## Abstract

Multi-conjugate adaptive optics (MCAO) is the most promising technique currently developed to enlarge the corrected field of view of adaptive optics for astronomy. In this paper, we propose a new configuration of solar MCAO based on high order ground layer adaptive optics and low order high altitude correction, which result in a homogeneous correction effect in the whole field of view. An individual high order multiple direction Shack-Hartmann wavefront sensor is employed in the configuration to detect the ground layer turbulence for low altitude correction. Furthermore, the other low order multiple direction Shack-Hartmann wavefront sensor supplies the wavefront information caused by high layers’ turbulence through atmospheric tomography for high altitude correction. Simulation results based on the system design at the 1-meter New Vacuum Solar Telescope show that the correction uniform of the new scheme is obviously improved compared to conventional solar MCAO configuration.

© 2017 Optical Society of America

## 1. Introduction

It is aspired to achieve solar images with a uniform high spatial resolution and high temporal resolution across the entire field of view (FoV) in solar physics. The development of solar adaptive optics (AO) aims at these requirements and it has revitalized ground-based solar astronomy at existing telescopes [1–4]. Besides the high quality image, the higher spectrograph and polarization measurements sensitivity, and more observing time of the telescope could be obtained with the help of AO. However, conventional AO technique has a very small corrected FoV (~10 arcsec for visible imaging), which is much smaller than the usual solar active region with 1~2 arcmin in FoV [5]. In this case, post-processing methods are generally used to correct the residual phase aberration and restore the image. However, this method is time-consuming and the temporal resolution will be decreased. Therefore, instead of the conventional AO and post-processing, the real-time correction at the entire FoV is of advantage and has been considered for large aperture solar telescope, such as ground layer adaptive optics (GLAO) and multi-conjugate adaptive optics (MCAO).

The MCAO, which deals with the wavefront aberration caused by atmospheric turbulence in several layers with different altitudes [6], is the most promising technique currently developed to enlarge the corrected FoV. Two kinds of the MCAO that are so called the “star oriented” and “layer oriented” MCAO were under development based on different wavefront sensors [7,8]. Originally, the concept of MCAO was proposed based on multiple artificial stars and solar observations for the requirements on several more beacons for three dimension wavefront sensing. The sun is an extended target, the small-scale structures, such as sunspots and granulations, which is everywhere on the surface of the sun, can be used as the “guide stars”, that is why almost all the meter-class solar telescopes have the plan to develop MCAO.

The first solar MCAO experimental system was installed on the 70cm aperture telescope VTT [9]. In this system, the conventional adaptive optics system was used to correct the low altitude turbulence, and a high altitude correction loop was added for high layer’s turbulence correction. A multiple direction Shack-Hartmann wavefront sensor (MD-WFS) with 7 subapertures and 19 subregions (like 19 guide stars) was employed to obtain wavefront of high altitude turbulence for deformable mirror (DM) that conjugated to the corresponding turbulence layer. On sky observation results showed that the FoV gain was about three times larger than conventional AO. Almost at the same time, another solar MCAO experimental system was built at the 76cm aperture telescope DST [10]. The conventional AO module (the AO76) was followed by a high altitude correction loop, which contained a MD-WFS with 21subapertures and 3 subregions (increased to 4 subregions in the later update) and a 97-element DM. The DM conjugated to an adjustable altitude between 2 and 10 km corrected the wavefronts with 15 to 20 KL modes. This system was also a successful sky observation, and the correction FoV was larger than conventional AO and ground layer adaptive optics [11]. For the new generation solar telescope, such as GREGOR and NST, MCAO has been considered as a conventional instrument based on the two MCAO systems.

In general, these two experimental MCAO systems have similar optical configuration, and this configuration could be considered as a cascade system, that is, an AO system plus a high altitude correction (HAC) loop, AO + HAC for short. The block diagram is shown in Fig. 2. The real-time controller of the two parts could be combined to achieve a modified AO + HAC configuration with the control algorithm update [12], the low order modes from MD-WFS and the high order modes from OAWFS are combined to control the ground layer conjugated DM. The advantage of AO + HAC configuration lies in simply switching the AO and MCAO system, and much lower cost. However, the correction effect is non-uniform, because the ground layer correction based on-axis wavefront sensor (OAWFS) is just beneficial for on-axis direction but sacrifices off-axis direction due to anisoplanatic error.

In this article, a new optical configuration for solar MCAO system is introduced. According to this configuration, an independent GLAO system with MD-WFS and DM is employed for ground layer aberration correction, and a high altitude correction followed GLAO is used just for low order correction. Our proposed MCAO configuration is called HOGLAO + LOHAC for short. Based on the design parameters of our MCAO system at the 1-meter New Vacuum Solar Telescope, the simulation results for different configurations are displayed, including conventional AO, GLAO, and MCAO with three configurations (AO + HAC configuration, the modified AO + HAC configuration and the HOGLAO + LOHAC configuration). The arrangement of this article is as follows: in section 2, the turbulence is estimated by several phase screens, and the characteristics of atmosphere are analyzed for multi-conjugated correction. Based on these discussions, different configurations of solar MCAO are displayed and compared in section 3. The simulation results for different configurations based on our own simulation software are produced in section 4, and the last section is the conclusion.

## 2. Corrected modes needed for ground layer and high altitude correction

Atmospheric turbulence is continuously distributed from the ground to tens of kilometers in altitude, and the refractive index structure constant ${\text{C}}_{n}^{2}$ is required to express the vertical distribution of atmospheric turbulence. Furthermore, the volume of turbulence could be modeled with a sequence of phase screens distributed along the optical axis with the knowledge of${\text{C}}_{n}^{2}$. The total number of phase screens *M* will vary for different application. The optimal method for obtaining the location and the strength of different phase screens is presented by Edward P. Wallner and Richard G. Paxman et al. [13, 14], the result is:

The strength of turbulence for different slabs is expressed as the Fried coherence length and given by:

The relationship between the total ${r}_{0}$ and the Fried coherence length in the ${m}^{th}$section ${r}_{0m}$ is:

For the Kolmogorov spectrum, if the first $J$modes of the aberrations are corrected, the remaining wavefronts are given by [16]:

It is found that the requirement of corrected orders is less for large${r}_{0}$. Considering the generic HV10-10 profile model [15] as an example, the root mean square error of the residual wavefront is 0.0965${\lambda}_{d}$ at 0.5 μm wavelength after the first 65 orders correction for AO system. If three DMs are used for the correction with the optimal altitudes 0.7km, 8.1km and 14.0km based on Eq. (1), the Fried coherence length at the different slabs are 10.8cm, 49.1cm, and 61.1cm. Most of the turbulence strength is concentrated in the first few kilometers, which means more orders are needed for ground layer correction, and a few orders are enough for high altitude correction. Selected 65 orders correction for ground layer and 35 orders correction for both of the higher layers, the root mean square error of the residual wavefront is 0.1005${\lambda}_{d}$.

The strength of the turbulence and the total number of the phase screens are decided by the observational site and the requirement of the residual error, which result in the different requirement of corrected orders for system design.

## 3. Different configurations of solar MCAO

The purpose of MCAO is to provide real-time diffraction limited imaging over an extended FoV. However, the correction FoV for AO + HAC configuration MCAO above-mentioned is non-homogeneous. Assuming the phase variation caused by atmospheric turbulence is modeled by two layers: the low layer ${\phi}_{L}$ and high layer ${\phi}_{H}$. The phase ${\Phi}_{{\alpha}_{i}}\left(r\right)$ on the telescope pupil in angular direction ${\alpha}_{i}$ is given by:

It is expected that phase is corrected in different layers. However, for the AO + HAC configuration, the AO system is used for low layer turbulence correction, the correction is given by:

Another configuration of solar MCAO is proposed in 2016 [17]. Only one MD-WFS placed at the hindmost and is used for the multiple direction wavefront sensing, and several DMs are controlled based on the wavefront information of atmospheric tomography. For this configuration, the MD-WFS needs many more subapertures to detect high order wavefront, and a large FoV for three dimension wavefront sensing. It is difficult because the detector with large size and high frame rate is a tough technical challenge. Meanwhile, it is not necessary to sense high order wavefront for high altitude correction. On the other hand, there are DMs conjugated to high altitude between low altitude DM and the OAWFS, which could introduce dynamic misregistration between the two [17,18].

In view of this, a new configuration for solar MCAO, the HOGLAO + LOHAC configuration, is proposed, as shown in Fig. 3. This configuration is based on high order GLAO and low order high altitude correction loop. New MD-WFS (MD-WFS1 in Fig. 3) is employed instead of the OAWFS of conventional AO. The average value of the measurement from different FoV is obtained for the low layer turbulence correction. Atmospheric tomography is used for the wavefront reconstruction caused by high-altitude turbulence based on measured value of MD-WFS2 in high altitude correction loop.

This structure combines the advantages of the star-oriented and layer-oriented MCAO system. The real layer-oriented solar MCAO has not been reported currently, although some theoretical analysis had been published based on cross-correlating Shack-Hartmann wavefront sensor [19,20]. While for star-oriented MCAO, the ground layer turbulence could be estimated by averaging the measurement from different FoV, and the residual aberration from high layers could be reconstructed through tomography for high layers’ correction. The average method needs more guide stars to obtain ground layer turbulence accurately. This is very suitable for solar MCAO, because more guide regions are needed to cover the meta-pupil on high altitude layer for the solar telescopes in contrast with the very large aperture night-time telescopes [21]. The fact lies in that the independent control system for GLAO is easy to implement. Compared with the MCAO with the AO + HAC configuration, the uniformity of the MCAO correction with the HOGLAO + LOHAC configuration over the whole FoV will be improved evidently.

In our MCAO configuration, moreover, it is beneficial to the MD-WFS of the high altitude turbulence when the GLAO works. And relatively large subaperture of the MD-WFS of the high altitude turbulence is advantageous when the solar granulation is used as the beacon for the wavefront sensing.

From Fig. 2 and Fig. 3, it can be seen that the optical setup is almost the same except that the MD-WFS1 is added to replace the OAWFS of conventional AO. Besides, the amount of computations will be increased to fit the multiple directions wavefront sensing and MCAO correction, which is a challenge for the real-time controller. Fortunately, the MCAO system [17] and our GLAO experimental system [22] had proved the performance of the real-time controller based on the different architectures.

## 4. Numerical simulation for different MCAO configurations

Under the support of the National Natural Science Foundation of China from 2012 to 2016, two generations of AO systems [4, 23] have been developed at the 1-meter New Vacuum Solar Telescope (NVST). Meanwhile, the development of the MCAO for the NVST is in progress. A MD-WFS has been designed and A GLAO prototype system based on the MD-WFS with 60 arcsec FoV had also been developed and tested at NVST [22]. The MCAO experiment is taken into account when designing the 151-element AO system, so it is convenient to install at the 1m NVST.

The open loop correction results for different configurations are provided in the numerical simulation, including the conventional AO, the GLAO, and the MCAO with the AO + HAC and the HOGLAO + LOHAC configurations. The MCAO with the modified AO + HAC configuration is also simulated and 3 to 5 Zernike modes detected by the OAWFS are replaced by the corresponded mean value from the MD-WFS.

The parameters for simulation are from system design or our actual system at the NVST, up to three deformable mirrors were considered, and three Shack-Hartmann WFSs are employed. Table 1 gives the simulation parameters for different system configurations and their combination of WFSs and DMs. The simulation software edited by ourselves is an end-to-end simulated tool for MCAO with modular framework, and the open loop diagram is shown in Fig. 4.

#### 4.*1 Model the atmosphere turbulence*

Normally, some classical and measured atmospheric turbulence models would be applied for analysis and simulation. Because the site of NVST has no measured profile information, the classical HV modified turbulence model is employed in this simulation. The HV10-10 and HV15-12 with the${r}_{0}$equal to 10cm and 15cm are divided into 6 layers. Table 2 shows the altitude and the ${r}_{0,m}$ for different turbulence slabs according to Eq. (1).

Six annular phase screens are placed at the height of the turbulence slabs in the simulation. All phase screens could be rotated to simulate the dynamic turbulence, as it is done in MAPS [24]. The phase screens would rotate a certain angle every frame to update the turbulence. Given the frame frequency of detector ${f}_{d}$ and the wind speed ${v}_{s}$ in simulation, one could obtain the rotating angle of phase screen per frame:

where $R$denotes the radius of the movement trajectory of the pupil in phase screens. In our simulation the frame frequency is 1s and the wind speed ${v}_{s}$ is 5m/s for all turbulence layers. To check the effect of the simulated turbulence, the wavefront of a certain FoV was obtained through six phase screens by ray tracing method, and the mean covariance of the Zernike coefficient for 1000 frames was calculated and compared with the theory result, as shown in Fig. 5. From 3rd to 65th order Zernike aberrations and 1-meter telescope aperture were considered in the test.#### 4.2 Wavefront sensing in multiple directions

There are three wavefront sensors in our simulations, all wavefront sensors are conjugated to the entrance pupil. The OAWFS with 13 × 13 subapertures corresponding with the conventional SHWFS in our 151-element AO system [4]. The other two MD-WFSs (MD-WFS1 and MD-WFS2) with 60 arcsec FoV for each subaperture, have 9 × 9 and 6 × 6 subapertures across the pupil respectively. A series of subregions are extracted from each subaperture for multiple direction wavefront sensing. Figure 6 shows the meta-pupil at 5km altitude and the arrangement of 7 and 19 subregions across 60 arcsec FoV. Wavefront propagation through the atmosphere is based on geometry optics, wavefront arrived at the wavefront sensor is obtained by adding linearly the fluctuations produced by each turbulent layer. Wavefront from different directions is divided by different subapertures. The slopes of different subapertures are calculated based on Fourier transform and centroid algorithm, and then transformed into Zernike coefficient for the wavefront reconstruction of different altitude layers. The first 65 order modes are used for the OAWFS and MD-WFS1, and the first 35 order modes for MD-WFS2.

#### 4.3 Reconstruction and correction

This part is simulated in the reconstructor module as shown in Fig. 4. The interaction matrix between the WFSs of different directions and the DMs that conjugated to different altitude layers is obtained first, and then the reconstructed matrix is acquired to seek the corrected wavefront from WFSs. Several works have focused on the wavefront reconstructor for MCAO, the knowledge of the statistics of phase perturbations and the turbulence vertical distribution was employed to derive optimal reconstructors operating in open-loop, such as the maximum a posteriori method and minimum mean square error method. In fact, the statistics of turbulence perturbations would be destroyed in close loop, and the least square estimator (LSE) reconstructor based on singular value decomposition was verified to work well in different experimental systems [25]. In our simulation system, all reconstructors are acquired through LSE estimation, the reconstructed matrix is the pseudo-inverse of the interaction matrix.

For MCAO system simulation, the wavefront aberration of ground layer is obtained using the conventional AO or GLAO reconstructor according to different MCAO configuration, and the mode tomography is employed to reconstruct the high altitude wavefront. The whole tip-tilt differences are ignored on account of the indetermination problem for tomography. In system, one could employ a fine tracking system to correct them as in our 151-element AO system. The DMs are assumed to be able to reproduce exactly the Zernike modes being reconstructed, that means the fitting error caused by DMs is not considered.

#### 4.4 Estimation

Considering the central symmetric arrangement of multiple guide stars, a series of targets arranged along the two lines in Fig. 6 are used to indicate the effect of correction in different FoV. One line is along the arrangement of sub-regions, and the other one has a 15-degree angle with the first, which is half of the angular separation of the guide stars ranged on the outer circle. There are two targets for a certain FoV, the root mean square error of the residual wavefront of the targets after correction is obtained and we take the average of 100 frames as the final result.

Typical index revealing the resolution of the telescope or AO systems are the Strehl ratios (SR) given by [26]:

#### 4.5 Simulation results

The SRs at the TiO band as the function of the FoV for the different conditions including the uncorrection, the AO correction, the GLAO correction and the MCAO corrections with the AO + HAC configuration, the modified AO + HAC configuration and our HOGLAO + LOHAC configuration are displayed in one figure for comparison, as shown in Fig. 7. Two and three layers corrections with the corrected altitude in Table 2 for the two HV modified models are considered in simulation, respectively.

According to the simulation results shown in Fig. 7, it is evident that the conventional AO correction has a typical isoplanatic region for different simulation conditions. The correction effect in central FoV is degraded for the MCAO compared to AO, but that becomes better as the FoV enlarged.

Especially, the corrected effect is non-homogeneous in the whole FoV for the different MCAO configurations. In contrast, the MCAO with the HOGLAO + LOHAC configuration has the better homogeneous correction effect in the whole FoV than the other MCAO configurations. As a first step, the GLAO also has a uniform correctional performance, however, It just enlarged the ${r}_{0}$ and the SR after GLAO correction will be lower compared to MCAO, because the wavefront aberration caused by high altitude turbulence was not corrected. Solar MCAO with the HOGLAO + LOHAC configuration would be more practical to obtain the solar images with high resolution of the entire FoV.

Besides, it can be seen from Fig. 7 that the three layers correction is much better than the two layers correction for all MCAO configurations, which means the turbulence at high altitude has a significant influence on the resolution of images. In view of this, three DMs will be employed in the design of our MCAO system.

Moreover, the correction effect for the HV15-12 turbulence model is better than that for the HV10-10. It is easy to understand due to the stronger atmospheric turbulence with HV10-10 model than the HV15-12 model.

It is noted that the maximal SR is limited by the cutoff error of the system. The cutoff error comes from the higher order aberrations than the corrected modes. In our simulation, the phase screens are produced based on the invert Fourier transform of a Kolmogorov spectrum, which contains infinite order Zernike modes in theory. The former 65 modes are corrected for the AO and ground layer turbulence, and the only former 35 modes are corrected for the high altitude correction. Assuming the first 65 and 35 order modes of the lowest altitude and the other five screens are removed completely, the residual wavefront of all targets is obtained based on 100 frames statistics, the average root mean square of the cutoff error are 0.12${\lambda}_{I}$ and 0.10${\lambda}_{I}$ at the TiO band for the HV10-10 and HV15-12 models.

It is also shown from the simulation results that the effect of GLAO is better than AO + HAC at some off-axis directions, the reason would be the weight of the ground layer in HV model is heavier, more than 80% aberration is converged at the first phase screen, the anisoplanatic error from Eq. (7) is ascendant in the residual wavefront error.

The simulation results show the correction performance of the modified AO + HAC is superior to that of AO + HAC. This is evident due to the consideration of the low order modes anisoplantism. In the modified AO + HAC simulation, just the 3 to 5 Zernike modes detected by the OAWFS are replaced by the corresponded mean value from the MD-WFS2, so the correction is not very good. With the modified Zernike modes increases, the correction will become better.

## 5. Conclusion

According to the characteristic of atmosphere, a new configuration of solar MCAO was proposed to realize a uniform correction across the entire FoV. The high order GLAO system was followed by a low order high altitude correction loop in this configuration. Based on the classical atmosphere models and the system parameter at the 1-meter NVST, different systems including AO, GLAO and MCAO with three configurations were simulated. The results were compared to demonstrate the advantage of the new configuration. This work might well guide the development of the solar MCAO system for 1m NVST.

It must be noted that, this paper concentrates on the uniformity of the correction for different MCAO configurations and the WFS noise isn’t considered. Some factors that influence the correction effect of MCAO such as the measurement noise of the WFS and the temporal control bandwidth of the system will further be considered in the future research. The arrangement of the guide stars and the altitude of corrections of the MCAO system are also needed to optimize according to the measured atmospheric turbulence profile.

## Funding

National Natural Science Foundation of China (NSFC) (11178004); Laboratory Innovation Foundation of the Chinese Academy of Sciences (YJ15K007)

## Acknowledgments

We would like to thank Dr. Yangyi Liu for checking up on language error. A special acknowledgment should be shown to Prof. Wenhan Jiang from the IOE, for his revision. The reviewers are also acknowledged for the very valuable suggestions and comments.

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