Abstract

Extremely Large Telescopes (ELTs) are very challenging with respect to their adaptive optics (AO) requirements. Their diameters and the specifications required by the astronomical science for which they are being designed imply a huge increment in the number of degrees of freedom in the deformable mirrors. Faster algorithms are needed to implement the real-time reconstruction and control in AO at the required speed. We present the results of a study of the AO correction performance of three different algorithms applied to the case of a 42-m ELT: one considered as a reference, the matrix-vector multiply (MVM) algorithm; and two considered fast, the fractal iterative method (FrIM) and the Fourier transform reconstructor (FTR). The MVM and the FrIM both provide a maximum a posteriori estimation, while the FTR provides a least-squares one. The algorithms are tested on the European Southern Observatory (ESO) end-to-end simulator, OCTOPUS. The performance is compared using a natural guide star single-conjugate adaptive optics configuration. The results demonstrate that the methods have similar performance in a large variety of simulated conditions. However, with respect to system misregistrations, the fast algorithms demonstrate an interesting robustness.

© 2010 Optical Society of America

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References

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2010

2009

C. Béchet, M. Tallon, and E. Thiébaut, “Comparison of minimum-norm maximum likelihood and maximum a posteriori wavefront reconstructions for large adaptive optics systems,” J. Opt. Soc. Am. A 26, 497–508 (2009).
[CrossRef]

C. Béchet, M. Tallon, and M. L. Louarn, “Very low flux adaptive optics using spatial and temporal priors,” presented at the Adaptive Optics for Extremely Large Telescopes Conference, Paris, France, 22–26 June 2009.

2008

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope Project,” Proc. SPIE 7012, 70121A-18 (2008).
[CrossRef]

M. Johns, “Progress on the GMT,” Proc. SPIE 7012, 70121B-15 (2008).
[CrossRef]

C. Béchet, M. L. Louarn, M. Tallon, and E. Thiébaut, “Performances of the fractal iterative method with an internal model control law on the ESO end-to-end ELT adaptive optics simulator,” Proc. SPIE 7015, 70151H-9 (2008).
[CrossRef]

2007

2006

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a FRactal Iterative Method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

2005

2004

S. M. Tulloch, “Application of L3 technology to wavefront sensing,” Proc. SPIE 5490, 1167–1176 (2004).
[CrossRef]

2003

L. A. Poyneer, “Advanced techniques for Fourier transform wavefront reconstruction,” Proc. SPIE 4839, 1023–1034 (2003).
[CrossRef]

2002

2000

E. Chu and A. George, Inside the FFT Black Box, Serial and Parallel Fast Fourier Transform Algorithms (CRC Press, 2000).

1999

F. Roddier, Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
[CrossRef]

1996

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

1994

1989

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

1986

1979

1977

Béchet, C.

C. Béchet, M. Tallon, and E. Thiébaut, “Comparison of minimum-norm maximum likelihood and maximum a posteriori wavefront reconstructions for large adaptive optics systems,” J. Opt. Soc. Am. A 26, 497–508 (2009).
[CrossRef]

C. Béchet, M. Tallon, and M. L. Louarn, “Very low flux adaptive optics using spatial and temporal priors,” presented at the Adaptive Optics for Extremely Large Telescopes Conference, Paris, France, 22–26 June 2009.

C. Béchet, M. L. Louarn, M. Tallon, and E. Thiébaut, “Performances of the fractal iterative method with an internal model control law on the ESO end-to-end ELT adaptive optics simulator,” Proc. SPIE 7015, 70151H-9 (2008).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a FRactal Iterative Method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Brase, J.

Chu, E.

E. Chu and A. George, Inside the FFT Black Box, Serial and Parallel Fast Fourier Transform Algorithms (CRC Press, 2000).

Freischlad, K. R.

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216231 (2005).
[CrossRef]

Gavel, D.

George, A.

E. Chu and A. George, Inside the FFT Black Box, Serial and Parallel Fast Fourier Transform Algorithms (CRC Press, 2000).

Gilles, L.

Gilmozzi, R.

R. Gilmozzi and J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Hudgin, R. H.

Hunt, B. R.

Johns, M.

M. Johns, “Progress on the GMT,” Proc. SPIE 7012, 70121B-15 (2008).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216231 (2005).
[CrossRef]

Koliopoulos, C. L.

Louarn, M. L.

C. Béchet, M. Tallon, and M. L. Louarn, “Very low flux adaptive optics using spatial and temporal priors,” presented at the Adaptive Optics for Extremely Large Telescopes Conference, Paris, France, 22–26 June 2009.

C. Béchet, M. L. Louarn, M. Tallon, and E. Thiébaut, “Performances of the fractal iterative method with an internal model control law on the ESO end-to-end ELT adaptive optics simulator,” Proc. SPIE 7015, 70151H-9 (2008).
[CrossRef]

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

Macintosh, B. A.

Marichal-Hernández, J. G.

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

Montilla, I.

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

Morari, M.

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

Nelson, J.

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope Project,” Proc. SPIE 7012, 70121A-18 (2008).
[CrossRef]

Parenti, R. R.

Poyneer, L.

Poyneer, L. A.

L. A. Poyneer, “Advanced techniques for Fourier transform wavefront reconstruction,” Proc. SPIE 4839, 1023–1034 (2003).
[CrossRef]

Reyes, M.

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
[CrossRef]

Rodríguez-Ramos, J. M.

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

Rodríguez-Ramos, L. F.

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Sanders, G. H.

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope Project,” Proc. SPIE 7012, 70121A-18 (2008).
[CrossRef]

Sasiela, R. J.

Spyromilio, J.

R. Gilmozzi and J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Tallon, M.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

C. Béchet, M. Tallon, and M. L. Louarn, “Very low flux adaptive optics using spatial and temporal priors,” presented at the Adaptive Optics for Extremely Large Telescopes Conference, Paris, France, 22–26 June 2009.

C. Béchet, M. Tallon, and E. Thiébaut, “Comparison of minimum-norm maximum likelihood and maximum a posteriori wavefront reconstructions for large adaptive optics systems,” J. Opt. Soc. Am. A 26, 497–508 (2009).
[CrossRef]

C. Béchet, M. L. Louarn, M. Tallon, and E. Thiébaut, “Performances of the fractal iterative method with an internal model control law on the ESO end-to-end ELT adaptive optics simulator,” Proc. SPIE 7015, 70151H-9 (2008).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a FRactal Iterative Method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Thiébaut, E.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “Comparison of minimum-norm maximum likelihood and maximum a posteriori wavefront reconstructions for large adaptive optics systems,” J. Opt. Soc. Am. A 26, 497–508 (2009).
[CrossRef]

C. Béchet, M. L. Louarn, M. Tallon, and E. Thiébaut, “Performances of the fractal iterative method with an internal model control law on the ESO end-to-end ELT adaptive optics simulator,” Proc. SPIE 7015, 70151H-9 (2008).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a FRactal Iterative Method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Tulloch, S. M.

S. M. Tulloch, “Application of L3 technology to wavefront sensing,” Proc. SPIE 5490, 1167–1176 (2004).
[CrossRef]

Veran, J.-P.

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Zafiriou, E.

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. IEEE

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216231 (2005).
[CrossRef]

Proc. SPIE

C. Béchet, M. L. Louarn, M. Tallon, and E. Thiébaut, “Performances of the fractal iterative method with an internal model control law on the ESO end-to-end ELT adaptive optics simulator,” Proc. SPIE 7015, 70151H-9 (2008).
[CrossRef]

I. Montilla, M. Reyes, M. L. Louarn, J. G. Marichal-Hernández, J. M. Rodríguez-Ramos, and L. F. Rodríguez-Ramos, “Performance of the Fourier Transform Reconstructor for the European Extremely Large Telescope,” Proc. SPIE 7015, 70152Y-10 (2008).
[CrossRef]

L. A. Poyneer, “Advanced techniques for Fourier transform wavefront reconstruction,” Proc. SPIE 4839, 1023–1034 (2003).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope Project,” Proc. SPIE 7012, 70121A-18 (2008).
[CrossRef]

M. Johns, “Progress on the GMT,” Proc. SPIE 7012, 70121B-15 (2008).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a FRactal Iterative Method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

S. M. Tulloch, “Application of L3 technology to wavefront sensing,” Proc. SPIE 5490, 1167–1176 (2004).
[CrossRef]

Other

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

F. Roddier, Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
[CrossRef]

C. Béchet, M. Tallon, and M. L. Louarn, “Very low flux adaptive optics using spatial and temporal priors,” presented at the Adaptive Optics for Extremely Large Telescopes Conference, Paris, France, 22–26 June 2009.

E. Chu and A. George, Inside the FFT Black Box, Serial and Parallel Fast Fourier Transform Algorithms (CRC Press, 2000).

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Figures (9)

Fig. 1
Fig. 1

Short-exposure Strehl versus iteration for a flux of 1000 ph . subaperture 1 . frame 1 .

Fig. 2
Fig. 2

Short-exposure Strehl versus iteration for a flux of 1 ph . subaperture 1 . frame 1 .

Fig. 3
Fig. 3

Long-exposure Strehl versus flux in photons . subaperture 1 . frame 1 .

Fig. 4
Fig. 4

Long-exposure Strehl versus r 0 .

Fig. 5
Fig. 5

Long-exposure Strehl versus τ 0 .

Fig. 6
Fig. 6

Case when the actuators are shifted from the subapertures’ corners. The actuators are the circles, and the grid represents the subapertures’ positions.

Fig. 7
Fig. 7

Comparison of short-exposure Strehl versus iteration number as a function of the actuators’ position misregistrations: top-left, 12.5% of a subaperture shift; top-right, 25% of a subaperture shift; lower-left, 37.5% of a subaperture shift; lower-right, 50% of a subaperture shift.

Fig. 8
Fig. 8

Long-exposure Strehl reached by the MVM method after 2000 loops, depending on the misregistration as percentage of the subaperture size. The different curves are obtained with different regularization strength and different number of corrected modes. Top curve, weak regularization and 5402 corrected modes; middle curve, weak regularization and only 3000 modes corrected; bottom curve, a stronger regularization and only 3000 modes corrected.

Fig. 9
Fig. 9

Approximate number of operations versus subapertures for the three algorithms under study.

Tables (4)

Tables Icon

Table 1 Configuration Parameters

Tables Icon

Table 2 Strehl Variations for Conditions r 0 = 13 cm and 1000  photons . subaperture 1 . frame 1

Tables Icon

Table 3 Strehl Variations for Conditions r 0 = 5 cm and 1000  photons . subaperture 1 . frame 1

Tables Icon

Table 4 Strehl Variations for Conditions r 0 = 13 cm and 1  photons . subaperture 1 . frame 1

Equations (22)

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

ϵ 2 = d x W ( x ) [ K i ( x ) j = 1 n c i ( j ) m j ( x ) ] 2 ,
c i = Γ 1 p i ,
Γ k l = d x W ( x ) m k ( x ) m l ( x ) , 1 k n , 1 l n ,
p i ( k ) = d x W ( x ) K i ( x ) m k ( x ) .
R MAP = ( A mod t C n 1 A mod + C VK 1 ) 1 A mod t C n 1 ,
C VK K frac K frac T ,
w = K frac u ,
R MAP = ( A frac t C n 1 A frac + I ) 1 A frac t C n 1 ,
( A frac t C n 1 A frac + I ) u ̂ = A frac t C n 1 ( s + G a ) ,
a k = a k 1 + δ ( K frac u ̂ k a k 2 ) ,
s x [ m , n ] = 1 2 ( ϕ [ m , n + 1 ] ϕ [ m , n ] ) + 1 2 ( ϕ [ m + 1 , n + 1 ] ϕ [ m + 1 , n ] ) ,
s y [ m , n ] = 1 2 ( ϕ [ m + 1 , n ] ϕ [ m , n ] ) + 1 2 ( ϕ [ m + 1 , n + 1 ] ϕ [ m , n + 1 ] ) .
ϕ ̂ [ k , l ] = { 0 , if k , l = 0 , N 2 F x [ k , l ] S x [ k , l ] + F y [ k , l ] S y [ k , l ] , otherwise } ,
F x [ k , l ] = exp ( j 2 π k N ) 1 T [ k , l ] ,
F y [ k , l ] = exp ( j 2 π l N ) 1 T [ k , l ] ,
T [ k , l ] = 1 4 [ sin 2 ( π k N ) + sin 2 ( π l N ) ] .
C [ k , l ] = I F T { ϕ ̂ [ k , l ] F T [ I ( m , n ) ] } ,
σ fitting 2 = 0.287 ( d r 0 ) 5 3 .
σ delay 2 = 0.962 ( τ τ 0 ) 5 3 ,
N op MVM 4 ( π 2 ( N 2 ) 4 π N 3 4 ) π ( N 2 ) 2 .
N op FTR N 2 ( 18 π 2 ) + 15 N 2 log 2 ( N ) .
N op FrIM ( 24 n iter + 13 ) n frac + ( 5 n iter + 17 ) π ( N 2 ) 2 ,

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