Abstract

A model-based wavefront sensorless (WFSless) adaptive optics (AO) system with a 61-element deformable mirror is simulated to correct the imaging of a turbulence-degraded extended object. A fast closed-loop control algorithm, which is based on the linear relation between the mean square of the aberration gradients and the second moment of the image intensity distribution, is used to generate the control signals for the actuators of the deformable mirror (DM). The restoration capability and the convergence rate of the AO system are investigated with different turbulence strength wave-front aberrations. Simulation results show the model-based WFSless AO system can restore those images degraded by different turbulence strengths successfully and obtain the correction very close to the achievable capability of the given DM. Compared with the ideal correction of 61-element DM, the averaged relative error of RMS value is 6%. The convergence rate of AO system is independent of the turbulence strength and only depends on the number of actuators of DM.

© 2015 Optical Society of America

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References

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2015 (2)

2014 (2)

2013 (1)

2011 (1)

2007 (2)

2006 (1)

2003 (1)

M. Zakynthinaki and Y. Saridakis, “Stochastic optimization for a tip-tilt adaptive correcting system,” Comput. Phys. Commun. 150, 274–292 (2003).
[Crossref]

2000 (1)

1990 (2)

N. A. Roddier, “Atmospheric wavefront simulation using zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

W. Jiang, N. Ling, X. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1990).
[Crossref]

Adler, J.

Antonello, J.

Ao, M.

Booth, M. J.

Carhart, G. W.

Cauwenberghs, G.

Cohen, M.

Geng, C.

Gerritsen, H. C.

Goodman, J.

J. Goodman, Fourier Optics (Roberts & Company, 2005).

Huang, L.-H.

L.-H. Huang, “Coherent beam combination using a general model-based method,” Chinese Phys. Lett. 31, 094205 (2014).
[Crossref]

Jia, J.

Jiang, W.

Keller, C. U.

Li, X.

Ling, N.

W. Jiang, N. Ling, X. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1990).
[Crossref]

Linhai, H.

Lipson, S. G.

Liu, H.

Liu, Y.

Luo, W.

Mu, J.

Rao, C.

Rao, X.

W. Jiang, N. Ling, X. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1990).
[Crossref]

Ribak, E. N.

Roddier, N. A.

N. A. Roddier, “Atmospheric wavefront simulation using zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

Saridakis, Y.

M. Zakynthinaki and Y. Saridakis, “Stochastic optimization for a tip-tilt adaptive correcting system,” Comput. Phys. Commun. 150, 274–292 (2003).
[Crossref]

Shi, F.

W. Jiang, N. Ling, X. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1990).
[Crossref]

Tan, Y.

Truong, H. H.

van Werkhoven, T.

Verhaegen, M.

Vorontsov, M. A.

Wang, M.

Wu, J.

H. Yang, Z. Zhang, and J. Wu, “Performance comparison of wavefront-sensorless adaptive optics systems by using of the focal plane,” International Journal of Optics, 1–8 (2015).
[Crossref]

Xu, B.

Yang, H.

H. Yang, Z. Zhang, and J. Wu, “Performance comparison of wavefront-sensorless adaptive optics systems by using of the focal plane,” International Journal of Optics, 1–8 (2015).
[Crossref]

Yang, P.

Yang, Q.

Zakynthinaki, M.

M. Zakynthinaki and Y. Saridakis, “Stochastic optimization for a tip-tilt adaptive correcting system,” Comput. Phys. Commun. 150, 274–292 (2003).
[Crossref]

Zhang, Z.

H. Yang, Z. Zhang, and J. Wu, “Performance comparison of wavefront-sensorless adaptive optics systems by using of the focal plane,” International Journal of Optics, 1–8 (2015).
[Crossref]

Zhao, J.

Zommer, S.

Chinese Phys. Lett. (1)

L.-H. Huang, “Coherent beam combination using a general model-based method,” Chinese Phys. Lett. 31, 094205 (2014).
[Crossref]

Comput. Phys. Commun. (1)

M. Zakynthinaki and Y. Saridakis, “Stochastic optimization for a tip-tilt adaptive correcting system,” Comput. Phys. Commun. 150, 274–292 (2003).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

N. A. Roddier, “Atmospheric wavefront simulation using zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Proc. SPIE (1)

W. Jiang, N. Ling, X. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1990).
[Crossref]

Other (2)

J. Goodman, Fourier Optics (Roberts & Company, 2005).

H. Yang, Z. Zhang, and J. Wu, “Performance comparison of wavefront-sensorless adaptive optics systems by using of the focal plane,” International Journal of Optics, 1–8 (2015).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Dependence of the second moment of the point source on the averaged square of the phase gradient, plotted for the different angular coordinates of the point source. In the simulation, the CCD size was 1024×1024 pixels, the 500 turbulent phase realisation has been used as described in Section 2.4. The position of the point source was chosen to be in the centre of the image (bottom plot), and sequentially shifted 5,10,…,50 % of the image size towards the image edge (bottom to top). The plots are artificially separated for better visibility. The green lines have slope of 1 4 π 2 . Please note the different behaviour for the two upper plots (corresponding to the point sources located 51 pixel from the image edge and on the image edge).
Fig. 2
Fig. 2 Extended objects to be imaged and their diffraction-limited images
Fig. 3
Fig. 3 Two examples of the wavefront aberrations, PSF and corresponding images. The wavefront scale is in λ; the PSF are normalized according to the ideal PSF of the imaging system.
Fig. 4
Fig. 4 MDSE plotted as a function of MSG for Satellite and Whirlpool. The green lines have slope of 1 4 π 2 .
Fig. 5
Fig. 5 Block diagram of WFSless AO system for extended sources
Fig. 6
Fig. 6 Averaged adaptation process over 200 different phase screens under different turbulence strengths for Satellite.
Fig. 7
Fig. 7 Averaged adaptation process over 200 different phase screens under different turbulence strengths for the Whirlpool.
Fig. 8
Fig. 8 Comparisons of initial RMS, fitted RMS and corrected RMS under different turbulences.
Fig. 9
Fig. 9 Correction effect of wavefront and imaging for D/r0 = 5
Fig. 10
Fig. 10 Correction effect of wavefront and imaging for D/r0 = 15
Fig. 11
Fig. 11 Correction effect of wavefront and imaging for D/r0 = 25

Equations (35)

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I ( u ) = | ( a P ( x ) e i φ ( x ) ) | 2 | ( A ( x ) ) | 2 ,
( f g ) = ( ( f ) ) * ( g ) ,
( f x ( x ) ) = 2 π i u ( f ( x ) ) ,
I ( u ) u 2 = | ( A ( x ) ) | 2 u 2 = ( ( A ( x ) ) u ) * ( ( A ( x ) ) u ) = 1 4 π 2 ( ( A x ( x ) ) ) * ( A x ( x ) ) = 1 4 π 2 ( A x ( x ) A x ( x ) ) .
2 ( f ) ( u ) d u = f ( 0 ) ,
2 I ( u ) u 2 d u = 1 4 π 2 2 ( A x ( x ) A x ( x ) ) d u = 1 4 π 2 ( A x A x ) ( 0 ) = 1 4 π 2 2 A * x ( x ) A x ( x + 0 ) d x = 1 4 π 2 2 | A x ( x ) | 2 d x ,
| A x ( x ) | 2 = | a P x ( x ) e i φ ( x ) i φ x ( x ) a P ( x ) e i φ ( x ) | 2 = | A ( x ) ( P x ( x ) P ( x ) i φ x ( x ) ) | 2 = | A ( x ) | 2 | P x ( x ) P ( x ) i φ x ( x ) | 2 = a 2 P 2 ( x ) ( ( P x ( x ) P ( x ) ) 2 + ( φ x ( x ) ) 2 ) = a 2 ( ( P x ( x ) ) 2 + ( P ( x ) φ x ( x ) ) 2 )
2 I ( u ) u 2 d u = a 2 4 π 2 2 ( ( P x ( x ) ) 2 + ( P ( x ) φ x ( x ) ) 2 ) d x .
2 I ( u ) u 2 d u = 2 I 0 ( u ) u 2 d u + a 2 4 π 2 2 ( P ( x ) φ x ( x ) ) 2 d x ,
2 ( I ( u ) I 0 ( u ) ) u 2 d u = a 2 4 π 2 2 ( P ( x ) φ x ( x ) ) 2 d x ,
2 ( I ( u ) I 0 ( u ) ) | u | 2 d u = a 2 4 π 2 2 P 2 ( x ) ( ( φ x ( x ) ) 2 + ( φ y ( x ) ) 2 ) d x , a 2 4 π 2 2 P 2 ( x ) | φ | 2 d x .
2 I ( u ) d u = 2 I 0 ( u ) d u = a 2 2 P 2 ( x ) d x ,
2 ( I ( u ) I 0 ( u ) ) | u | 2 d u 2 I ( u ) d u = 1 4 π 2 2 P 2 ( x ) | φ | 2 d x 2 P 2 ( x ) d x .
2 ( I ( u ) I 0 ( u ) ) u d u 2 I ( u ) d u = 1 2 π 2 P 2 ( x ) φ d x 2 P 2 ( x ) d x ,
m ( u ) = { ( 1 | u | 2 / R 2 ) if | u | R 0 if | u | > R ,
| u | R I ( u ) d u | u | R I 0 ( u ) d u a 2 2 P 2 ( x ) d x ,
MDS = | u | R I ( u ) ( 1 | u | 2 R 2 ) d u | u | R I ( u ) d u
MDS = 1 1 R 2 a 2 2 P 2 ( x ) d x | u | R I ( u ) | u | 2 d u 1 1 R 2 a 2 2 P 2 ( x ) d x ( | u | R I 0 ( u ) | u | 2 d u + a 2 4 π 2 2 ( P ( x ) φ x ( x ) ) 2 d x ) 1 | u | R I 0 ( u ) | u | 2 d u R 2 a 2 2 P 2 ( x ) d x 1 4 π 2 R 2 2 P 2 ( x ) d x 2 ( P ( x ) φ x ( x ) ) 2 d x = MDS 0 c 0 ( R ) 2 ( P ( x ) φ x ( x ) ) 2 d x ,
I 0 , ξ ( u ) = | ( a P ( x ) e i k ξ x ) | 2 ,
I 0 , ξ ( u ) = I 0 ( u ξ ) .
2 I 0 , ξ ( u ) | u | 2 d u = 2 I 0 ( u ) | u | 2 d u + a 2 4 π 2 2 ( P ( x ) k | ξ | ) ) 2 d x
2 φ x ( x ) P 2 ( x ) d x = 2 φ y ( x ) P 2 ( x ) d x = 0.
I ξ ( u ) = I ( u ξ ) | ( a P ( x ) e i ( φ ( x ) + k ξ x ) ) | 2 .
2 I ξ ( u ) | u | 2 d u = 2 I 0 ( u ) | u | 2 d u + a 2 4 π 2 2 ( P ( x ) ( φ ( x ) + k | ξ | ) ) 2 d x = 2 I 0 ( u ) | u | 2 d u + a 2 k 2 4 π 2 2 P 2 ( x ) | ξ | 2 d x + 2 a 2 k 4 π 2 2 P 2 ( x ) | ξ | φ ( x ) d x + a 2 4 π 2 2 ( P ( x ) φ ( x ) ) 2 d x = 2 I 0 , ξ ( u ) | u | 2 d u + 0 + a 2 4 π 2 2 ( P ( x ) φ ( x ) ) 2 d x ,
2 ( I ξ ( u ) I 0 , ξ ( u ) ) | u | 2 d u = a 2 4 π 2 2 P 2 ( x ) | φ | 2 d x .
i ( 0 ) ( u ) = 2 I ( 0 , ) ξ d ξ ,
2 ( i ( u ) i 0 ( u ) ) | u | 2 d u = 2 2 ( I ξ ( u ) I 0 , ξ ( u ) ) | u | 2 d u d ξ = 1 4 π 2 2 o ( ξ ) d ξ 2 ( P ( x ) | φ | ) 2 d x ,
2 i ( u ) du = 2 i 0 ( u ) d u = 2 o ( ξ ) 2 d ξ 2 P 2 ( x ) dx ,
2 ( i ( u ) i 0 ( u ) ) | u | 2 d u 2 i ( u ) d u = 1 4 π 2 2 ( P ( x ) φ x ( x ) ) 2 d x 2 P 2 ( x ) d x ,
MDSE = 2 i ( u ) | u | 2 d u 2 i ( u ) d u ,
MDSE MDSE 0 = c 0 MSG ,
E k ( r ) = E k ( x , y ) = e ln ω ( ( x x k ) 2 + ( y y k ) 2 / d ) α ,
S ( i , j ) = D 1 D ( x E i ( x , y ) x E j ( x , y ) + y E i ( x , y ) y E j ( x , y ) ) d x d y .
ν = S 1 M 2 c 0 β β S 1 S m 2 ,
M = ( MDSE 1 MDSE i n i t MDSE 2 MDSE i n i t MDSE k MDSE i n i t ) ,

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