isoplanatic angle in the same condition.

## 5. Conclusion

In this paper, the AO images of the 37-element AO system are reconstructed using the speckle reconstruction technique. It is visible that the reconstruction improved the image quality at the whole FOV. The field dependent of the relative improvement of the contrast are investigated by the simulations of the anisoplanatism imaging and AO correction with different correction numbers. The simulations shown much larger disparities between the calculated STFs and the AO STFs when the patches within the isoplanatic angle. As the patches beyond the range of the isoplanatic angle, the consistency between the AO STFs and the calculated STF becomes better. This is explained by the less effective correction of the AO system at those regions and the statistical information of the residual wavefront error are more alike the Kolmogorov spectrum. Thus the reconstruction method is approximately more effective and tends to identical in the regions far away the isoplanatic patch.

## Acknowledgments

This work was supported by the Natural Science Foundation of China No.11178004.

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