Synchronous model-based approach for wavefront sensorless adaptive optics system

Wen Lianghua; Ping Yang; Yang Kangjian; Chen Shanqiu; Wang Shuai; Liu Wenjing; Bing Xu

doi:10.1364/OE.25.020584

1. Introduction

Wavefront sensorless adaptive optics(WFSless AO) system has been widely used in some applications, such as inertial confinement fusion(ICF) [1,2], microscopy [3,4], human eyes imaging [5], optical tracking [6], and free-space laser communication [7]. The performance of WFSless AO system is greatly dependent on control algorithms. Although there are many approaches for WFSless AO system, such as hill climbing [8,9], SPGD [10,11], simulated annealing(SA) [12], genetic algorithm(GA) [13], trust region method [14], and so on. In general, they could be divided into blind optimization search method and approximately solving approach according to implementing procedure. The convergence speed of approximately solving algorithm is generally faster than that of search method owing to directly acquiring the approximate solutions of control signal from far-field. This kind algorithm is first put forward by M. J. Booth [15, 16]. Sequentially, a general model-based approach proposed by L. H. Huang is fast and stable with only N + 1 measurements for the N aberration modes as the predetermined bias functions [17]. This method is successfully applied in the coherent beam combination with segment deformable mirror (DM) [18], and extended objects imaging simulation [19]. The correction capability and convergence speed of WFSless AO system based on this approach is more excellent than that of SPGD algorithm [20]. However the performance of this method gradually becomes worse as increasing N, eventually it fails to close loop in the practical systems under the poor conditions such as the intensity fluctuation of far-field. This degradation is mainly caused by the long interval time between adjacent corrections in practical AO systems, which work as that an aberration correction is only performed after applying N modes aberration perturbations on DM.

In this paper, a high speed and general model-based approach is proposed to improve the temporal property in synchronous correction mode. This approach can implement a mode aberration correction after imposing the same mode aberration perturbation on DM. Firstly, the analysis of the correlations among the gradients of various Zernike functions is presented. Then a new aberration modes function set is reconstructed by singular value decomposition (SVD), whose gradients are orthogonal. Sequentially, the synchronous correction algorithm is presented through theory derivation. To validate this algorithm, a Fresnel zone plates (FZP) imaging system with WFSless AO is set up in laboratory. Then the aberration corrections experiments of FZP imaging are completed by the proposed method and SPGD algorithm. The improvement of the FZP imaging system with WFSless AO is demonstrated by point spread function (PSF), full width half maximum (FWHM) value of far-field intensity and modulation transform function (MTF). The convergence process is indicated by several metrics of far-field intensity.

The contribution of our work is that a synchronous model-based method is presented, and validated in the practical WFSless AO system for aberration correction of the FZP imaging. Furthermore, WFSless AO system is firstly employed in the FZP imaging, which is a potential application in lightweight and large diameter space telescope [21,22]. The remains of this paper are organized as follows. In section 2, the limitations of the previous method in [17] are analyzed, a new aberration modes set is reconstructed, and a novel model-based algorithm for the WFSless AO system is proposed. In section 3, the experiments about the FZP imaging with aberration correction are completed. Finally, the result analysis of experiment is given.

2. Theoretical basis

The schematic of a WFSless AO system can be depicted as shown in Fig. 1. The incident wavefront is corrected by DM, and then focused on the focal plane through lens and a pinhole. The CCD camera samples the far-field intensity and forms feedback signals to the controller. The controller manipulates DM by driver voltages generated through the various optimization algorithms. During iterations, the expected control objective is to minimize the residual aberration $R (x, y) = ϕ (x, y) - ψ (x, y)$ , and to optimize the far-filed intensity distribution. $ϕ (x, y)$ and $ψ (x, y)$ are the aberration of incident wavefront and DM’s correction shape respectively. The previous method in [17] implements an aberration correction after applying N modes perturbations on DM, which is based on the approximately linear relation between the mean square of the aberration gradients and the second moment of far-field intensity distribution. However the long interval time (correction delay) between aberration correction and second moment detection degrades the performance due to the coupling of aberration modes’ gradients.

Fig. 1 The schematic of WFSless AO system. The light is depicted as dot line. The control loop is shown as solid line.

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2.1 model-based aberration corrections

The $ϕ (x, y)$ and $ψ (x, y)$ can be expressed as the series of M orthonormal functions and N orthonormal functions whose coefficients are $β_{i}$ and $p_{i}$ respectively. $F_{i} (x, y)$ could be all the kinds of orthonormal basis functions named general modes, such as Zernike functions, L-Z functions, and so on.

ϕ (x, y) = \sum_{i = 1}^{M} β_{i} F_{i} (x, y)

ψ (x, y) = \sum_{i = 1}^{N} p_{i} F_{i} (x, y)

The mean square of aberration gradient (SM) is defined as following.

S M = S^{- 1} {{\iint_{S} [\frac{\partial ϕ (x, y)}{\partial x}]}^{2} d x d y + {\iint_{S} [\frac{\partial ϕ (x, y)}{\partial y}]}^{2} d x d y}

where

S

is the area of incident light pupil plane, whose coordinate is indicated as (x, y).

The second moment of far-field intensity distribution is expressed with MDS.

M D S = \frac{\iint I (x_{1}, y_{1}) [1 - \frac{(x_{1}^{2} + y_{1}^{2})}{R^{2}}] d x_{1} d y_{1}}{\iint I (x_{1}, y_{1}) d x_{1} d y_{1}} = \frac{\iint I (x_{1}, y_{1}) [1 - \frac{r^{2}}{R^{2}}] d x_{1} d y_{1}}{\iint I (x_{1}, y_{1}) d x_{1} d y_{1}}

where

(x_{1}, y_{1})

is the coordinate of the focal plane, and

I (x_{1}, y {}_{1})

is the far-filed intensity distribution sampled by CCD camera. R is the integral radius of far-field intensity. During the MDS calculations,

I (x_{1}, y {}_{1})

is weighted by

1 - r^{2} / R^{2} = 1 - (x_{1}^{2} + y_{1}^{2}) / R^{2}

(

r = 0

, when

x_{1}^{2} + y_{1}^{2} > R^{2}

), which is easy to be realized by program in controller. The linearity between SM and MDS is derived from Geometrical Optics and Physical Optics in [17,19].

S M \approx C_{0} (1 - M D S)

The voltages corresponding to every $α F_{i} (x, y)$ aberration mode are imposed on DM in sequence, and the far-field intensity is sampled by camera. Then the corresponding $M D S_{i}$ is calculated and recorded sequentially. The increment $w_{i}$ of SM is obtained after imposing the perturbations in line with $α F_{i} (x, y)$ on DM.

\begin{array}{l} w_{i} = S M_{i} - S M_{0} = S^{- 1} \iint_{S} 2 α [\frac{\partial F_{i} (x, y)}{\partial x} \frac{\partial ϕ (x, y)}{\partial x} + \frac{\partial F_{i} (x, y)}{\partial y} \frac{\partial ϕ (x, y)}{\partial y}] d x d y \\ + S^{- 1} \iint_{S} α^{2} {{[\frac{\partial F_{i} (x, y)}{\partial x}]}^{2} + {[\frac{\partial F_{i} (x, y)}{\partial y}]}^{2}} d x d y \end{array}

w_{i} = S M_{i} - S M_{0} \approx C_{0} (M D S_{0} - M D S_{i}) = C_{0} m_{i}_{(i = 1, 2 \dots M)}

When all the perturbation voltages in accordance to M aberration modes are imposed sequentially on DM, $Δ M = {(m_{1}, m_{2} \dots m_{M})}^{T}$ consisting of the increment of $M D S_{i}$ is obtained. Then the aberration modes coefficient vector $β = {(β_{1}, β_{2} \dots β_{M})}^{T}$ is approximately solved according to Eq. (8). Finally, the control voltages are obtained according to Eq. (9). The formula derivation is given in [17].

β = \frac{K^{- 1} (W - α^{2} K_{M})}{2 α} \approx \frac{K^{- 1} (C_{0} Δ M - α^{2} K_{M})}{2 α}

v = C_{z v} β

where

W = {(w_{1}, w_{2} \dots w_{M})}^{T}

is a vector consisting of the increment of

S M_{i}

,

C_{0}

is a linearity constant,

C_{z v}

is the correlation matrix between the aberration modes and the influence functions of DM.

K

is the correlation matrix of the basis functions’ gradients,

K_{M}

is the vector constituted with the diagonal elements of K. Generally, the gradients of aberration mode functions are not always orthonormal. In other words, K is not a diagonal matrix.

K = (\begin{matrix} K_{11} & K_{12} & \dots & K_{1 N} & \dots & K_{1 M} \\ K_{21} & K_{22} & \dots & K_{2 N} & \dots & K_{2 M} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ K_{N 1} & K {}_{N 2} & \dots & K_{N N} & \dots & K_{M N} \end{matrix})

K_{i, j} = S^{- 1} \iint_{S} \frac{\partial F_{i} (x, y)}{\partial x} \frac{\partial F_{j} (x, y)}{\partial x} d x d y + S^{- 1} \iint_{S} \frac{\partial F_{i} (x, y)}{\partial y} \frac{\partial F_{j} (x, y)}{\partial y} d x d y

Although the numeral simulation results of this algorithm show the Strehl ratio could be raised up to 0.9 after an aberration correction cycle. The performance degrades sharply in the practical WFSless AO systems. Indeed, the WFSless AO system with this method fails to close loop, especially under the detrimental conditions such as the intensity fluctuation, changing aberration, vibration of optics platform, and so on. The significant cause is that correction delay NT is too long. The aberration correction timing chart is shown in Fig. 2.

Fig. 2 The correction timing of the previous method in [17]: a correction cycle needs N + 1 measurements for N aberration modes correction; The aberration frozen time window is sized about (N + 1)T, and the correction delay is about NT.

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During the N + 1measurements of a correction cycle, the aberration is supposed to be invariant to ensure that the MDS₀ is a constant. It seems that the aberration is frozen during this time (N + 1)T, which may be called aberration frozen window. In practical systems, this frozen window is difficult to exist under the changing aberration. Actually, it is hard to keep MDS₀ invariant in a correction cycle owing to the absence of aberration frozen window and some others factors. The fluctuations of $M D S_{0}$ calculated from actual far-field intensity images are shown in Fig. 3, which changes randomly ranging from 0.9865 to 0.9890 during 200 sample periods. Although the absolute value of fluctuations is very small, it is nearly equal to the increment $m_{i}$ in line with applying aberration perturbations on DM.

Fig. 3 The MDS fluctuation in real system

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In a correction cycle, an aberration correction is only operated after applying N modes aberration perturbations on DM. In N times perturbation procedure, the $m_{i}$ (i = 1, 2…N) is measured and calculated referring to the same $M D S_{0}$ supposed to be invariant, which is actually varied owing to all the kinds of factors mentioned previously. As the increasing of N, the errors of $m_{i}$ (i = 2, 3…N) gradually become larger, the $β_{i}$ deviates from the real value, and the algorithm is failed. For roughly estimating, under the parameters that $C_{0} = 40$ and $α = 0.01$ , if the error of $m_{i}$ is 0.001, the error of $β_{i}$ will be up to 0.1, which leads enormously large aberration. This failure in practical WFSless AO system suggests that it is significantly important to shorten the correction delay, which is nearly equal to NT owing to the coupling of aberration modes’ gradient.

2.2 Decoupling and Reconstruction

The algorithm degrades sharply in real systems owing to the coupling of aberration’s gradients. Therefore, it is significant to remove the coupling and to reconstruct new predetermined aberration modes basis functions, whose gradients are orthogonal. In algorithm implementation, there is a hypothesis that M = N in Eq. (1) and Eq. (2), whose matrix expressions are following.

ϕ (x, y) = V_{N} F_{N} = (V_{N} C_{N}) (C_{N}^{- 1} F_{N})

ψ (x, y) = P_{N} F_{N} = (P_{N} C_{N}) (C_{N}^{- 1} F_{N})

where

F_{N} = {(F_{1} (x, y), F_{2} (x, y), \dots F_{N} (x, y))}^{T}

is a vector consisting of modes basis functions, which are Zernike functions in this paper.

β_{N} = (β_{1}, β_{2}, \dots β_{N})

and

p_{N} = (p_{1}, p_{2}, \dots p_{N})

are composed by the mode coefficients of wavefront and DM respectively.

C_{N}

is obtained by the singular value decomposition (SVD) of gradient correlation matrix K.

L = C_{N}^{- 1} K {(C_{N}^{- 1})}^{T}

Then, a new aberration mode function set is represented as $G_{N} (x, y)$ , whose gradients of modes basis functions are orthogonal. Therefore the correlation matrix L transformed from K is diagonal.

G_{N} (x, y) = C_{N}^{- 1} F_{N}

L = (\begin{matrix} l_{1, 1} & 0 & \dots & 0 \\ 0 & l_{2, 2} & \dots & 0 \\ 0 & \dots & ⋱ & 0 \\ 0 & \dots & 0 & l_{N, N} \end{matrix})

l_{i, j} = s^{- 1} \iint_{s} \frac{\partial G_{i} (x, y)}{\partial x} \frac{\partial G_{j} (x, y)}{\partial x} d x d y + s^{- 1} \iint_{s} \frac{\partial G_{i} (x, y)}{\partial y} \frac{\partial G_{j} (x, y)}{\partial y} d x d y

According to Eqs. (14) and (15), a new 33 aberration modes are drawn in Fig. 4(b), which are reconstructed from 33 Zernike modes drawn in Fig. 4(a). The inverse matrixes $K^{- 1}$ and $L^{- 1}$ of gradient correlation matrixes of Zernike and new aberration modes are shown in Fig. 5 respectively. It is apparent that the matrix $L^{- 1}$ is diagonal by decoupling correlations among the aberration modes’ gradients.

Fig. 4 Aberration modes: (a) 33 Zernike modes (3-35); (b) 33 reconstructed modes

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Fig. 5 The inverse matrixes $K^{- 1}$ and $L^{- 1}$ of the correlation matrixes of Zernike and new aberration modes’ gradients: (a) 33 Zernike modes (3-35); (b) 33 new aberration modes.

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2.3 Synchronous correction

The $F_{i} (x, y)$ in Eq. (6) is replaced with $G_{i} (x, y)$ , which is the element of aberration mode basis functions set $G_{N} (x, y)$ , and the new relations are expressed as follows.

\begin{array}{l} w_{i} = S M_{i} - S M_{0} = S^{- 1} \iint_{S} 2 α [\frac{\partial G_{i} (x, y)}{\partial x} \frac{\partial ϕ (x, y)}{\partial x} + \frac{\partial G_{i} (x, y)}{\partial y} \frac{\partial ϕ (x, y)}{\partial y}] d x d y \\ + S^{- 1} \iint_{S} α^{2} {{[\frac{\partial G_{i} (x, y)}{\partial x}]}^{2} + {[\frac{\partial G_{i} (x, y)}{\partial y}]}^{2}} d x d y \end{array}

The perturbation voltages corresponding to every $α G_{i} (x, y)$ mode are imposed on DM, and far-field intensity is sampled by camera. Then $M D S_{i}$ , $M_{i}$ and $w_{i}$ are obtained according to Eqs. (4)-(7). The Eq. (8) is rearranged as following.

\frac{(C_{0} Δ M - α^{2} K_{m})}{2 α} = \frac{1}{2 α} (\begin{array}{l} C_{0} m_{1} - α^{2} K_{11} \\ C_{0} m_{2} - α^{2} K_{22} \\ ⋮ \\ C_{0} m_{N} - α^{2} K_{N N} \end{array})

(\begin{array}{l} β_{1} \\ β_{2} \\ ⋮ \\ β_{N} \end{array}) \approx \frac{1}{2 α} (\begin{matrix} K^{'}_{1, 1} & K^{'}_{12} & \dots & K^{'}_{1 N} \\ K^{'}_{21} & K^{'}_{2, 2} & \dots & K^{'}_{2 N} \\ ⋮ & ⋮ & \dots & ⋮ \\ K^{'}_{N 1} & K^{'}_{N 2} & \dots & K^{'}_{N, N} \end{matrix}) (\begin{array}{l} C_{0} m_{1} - α^{2} K_{11} \\ C_{0} m_{2} - α^{2} K_{22} \\ ⋮ \\ C_{0} m_{N} - α^{2} K_{N N} \end{array})

where

K^{- 1} = (\begin{matrix} K^{'}_{1, 1} & K^{'}_{12} & \dots & K^{'}_{1 N} \\ K^{'}_{21} & K^{'}_{2, 2} & \dots & K^{'}_{2 N} \\ ⋮ & ⋮ & \dots & ⋮ \\ K^{'}_{N 1} & K^{'}_{N 2} & \dots & K^{'}_{N, N} \end{matrix})

is the inverse matrix of K.

β_{i}

can be obtained asfollowing.

β_{i} = \frac{1}{2 α} (K^{'}_{i 1}, K^{'}_{i 2} \dots K^{'}_{i N}) (\begin{array}{l} C_{0} m_{1} - α^{2} K_{11} \\ C_{0} m_{2} - α^{2} K_{22} \\ ⋮ \\ C_{0} m_{N} - α^{2} K_{N N} \end{array})

If matrix K and its inverse matrix $K^{- 1}$ are diagonal, Eq. (21) will be simplified as the scalar computation.

β_{i} = \frac{1}{2 α} K^{'}_{i i} (C_{0} m_{i} - α^{2} K_{i i})

The matrix L and

L^{- 1}

are the diagonal transformations of K and its inverse matrix. The rearrangement of Eq. (22) is as following.

β_{i} \approx \frac{l_{i i}^{- 1} (C_{0} m_{i} - α^{2} l_{i i})}{2 α}

where

l_{i i}

is the diagonal element of matrix L, the other symbols in Eq. (23) are the same as above described. The coefficient of mode aberration

G_{i} (x, y)

included in the wavefront aberration is acquired according to Eq. (23), which means that a mode aberration can be compensated immediately after applying the same mode aberration perturbations on DM. This synchronous increment correction is worked as shown in Fig. 6.

Fig. 6 The correction timing of synchronous correction: a correction cycle for N aberration modes needs 2N measurements, The aberration frozen time window is sized about 2T, and the correction delay is about T.

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Comparing with the previous method [17], the measurements are increased by N-1. However, the correction delay is nearly fixed to $Δ t = T$ only determined by sampling and perturbation period. The aberration frozen window size is reduced to about 2T, which is more beneficial to correct the dynamic aberration and to keep MDS₀ invariant comparing with (N + 1)T. The more important is that the linear relation between SM and MDS gradually becomes better, when the aberration is corrected incrementally as one correction one perturbation [17]. On the other hand, the correction speed and capability are enhanced greatly due to the improvement of linear relation according to Eq. (8) or Eq. (23). This method is based on general aberration mode such as Zernike functions, L-Z functions and so on, so it can be called “AOG” algorithm that means WFSless AO based on general modes. The merits of AOG are to improve the correlation between the control and measurement by reducing the delay, and to reduce the computing burden by transforming matrix operation in Eq. (8) into scalar form in Eq. (23). The mainly reduction of computation is that the number of multiplications is reduced by $N * (N - 2)$ , and the addition is reduced by $N^{2}$ in a correction cycle.

3. Experiment

To verify the correction capability of this synchronous model-based method, a WFSless AO system is used to correct aberration of the FZP imaging system. The FZP is the membrane diffraction optics component, which will be potentially applied in next generation space telescope featured with lightweight and large diameter [21].The unavoidable aberration of the FZP imaging caused by the substructure defects, space environmental factors, and stability of membrane materials are supposed to be corrected by adaptive optics [22]. However the space arrangement and launch load are very limited in space telescope application. Comparing with the conventional AO with wavefront sensor, WFSless AO is the most feasible to compensate the aberration for FZP imaging due to its simple structure, which are comprised of static and varying slowly aberration. Additionally, the FZP imaging is low signal to noise ratio scenario for AO correction, which could be used to verify the anti-noise characteristics of this method.

An experiment system incorporating the FZP with WFSless AO is set up as shown in Fig. 7. The system is mainly composed by a collimated light source, a FZP, an optical beam shrinker, two tilt/tip mirrors (TM1 and TM2), a deformable mirror (DM), a computer control system, and a combination sensor. The far-field detection and the feedback signal to the control computer are completed by the far-field camera of the combination sensor. Due to abnormal trigger function of CCD camera, the delay (t<<T shown in Fig. 2, and Fig. 6) between measurement and applying voltages on DM is set to the sample period of CCD camera. However the aberration frozen window and correction delay are both increased by a sample period, the correction timing is assured strictly. There are two measurements and two applying voltages on DM during one algorithm iteration. So it takes four sample periods to implement a correction.

Fig. 7 The experimental system for correcting aberration of the FZP imaging

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For analyzing the correction ability and the convergence speed of control algorithms, the WFSless AO system is manipulated by SPGD algorithm based on aberration modes, SPGD algorithm based on voltages, and AOG method respectively in aberration correction experiments. To acquire the correction performance, 1024 frames image of far-field intensity are sampled and recorded in real time.

3.1 PSF and MTF

The imaging is the convolution of the object transmission and the intensity of the focal spot. Therefore the shape of focal spot determines the imaging quality. Before aberration correction, there are many side lobes in focal plane spot of the FZP imaging with aberration. When the aberration is corrected by WFSless AO with the SPGD algorithms, and AOG method respectively, the peak value of far-field intensity is doubled, and the spot shape is improved to near diffraction limit (1.2DL). However the results of the different algorithms are distinct as shown in Fig. 8 and listed in Table 1. It is evident that the maximum value of far-filed intensity (“Imax” in Table 1), the shape of focal plane spot, and the FWHM of far-filed intensity are the most optimum, when the aberration is compensated by AOG algorithm.

Fig. 8 The focal plane intensity distribution: (a) The intensity with aberration; (b) The intensity corrected by SPGD based on modes; (c) The intensity corrected by SPGD based on voltages; (d) The intensity corrected by AOG. The corresponding focal spots are shown in (e), (f), (g), and (h) in sequence.

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Table 1. The maximum value and FWHM of the far-field intensity

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As shown in Fig. 9, the moderate frequency components of the MTF are decreased sharply due to the wavefront aberration of imaging beam, which are enhanced apparently after aberration correction. Generally, the imaging performance becomes better as the increasing of the area surrounded by MTF curve and coordinate axis. From the PSFs, FWHM values and MTFs of the FZP imaging system with aberration correction, the correction ability of AOG algorithm is more excellent than that of SPGD approaches.

Fig. 9 The MTF curves of FZP imaging system. The curve represented by legend “aog” is the result of AOG algorithm. The “spgdm” stands for the result of SPGD algorithm based on modes. The “spgdv” indicates the result of SPGD method based on voltages. The “original” is the curve without correction (only in Fig. 9).The same legends are adopted in following figure.

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3.2 Convergence characteristics

In aberration correction of the FZP imaging, the convergence speed of AOG method and SPGD algorithms is analyzed from efficient radius(R₀), secondary moment( $M D S$ ), and image sharpness(J). The efficient radius and the image sharpness are defined respectively as follows.

R_{0} = \frac{\iint_{s} \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}} I (x, y) d x d y}{\iint_{s} I (x, y) d x d y}

x_{0} = \frac{\iint_{s} x . I (x, y) d x d y}{\iint_{s} I (x, y) d x d y}, y_{0} = \frac{\iint_{s} y . I (x, y) d x d y}{\iint_{s} I (x, y) d x d y}

J = \frac{\iint_{s} I {(x, y)}^{2} d x d y}{(\iint_{s} I (x, y) d x d y)^{2}}

Where is the intensity distribution of the focal plane spot, whose centroid is expressed with . The MDS curves of far-filed is shown as Fig. 10, when WFSless AO is running with AOG algorithm, SPGD algorithm based on modes, and SPGD algorithm based on voltages. The corresponding efficient radius curves and image sharpness curves of far-filed intensity are indicated in Figs. 11 and 12 respectively. From these characteristics curves of far-field intensity, it is distinct that the aberration correction speed with AOG algorithm is the fastest, and the corresponding metrics are the most optimum.

Fig. 10 The secondary moment of the focal spot

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Fig. 11 The efficient radius of the focal spot

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Fig. 12 The image sharpness of the focal spot

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From the results listed in Table 2, the AOG algorithm is converged through 51 iterations, which is closed loop at frame 96 and nearly converged at frame 302. The value of MDS has improved from 0.9901 to 0.9993.The image sharpness J raises from 4.02 × 10⁻³ to 9.91 × 10⁻³. The efficient radius is reduced from 6.450 to 3.700. By contrast, attaining the near correction, the iteration number of the SPGD algorithm based on modes is 202, which is closed loop at frame 96, and nearly converged at frame 904. The correction speed of AOG method is about 4 times than that of SPGD algorithm based on modes. Additionally the SPGD algorithm based on voltages is still optimizing, even if it is operated prior to the other two methods. It is necessary to explain that although the absolute value of the metrics (MDS, J and R₀) is very small, the very subtle changes of these metrics indicate the huge changes of the system’s performance owing to the normalizing in their definitions. According to the above analysis and the characteristics curves of far-field intensity, the performance of wavefront aberration corrections with AOG algorithm is much more excellent than SPGD method, especially in correction speed.

Table 2. The convergence characteristics of different algorithms

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4. Conclusion

According the linearity between SM and MDS, a synchronous model-based algorithm for WFSless AO system is set up by orthogonalizing aberration modes’ gradients and reconstructing new aberration modes set. This algorithm is distinct from conventional optimum algorithms such as SPGD due to directly deriving the approximate solutions of control signals from the far-filed intensity. To verify the performance of this algorithm, the aberration corrections experiments for FZP imaging system with WFSless AO are completed. The results indicate that this method is excellent control algorithm for WFSless AO system. It is better than SPGD algorithm in correction speed, and correction capability. Especially, the correction speed is about 4 times than that of SPGD algorithm, which is very important to compensate varying aberration. In addition, the effectiveness of aberration correction for the FZP imaging by WFSless AO is verified firstly in practical system, which is significant for membrane Fresnel diffract imaging in the space telescope in future.

Funding

The work is sponsored by National Natural Science Foundation of China (Grant No.60978049), Youth Innovation Promotion Association, Scientists of Chinese Academy of Sciences (Grant No.2012280), Foundation for Outstanding Young Scientists of Chinese Academy of Sciences, and Science and Technology Innovation Fund, Chinese Academy of Sciences (Grant No.CXJJ-16M208).

Acknowledgments

The authors would like to acknowledge Doctor L. H. Huang from the key laboratory of Adaptive Optics, Chinese Academy of Sciences for help with the algorithms’ simulation work; Doctor W. Yang from Institute of Optics and Electronics, Chinese Academy of Sciences for the design of FZP.

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algorithms	Imax(ADU)	FWHM(pixel)
SPGD based on modes	1714	8.05
SPGD based on voltages	1658	8.16
AOG algorithm	1811	7.645

algorithms	Imax(ADU)	FWHM(pixel)
SPGD based on modes	1714	8.05
SPGD based on voltages	1658	8.16
AOG algorithm	1811	7.645

Synchronous model-based approach for wavefront sensorless adaptive optics system

Abstract

1. Introduction

2. Theoretical basis

2.1 model-based aberration corrections

2.2 Decoupling and Reconstruction

2.3 Synchronous correction

3. Experiment

3.1 PSF and MTF

3.2 Convergence characteristics

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (12)

Tables (2)

Equations (26)

Optics Express

Algorithms	Convergence	MDS	R₀	J( × 10⁻³)
SPGD based on modes	202	0.9972	3.750	9.89
SPGD based on voltages	~	0.9960	4.015	8.11
AOG algorithm	51	0.9993	3.700	9.91