Modulation-nonmodulation pyramid wavefront sensor with direct gradient reconstruction algorithm on the closed-loop adaptive optics system

Shengqian Wang; Kai Wei; Wenjia Zheng

doi:10.1364/OE.26.020952

1. Introduction

The pyramid wavefront sensor (PWFS) has the special characteristics of variable gain and adjustable sampling in real time in order to optimize performance, which makes it a promising option for next generation adaptive optics systems [1–3]. The adaptive optics module of the Telescopio Nazionale Galileo has successfully used the PWFS to measure the aberrations introduced by atmospheric turbulence and encouraging on-sky results have been achieved [4,5]. The PWFS has also been mounted on the adaptive optics system at the 3.5m telescope of the Calar Alto Observatory and an on-sky test was carried out [6]. The sensor was also investigated for implementation on the Very Large Telescope Optics Facility [7]. In particular an adaptive optics system based on the PWFS is installed on the Large Binocular Telecsope and has obtained a new level of performance with 83% Strehl Ratio in R band for 8.5 magnitude stars [8]. Furthermore, the PWFS is a new and promising approach for MCAO (multi-layer conjugate adaptive optics) to solve the problems related with limited field of view [9,10]. The PWFS’s ability to phase and align segmented mirrors has also been demonstrated [11,12]. The modulation mode is often used for the PWFS, where the pyramid itself moves in a circular motion or where a tilting mirror is used to provide modulation. There are also some scholars put forward a static modulation mode, where for example atmosphere’s residual error or a diffusion plate provide the same effect as dynamic modulation [13–15]. Costa’s research shows that higher order uncompensated residuals of the turbulence can be considered as a form of modulation [16].

The interaction matrix of the adaptive optics system describes the control relationship between the deformable mirror (DM) and the PWFS. As happens with every wavefront sensor, the PWFS has a range where the response signal is linear, and the system should be calibrated within this linear range. According to geometrical optics theory and the Foucault knife-edge effect, the linear range of the non-modulation PWFS measurement is very small and signal is easy saturated [17,18]. Therefore, if the non-modulation mode is used in the measurement of the interaction matrix, the amplitude is small and the identification of the noise propagation coefficients is noisy. This noisy measurement decreases the quality of the interaction matrix.

How to improve the performance of closed-loop operation is also a key problem for an adaptive optics system. It has been found that the sensitivity of the PWFS is inversely proportional to the modulation amplitude, namely the sensitivity will decrease with the modulation amplitude [2,3]. The modulation aspect also increases the complexity of the closed-loop system. Since the modulation period must be, at least, as fast as the PWFS detector, the modulator is more affecting cost and complexity using the most recent technology. Therefore, if the modulation mode is adopted in the closed-loop operation, the sensitivity of the wavefront detection will decrease and the complexity of the system will also increase.

In the above described content an attractive idea that rises with the use of the PWFS based on modulation-nonmodulation method. According to this method, the interaction matrix of the PWFS is determined with modulation but the sensor without modulation is used in the closed-loop process. In this paper, the direct gradient reconstruction (DGR) algorithm [19,20] is used for the modulation-nonmodulation PWFS. Firstly, the theoretical basis and simulation analysis are shown in detail. Secondly, the experimental results of the adaptive optics system based on the PWFS and the liquid-crystal spatial light modulator are given. Thirdly, the results using the PWFS and the 127-element discrete piezoelectric DM are also described. Finally, the field experiment on a 1.8m telescope has been carried out in the actual atmospheric turbulence.

2. Theory and simulation analysis

2.1 Mathematical model

The basic principle of the PWFS is illustrated in Fig. 1. The refractive pyramid is placed in the position of the focus to drive the light in four separate beams, and then through a relay lens the four sub-pupil images are obtained in the detector camera.

Fig. 1 Basic principle of the PWFS: (a) schematic diagram of the PWFS, and (b) four sub-pupil images in the detector camera

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The PWFS signals $S_{X} (x, y)$ and $S_{Y} (x, y)$ can be calculated according to the intensity difference of the four separated sub-pupil images ( $I_{1} (x, y)$ , $I_{2} (x, y)$ , $I_{3} (x, y)$ and $I_{4} (x, y)$ ).

\begin{array}{l} S_{X} (x, y) = \frac{I_{1} (x, y) + I_{4} (x, y) - I_{2} (x, y) - I_{3} (x, y)}{I_{1} (x, y) + I_{2} (x, y) + I_{3} (x, y) + I_{4} (x, y)} \\ S_{Y} (x, y) = \frac{I_{1} (x, y) + I_{2} (x, y) - I_{3} (x, y) - I_{4} (x, y)}{I_{1} (x, y) + I_{2} (x, y) + I_{3} (x, y) + I_{4} (x, y)} \end{array}

The complex amplitude in the focal plane of the PWFS is the Fourier transform of the complex amplitude $E (ε, η)$ in the entrance-pupil plane. Every refracting surface of the pyramid can be regarded as a filter $H_{i} (u, v)$ (when $(u, v)$ is in the $i t h$ quadrant, $H_{i} (u, v)$ = 1; otherwise $H_{i} (u, v)$ = 0), and the filtered field is Fourier transform again by the relay-lens and finally imaged in the detector camera to obtained the intensity signal $I_{i} (x, y)$ . Therefore, the whole simulation calculation process of the PWFS detection signal is shown in Fig. 2 (where FT is the Fourier transform, $f_{1}$ is the focal length of the focusing lens, $f_{2}$ is the focal length of the relay lens, and $λ$ is the wavelength). When the PWFS is used in the modulation mode, the pyramid filter factor $H_{i} (u, v)$ forms a periodic change, and the corresponding signal $I_{i} (x, y)$ also needs to be integrated in the period (the derivation process of modulation is no longer discussed in detail in the paper).

Fig. 2 Simulation calculation process of the PWFS signal

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The DM is the key element to compensate the desired wavefront error of the adaptive optics system, and the analytic form for the driver influence function of the DM can be expressed as the following Gauss function:

V_{j} (ε, η) = \exp [\ln ω {(\sqrt{{(ε - ε_{j})}^{2} + {(η - η_{j})}^{2}} / d)}^{α}]

where

(ε_{j} η_{j})

is the spatial coordinates of the

j t h

driver,

d

is the distance between drivers,

α

is the Gaussian exponent value, and

ω

is the cross-linking value of the drivers. When the DM is placed in the entrance-pupil plane of the PWFS, the complex amplitude field can be expressed as the following function:

E (ε, η) = \exp (i \frac{2 π}{λ} V_{j} (ε, η)) \cdot P (ε, η)

where

i

is the imaginary unit, and

P (ε, η)

is the aperture function.

Based on the above formula, the characteristic of the PWFS corresponding to the DGR algorithm is analyzed in the following section.

2.2 DGR algorithm

The process of the DGR algorithm is described as follows. Firstly, the influence function of each actuator of the DM is used as an input mode function, and then the PWFS detection signals corresponding to each mode function are calculated. Secondly, these signals are combined to form the interaction matrix $M$ . Thirdly, the reconstruction matrix $R$ is obtained with the calculation of an inverse matrix of matrix $M$ . Fourthly, by multiplying the reconstruction matrix $R$ with the PWFS detected signal $s$ , the control voltage vector $v$ of the DM is calculated with $v = R \cdot s$ . Finally, the measurement and correction continues repeated until the closed-loop control system is stabilized. In order to analyze the DGR algorithm, an adaptive optics simulation platform based on 127-element DM model is set up and the following results are obtained.

Firstly, the PWFS signal characteristic corresponding to the single driver influence function for different modulation amplitude is analyzed. As shown in Fig. 3, PWFS sub-pupil images with different modulation amplitude are given when the control voltage is applied to one actuator (the PV value of the DM wavefront is 0.5λ), and the calculated PWFS signal Sx and Sy are also illustrated in the figure. From the simulation images, it can be seen that the PWFS signal is in a saturated state (when there is no modulation) and the dispersion degree in X and Y directions is large. As the modulation amplitude increases, the PWFS signal gradually de-saturates and shifts to the linear region.

Fig. 3 PWFS signals of the actuator of the DM with different modulation amplitude: (a) the wavefront map of the actuator, (b) the four PWFS sub-pupil images of the actuator, and (c) the calculated PWFS signals Sx and Sy.

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Secondly, the condition number of the reconstruction matrix for different modulation amplitude is analyzed. It is well known that the effect of PWFS measurement noise on the result of recovery voltage can be measured by the condition number of the reconstruction matrix R. If the condition number is represented by the parameter cond(R), the relationship between the recovery voltage jitter and the detection signal jitter is given by:

\frac{{‖ Δ v ‖}_{2}}{{‖ v ‖}_{2}} \leq c o n d (R) \leq \frac{{‖ Δ s ‖}_{2}}{{‖ s ‖}_{2}}

where ||•|| is the euclidean norm. It can be seen from the formula that the condition number is large, and the influence of measurement noise on the restoration voltage is more serious. Therefore, the condition number of reconstruction matrix can be used as a measure factor of noise influence degree. The reconstruction matrices using DGR method with different modulation amplitude (0, 1 2, 3, 4 and 5 times diffraction limit width respectively) are obtained, and the condition numbers are calculated. As shown in Fig. 4, it can be seen that the condition number gradually decreases with the increase of the modulation amplitude, so the reconstruction matrix measured by modulation mode can reduce the influence of the noise and improve the stability of the closed-loop system.

Fig. 4 Condition number of construction matrix with different modulation amplitude.

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Thirdly, the correlation between the driver’s corresponding detection signals for different modulation amplitude of PWFS is analyzed. In order to analyze the problem more clearly, the correlation coefficient matrix between the driver influence function is calculated, and the result is shown in Fig. 5 (a). Subsequently, the correlation coefficient matrix corresponding to the PWFS signal of the 127-element DM with different modulation amplitude is deduced, which is shown in Fig. 5 (b). Obviously, the correlation between different drivers is relatively small, and generally there is correlation between neighboring drivers. When the modulation amplitude is large, the same correlation phenomenon appears between the driver’s detection signals. However, the modulation amplitude is smaller, and the correlation between the driver’s detection signals is greater. It can also be seen from Fig. 4 that the extension of the cross line region is greater in the non-modulation condition. Therefore, using modulation mode in the calibration process, we can get the interaction matrix closer to the correlation relationship between the drivers of DM.

Fig. 5 (a) the correlation matrix of the driver influence function, and (b) the correlation matrix of the PWFS signals with different modulation amplitude for the 127-element DM.

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Fourthly, the cross-coupling error of the modulation-nonmodulation mode is analyzed. It is obvious that the modulated and non-modulated PWFS signals from a given input wavefront are not simply different due to a multiplication factor that depends on the modulation amplitude. The signals change in shape with different modulation amplitude, and the change of shape produces a cross-coupling between the reconstructed actuator-voltage when the modulated reconstruction matrix is used on a non-modulated PWFS closed-loop system. Setting $M_{m}$ and $M_{0}$ the interaction matrices with and without modulation, $R_{m}$ and $R_{0}$ being the respective reconstruction matrices, the cross-coupling error is given by (no noise, linear range):

Δ v = (R_{m} \cdot M_{0} - I) \cdot v

where

v

is the column vector of the actuator-voltage coefficients of the input DM wavefront,

Δ v

is the coupling error in the reconstruction process and

I

is the identity matrix. Setting

(R_{m} \cdot M_{0} - I) = Q_{m}

, the coupling error for matrix

Q_{m}

with increasing modulation amplitude is given by:

Δ v = Q_{m} \cdot v

Using the above notation, the residual error

Δ v

after

N

iteration of the closed-loop correction is given by:

Δ v = Q_{m}^{N} \cdot v

On the basis of matrix theory, if the absolute values of the eigenvalues of matrix

Q_{m}

are less than 1, the matrix

Q_{m}^{N}

approaches zeros when the parameter

N

increases gradually. The matrices

M_{m}

and

R_{m}

using direct gradient method with different modulation amplitude are calculated according to the above simulation system, and then the matrices

Q_{m}

are obtained. The maximum values of the absolute value of the eigenvalues of the matrixes

Q_{m}

with different modulation amplitude (1 2, 3, 4 and 5 times diffraction limit width respectively) are given in Fig. 6. According to the simulation results, the following qualitative conclusions can be obtained. ① It is clear that the norm of

Q_{m}

is less than 1 to have cross-coupling error

Δ v

converging to zero after a suitable number of iterations, so the modulation-nonmodulation method can be applied to the closed-loop correction system. ② The maximum absolute value of the eigenvalue of the matrix

Q_{m}

increases with the increase of the modulation amplitude

m

, and will gradually tighten to 1 but not more than 1. This means that when the value of the modulation amplitude

m

is greater, the cross-coupling error is greater and the more iterations of the closed-loop is required.

Fig. 6 Maximum absolute value of the eigenvalues with different modulation amplitude.

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Fifthly, the coupling effect of detection noise on restoration voltage is analyzed. In the closed-loop process of the actual system, the detection PWFS signal is the superposition of the real signal and the measurement noise. That is: $s^{'} = s + s_{n}$ , where $s^{'}$ is the detection signal, $s$ is the real signal and $s_{n}$ is the measurement noise. In general, it can be considered that measuring noise is a white noise, which is not related to the real signal, and the noise is not related to each other, that is, the following relationship is satisfied.

〈 s_{n} \cdot s_{n}^{T} 〉 = I \cdot σ_{n}^{2} 〈 s \cdot s_{n}^{T} 〉 = 〈 s_{n} \cdot s^{T} 〉 = 0

where

I

is the identity matrix and the superscript

T

represents a transposition of the matrix. The above formula assumes that the variance of each sub-aperture measurement noise is equal to

σ_{n}^{2}

. The correlation matrix between the restoration voltages

v

is calculated as below:

C_{V} = 〈 v \cdot v^{T} 〉 = R 〈 s \cdot s^{T} 〉 R^{T} + R \cdot R^{T} \cdot σ_{n}^{2}

It is obviously that

R \cdot R^{T}

is not a diagonal matrix, so the measurement noise will cause the coupling effect of restoration voltages. In order to reduce the coupling effect of noise on the recovery process, we need to orthogonal the effect of the noise on the recovery voltage, and then eliminate the terms seriously affected by the noise with the truncated filtering. The matrix eigenvalue decomposition of the matrix

R \cdot R^{T}

is calculated:

U \cdot Γ \cdot U^{T} = R \cdot R^{T}

Where

Γ

is a diagonal matrix and the diagonal elements are the eigenvalues of the matix

R \cdot R^{T}

. Converting the reconstruction formula

v = R \cdot s

into

U^{T} \cdot v = U^{T} \cdot R \cdot s

, and the correlation matrix

R \cdot R^{T} \cdot σ_{n}^{2}

caused by noise is converted to

U^{T} R \cdot R^{T} \cdot U \cdot σ_{n}^{2} = Γ \cdot σ_{n}^{2}

. It is clear that

Γ \cdot σ_{n}^{2}

is a diagonal matrix, so the influence of detection noise on every recovery mode is orthogonal. By introducing the truncated matrix

H

(which is illustrated in Fig. 7), we can eliminate those modes which are seriously affected by noise, thus improving the closed-loop stability and correction effect.

Fig. 7 The format of the truncated matrix.

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Finally, the whole process of the DGR algorithm for the modulation-nonmodulation PWFS using in the closed-loop system is summarized and shown in Fig. 8.

Fig. 8 The calculation process of DGR algorithm.

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Of course, in closed-loop application the amount of iteration to converge depends not only on the amount of modal cross-coupling due to the mis-reconstruction process for the nonmodulation-modulation mode, but also on the presence of initial saturation due to the nonmodulation closed-loop operation. Therefore, there are still many technical problems to be faced in practical closed-loop applications, and more reliable and accurate control strategies need to be established. In our closed-loop control system, a signal indicator is introduced to determine whether the initial saturation is playing an important role, or the cross-coupling is the main role. (for instance, using Strehl ratio as a indicator, and supposing that saturation effect is no longer dominant when Strehl ratio is equal or large than 70%.). According to the objective judgment of the signal indicator, a simple and effective variable-proportion control method can be used for the closed-loop operation. We set up a closed-loop simulation model with 127-element DM and 13 × 13 PWFS detection sub-apertures, and then the closed-loop correction process for turbulence phase-screen with D/r₀ = 13 (D is the input aperture diameter, and r₀ is the Fried parameter) is carried out with the above DGR algorithm. The results are shown in Fig. 9. The red curve indicates the traditional proportional-integral control with fixed-proportional coefficient, and the blue curve represents the variable-proportional control using the signal indicator. The closed-loop effects obtained by these two control methods are the same, but the number of iterations in the convergence process is quite different. The variable-proportion control method can converge quickly, while the traditional fixed-proportion control needs more iterations to achieve convergence.

Fig. 9 Plot of RMS with the number of iteration for turbulence phase-screen.

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3. Experiment results

3.1 Closed-loop system with the PWFS and the LC-SLM

A closed-loop experiment based on the liquid-crystal spatial light modulator (LC-SLM) and PWFS was performed. The purpose of this experiment is to verify the effectiveness of this method for multi-unit driver AO system, and to provide experimental basis for the application of large aperture astronomical telescope. The photography of the closed-loop laboratory setup is shown in Fig. 10. The pyramid optic is the double refractive pyramid that behaves as a single pyramid having a base angle equal to the difference between the two base angles of these two pyramids [21,22]. The LC-SLM is an x-y phase series spatial light modulator and has an array of 256 × 256 pixels across a 4.61 mm square aperture. In our case the LC-SLM is used as the DM.

Fig. 10 The photograph of AO system based on PWFS and LC-SLM.

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The SLM is used to mimic a DM with 595 actuators, and the detected sub-pupil image of the PWFS is sampled with 28 × 28 sub-apertures (similar to 28 × 28 sub-apertures sampling of the Hartmann wavefront sensor). The relationship of the 595 actuators and the corresponding 28 × 28 sub-apertures is shown in Fig. 11. The tilting mirror is placed on the conjugated pupil relay image position. For the calibration operation the tip-tilt mirror is used to produce the cycle movement with a modulation diameter on the tip of the pyramid of 3 λ/D (where λ is the detection wavelength and D is the input aperture diameter), but in the closed-loop process for correcting the aberration, the PWFS is working without modulation.

Fig. 11 The configuration diagram of the 595 actuators and the 28 × 28 pupil sampling.

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The experimental results in closed-loop compensation for the Kolmogorov turbulence phase screen with D/r₀ = 30 (D is the input aperture diameter, and r₀ is the Fried parameter) is shown in Fig. 12 and Fig. 13. The results show that the corrected wavefront has a RMS of about 0.0676λ referring to initial RMS value 1.2867λ, and the far-field spot energy is concentrated after closed-loop correction.

Fig. 12 The wavefront images for turbulence phase-screen (D/r₀ = 30): (a) the initial wavefront, and (b) the corrected wavefront.

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Fig. 13 The far-field images for turbulence phase-screen (D/r0 = 30): (a) the initial image, and (b) the corrected image.

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In order to analyze the closed-loop ability of the system, the intensity value of the corrected far-field image is limited within [0 500] and the corresponding image is shown in Fig. 14(a). The center region is saturated, but it is easy to see the more detailed information of the image. For example, as can be seen from the figure, the first-order diffraction ring of the far-field image is completed. The FWHM (full width at half maximum) is used to quantitative analyze the correction effect, and Fig. 14(b) shows the profile chart in the center of the far-field image. A rectangular area in this figure illustrates the FWHM with 19 pixels of the detection camera (the diffraction-limited region occupied 18.2 pixels). Consequently, the performance of the system after correction is close to the diffraction-limited condition.

Fig. 14 (a) The corrected far-field images (intensity value is limited within 0 to 500), and (b). The profile chart of the corrected far-field image (the rectangular region in the figure is to compute the FWHM. The x-coordinate represents the pixel of the detection camera, and the y-coordinate shows the intensity value).

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3.2 Closed-loop system with the PWFS and the continuous surface discrete DM

Since the LC-SLM generally operates at a single wavelength and the operating frequency of the SLM is relatively low, we have also built a closed-loop AO system based on the continuous surface discrete 127-element DM and the 13 × 13 PWFS detection sub-apertures. The laboratory setup is shown in Fig. 15. The configuration of the 127 actuators and the 13 × 13 sub-apertures is shown in Fig. 16. As mentioned above, the closed-loop working principle based on the modulation-nonmodulation PWFS with DGR algorithm is still used in this experimental system. The open-loop and closed-loop far-field images are shown in Fig. 17. Obviously, the closed-loop spot is close to the diffraction-limited level.

Fig. 15 The photograph of AO system based on PWFS and the discrete DM

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Fig. 16 The configuration diagram of the 127 actuators and the 13 × 13 pupil sampling.

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Fig. 17 The far-field images: (a) the open-loop image, and (b) the closed-loop image.

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3.3 Closed-loop system for a 1.8m telescope

The effectiveness of the closed-loop control method is fully verified by the experimental results above, but it is undeniable that the effect of the actual atmospheric turbulence is more complex. Therefore, it is necessary to verify whether this mode of work is still valid in the actual atmospheric turbulence. In order to achieve this goal, a PWFS system is installed on the AO system of 1.8 m telescope at the Yunnan Observatory [23], and the optical setup is shown in Fig. 18. The PWFS detection band is from 450nm to 700nm, and the far-field imaging band is from 700nm to 900nm. Using the above modulation-nonmodulation control method with DGR algorithm, the astronomical star in real atmospheric turbulence is observed, and the results are shown in Fig. 19. It is obvious that the light spot energy was concentrated after closed-loop correction. (FWHM is about 1.2 times the diffraction-limited value.)

Fig. 18 The photograph of installed PWFS system for 1.8m telescope.

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Fig. 19 The star image (Name: HIP116307, Magnitude: 6.1): (a) the open-loop image, and (b) the closed-loop image.

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4. Conclusions

An adaptive optics system based on the modulation-nonmodulation PWFS with the DRG algorithm is theoretically analyzed and experimentally demonstrated. Using this modulation-nonmodulation combination method, the modulation mode is used to obtain the interaction matrix in the calibration process, while the non-modulation mode is used in the closed-loop operation. The experiment results show that this method is effective in the closed-loop adaptive optics system.

However, what must be explained here is that the above-mentioned work is just an experimental demonstration of the feasibility of this work model, and there are many meticulous tasks that need to be gradually developed. For example, the influence of non-linearity error caused by non-modulated closed-loop process, the closed-loop control bandwidth analysis, and the comparison analysis with the traditional modulation mode. These issues will be analyzed in detail in the later work.

Funding

National Natural Science Foundation of China (NSFC) (61008038), Innovation Fund of Key Laboratory of Chinese Academy of Sciences (17S053); National Key Research and Development Program of China (2016YFB0501100).

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Modulation-nonmodulation pyramid wavefront sensor with direct gradient reconstruction algorithm on the closed-loop adaptive optics system

Abstract

1. Introduction

2. Theory and simulation analysis

2.1 Mathematical model

2.2 DGR algorithm

3. Experiment results

3.1 Closed-loop system with the PWFS and the LC-SLM

3.2 Closed-loop system with the PWFS and the continuous surface discrete DM

3.3 Closed-loop system for a 1.8m telescope

4. Conclusions

Funding

References and links

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Figures (19)

Equations (10)

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