Abstract

We propose a spline-based aberration reconstruction method through moment measurements (SABRE-M). The method uses first and second moment information from the focal spots of the SH sensor to reconstruct the wavefront with bivariate simplex B-spline basis functions. The proposed method, since it provides higher order local wavefront estimates with quadratic and cubic basis functions can provide the same accuracy for SH arrays with a reduced number of subapertures and, correspondingly, larger lenses which can be beneficial for application in low light conditions. In numerical experiments the performance of SABRE-M is compared to that of the first moment method SABRE for aberrations of different spatial orders and for different sizes of the SH array. The results show that SABRE-M is superior to SABRE, in particular for the higher order aberrations and that SABRE-M can give equal performance as SABRE on a SH grid of halved sampling.

© 2017 Optical Society of America

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References

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2016 (2)

C. C. de Visser, E. Brunner, and M. Verhaegen, “On distributed wavefront reconstruction for large-scale adaptive optics systems,” J. Opt. Soc. Am. A 33, 817–831 (2016).
[Crossref]

H. J. Tol, C. C. de Visser, and M. Kotsonis, “Model reduction of parabolic PDEs using multivariate splines,” Int. J. Control 7179, 1–16 (2016).
[Crossref]

2015 (1)

2013 (1)

2012 (1)

2011 (2)

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[Crossref]

H. Linhai and C. Rao, “Wavefront sensorless adaptive optics: a general model-based approach,” Opt. Express 19, 371–379 (2011).
[Crossref] [PubMed]

2010 (1)

2007 (1)

2006 (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

2002 (1)

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[Crossref] [PubMed]

1998 (1)

1996 (1)

1980 (1)

1977 (1)

1976 (1)

Artal, P.

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[Crossref] [PubMed]

Azucena, O.

Benito, A.

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[Crossref] [PubMed]

Bjorck, A.

A. Bjorck, Numerical Methods for Least Squares Problems (SIAM, 1996).
[Crossref]

Booth, M. J.

Brunner, E.

Castejón-Mochón, J. F.

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[Crossref] [PubMed]

Chen, D. C.

Chu, Q. P.

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[Crossref]

Crest, J.

Dai, G.-m.

Dainty, C.

de Visser, C. C.

H. J. Tol, C. C. de Visser, and M. Kotsonis, “Model reduction of parabolic PDEs using multivariate splines,” Int. J. Control 7179, 1–16 (2016).
[Crossref]

C. C. de Visser, E. Brunner, and M. Verhaegen, “On distributed wavefront reconstruction for large-scale adaptive optics systems,” J. Opt. Soc. Am. A 33, 817–831 (2016).
[Crossref]

C. C. de Visser and M. Verhaegen, “Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines,” J. Opt. Soc. Am. A 30, 82 (2013).
[Crossref]

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[Crossref]

Fried, D. L.

Fusco, T.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Kotadia, S.

Kotsonis, M.

H. J. Tol, C. C. de Visser, and M. Kotsonis, “Model reduction of parabolic PDEs using multivariate splines,” Int. J. Control 7179, 1–16 (2016).
[Crossref]

Kubby, J.

Lai, M. J.

M. J. Lai and L. L. Schumaker, Spline Functions on Triangulations (Camebridge University, 2007).
[Crossref]

Leroux, C.

Linhai, H.

López-Gil, N.

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[Crossref] [PubMed]

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Mulder, J. A.

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[Crossref]

Nicolle, M.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Noll, R. J.

Rao, C.

Robert,

K. Tyson and Robert, Principles of Adaptive Optics (Academic, Inc, 1991).

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Camebridge University, 1999).
[Crossref]

Roggemann, M. C.

Rousset, G.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Schulz, T. J.

Schumaker, L. L.

M. J. Lai and L. L. Schumaker, Spline Functions on Triangulations (Camebridge University, 2007).
[Crossref]

Soloviev, O.

Southwell, W.

Sullivan, W.

Tao, X.

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Tol, H. J.

H. J. Tol, C. C. de Visser, and M. Kotsonis, “Model reduction of parabolic PDEs using multivariate splines,” Int. J. Control 7179, 1–16 (2016).
[Crossref]

Tyson, K.

K. Tyson and Robert, Principles of Adaptive Optics (Academic, Inc, 1991).

Verhaegen, M.

Yang, H.

Appl. Opt. (1)

Automatica (1)

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[Crossref]

Int. J. Control (1)

H. J. Tol, C. C. de Visser, and M. Kotsonis, “Model reduction of parabolic PDEs using multivariate splines,” Int. J. Control 7179, 1–16 (2016).
[Crossref]

J. Opt. Soc. Am. (3)

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

Mon. Not. Roy. Astron. Soc. (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. Roy. Astron. Soc. 371, 323–336 (2006).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Vision Res. (1)

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[Crossref] [PubMed]

Other (4)

K. Tyson and Robert, Principles of Adaptive Optics (Academic, Inc, 1991).

M. J. Lai and L. L. Schumaker, Spline Functions on Triangulations (Camebridge University, 2007).
[Crossref]

F. Roddier, Adaptive Optics in Astronomy (Camebridge University, 1999).
[Crossref]

A. Bjorck, Numerical Methods for Least Squares Problems (SIAM, 1996).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Type-II triangulations (black lines) for first moment (left) and second moment (right) measurements of a 3 × 3 SH array (blue lines) for the SABRE and the moment-based SABRE.

Fig. 2
Fig. 2

Performance of the SABRE and SABRE-M on a 10 × 10 SH array, using a second degree spline, for different order Zernike input aberrations. The best fit of the aberration shows the maximal performance that can be obtained with a second degree spline.

Fig. 3
Fig. 3

Turbulent phase screens from Zernike modes of different polynomial order n and of different aberration strength D/r0, and corresponding SH patterns. (a) and (b): weak aberration with RMS ≈ 3 rad; (c) and (d): strong aberration with RMS ≈ 20 rad.

Fig. 4
Fig. 4

Performance of the SABRE and SABRE-M on a 10 × 10 SH array, using respectively a spline model of second and third degree. Stochastic results are shown, for input aberrations with Kolmogorov statistics created using Zernike modes up to different orders. The best fit of the aberration shows the maximal performance that can be obtained with a second, respectively third, degree spline.

Fig. 5
Fig. 5

Cross-section of Fig. 4, showing the performance of SABRE and SABRE-M on a 10 × 10 SH array, for Kolmogorov models including 25 orders of Zernike modes, at different aberration strengths D/r0. Both methods show to be nearly independent of aberration strength.

Fig. 6
Fig. 6

Performance of the SABRE and SABRE-M for a 10 × 10, 15 × 15 and 20 × 20 SH array. Results are obtained for phase screens including up to 25 orders of Zernike modes with aberration strength D/r0 = 40.

Fig. 7
Fig. 7

Performance of the SABRE and SABRE-M on a 10 × 10 SH array at different SNR levels. The dashed line shows the performance for the noiseless case. Results are obtained for phase screens including Zernike modes of polynomial order up to 25 with aberration strength D/r0 = 40.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

I x ( n ) ϕ x ( x n ) , n = 1 , , N ,
p ( b ( x ) ) = { | κ | = d c κ t B κ d ( b ( x ) ) , x t 0 , x t ,
ϕ ( x ) s d r ( b ( x ) ) = B d ( b ( x ) ) c , x T ,
Ac = 0
I x 2 : = 2 ( I ( u ) I 0 ( u ) ) u 2 d u 2 I ( u ) d u = 1 4 π 2 2 P 2 ( x ) ϕ x 2 ( x ) d x 2 P 2 ( x ) d x ,
I x ( n ) = c 1 P n ( x ) ϕ x ( x ) d x , n = 1 , 2 , , N ,
I x 2 ( n ) = c 2 P n ( x ) ϕ x 2 ( x ) d x I x ( n ) 2 , n = 1 , 2 , , N ,
I x y ( n ) = c 2 P n ( x ) ϕ x ( x ) ϕ y ( x ) d x I x ( n ) I y ( n ) , n = 1 , 2 , , N .
D e 1 p ( b ( x ) ) = d ! ( d 1 ) ! B t d 1 ( b ( x ) ) P d 1 , d ( a e ) c t ,
t B κ d ( b ( x ) ) d x = A t d ^ ,
t B κ d 1 ( b ( x ) ) B γ d 2 ( b ( x ) ) d x = d 1 ! d 2 ! ( d 1 + d 2 ) ! ( γ + κ ) ! γ ! κ ! A t d 1 + d 2 ^ ,
ϕ n ( x ) B n d ( b ( x ) ) c n , x T n ,
t ϕ x ( x ) d x = t d ! ( d 1 ) ! B t d 1 ( b ( x ) ) P d 1 , d ( a x ) c t d x
= d ! ( d 1 ) ! A t d 1 ^ 𝟙 P d 1 , d ( a x ) c t
= 2 A t d + 1 𝟙 P d 1 , d ( a x ) c t ,
I x ( n ) = 1 2 π 2 J n ( d + 1 ) j = 1 J n 𝟙 P j d 1 , d ( a x ) c t j ,
I x ( n ) = 1 2 π 2 J n ( d + 1 ) I 1 , n ( a x ) c n + η x ( x ) , I y ( n ) = 1 2 π 2 J n ( d + 1 ) I 1 , n ( a y ) c n + η y ( x ) ,
t ϕ x 2 ( x ) d x = t ( d ! ( d 1 ) ! B d 1 ( b ( x ) ) P d 1 , d ( a x ) c t ) 2 d x = ( d ! ( d 1 ) ! ) 2 c t P d 1 , d ( a x ) ( t B d 1 ( b ( x ) ) B d 1 ( b ( x ) ) d x ) P d 1 , d ( a x ) c t = ( d ! ( d 1 ) ! ) 2 c t P d 1 , d ( a x ) A t 2 ( d 1 ^ ) B I γ , κ d 1 P d 1 , d ( a x ) c t = d A t ( 2 d 1 ) c t P d 1 , d ( a x ) B I γ , κ d 1 P d 1 , d ( a x ) c t .
I x 2 ( n ) = 1 4 π 2 ( d J n ( 2 d 1 ) j = 1 J n c t P j d 1 , d ( a x ) B I γ , κ d 1 P j d 1 , d ( a x ) c t ( 2 J n ( d + 1 ) j = 1 J n 𝟙 P j d 1 , d ( a x ) c t ) 2 ) .
I x 2 ( n ) = 1 4 π 2 c n   ( d J n ( 2 d 1 ) I 2 , n ( a x , x ) 4 J n 2 ( d + 1 ) 2 I 1 , n   ( a x ) I 1 , n ( a x ) ) c n + η x 2 ( x ) , I y 2 ( n ) = 1 4 π 2 c n   ( d J n ( 2 d 1 ) I 2 , n ( a y , y ) 4 J n 2 ( d + 1 ) 2 I 1 , n   ( a y ) I 1 , n ( a y ) ) c n + η y 2 ( x ) , I x y ( n ) = 1 4 π 2 c n   ( d J n ( 2 d 1 ) I 2 , n ( a x , y ) 4 J n 2 ( d + 1 ) 2 I 1 , n   ( a x ) I 1 , n ( a y ) ) c n + η x y ( x ) ,
b n : = [ I x ( n ) , I y ( n ) , I x 2 ( n ) , I y 2 ( n ) , I x y ( n ) ] 5 × 1
[ r n ( c n ) ] m : = b n , m ( l n , m T c n + c n T Q n , m c n ) , m = 1 , , 5 ,
min c J d ^ f ( c ) = n = 1 N f n ( c n ) = 1 2 n = 1 N r n ( c n ) 2 2
s . t . Ac = 0 ,
i n = | e i k ( u 2 + v 2 ) / ( 2 f ) i λ f ( P n e i ϕ n ) δ 2 | 2 M CCD × M CCD ,
R M S E r e l = ϕ ϕ ^ ϕ ,

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