Abstract

The Cumulative Reconstructor is an accurate, extremely fast reconstruction algorithm for Shack–Hartmann wavefront sensor data. But it has shown an unacceptable high noise propagation for large apertures. Therefore, in this paper we describe a domain decomposition approach to deal with this drawback. We show that this adaptation of the algorithm gives the same reconstruction quality as the original algorithm and leads to a significant improvement with respect to noise propagation. The method is combined with an integral control and compared to the classical matrix vector multiplication algorithm on an end-to-end simulation of a single conjugate adaptive optics system. The reconstruction time is 20n (number of subapertures), and the method is parallelizable.

© 2012 Optical Society of America

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References

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  2. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  3. C. Béchet, M. Tallon, and E. Thiébaut, “Frim: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
    [CrossRef]
  4. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861(1986).
    [CrossRef]
  5. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111(2002).
    [CrossRef]
  6. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier-transform wavefront control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005).
    [CrossRef]
  7. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
    [CrossRef]
  8. L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wavefront reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
    [CrossRef]
  9. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
  10. M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
    [CrossRef]
  11. M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for Extremely Large Telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
    [CrossRef]
  12. D. G. MacMartin, “Local, hierarchic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093(2003).
    [CrossRef]
  13. C. Béchet, M. Tallon, and É. 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]
  14. M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).
  15. M. Le Louarn, “Multi-conjugate adaptive optics with laser guide stars: performance in the infrared and visible,” Mon. Not. R. Astron. Soc. 334, 865–874 (2002).
    [CrossRef]

2011 (2)

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
[CrossRef]

M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for Extremely Large Telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
[CrossRef]

2009 (1)

2008 (1)

2006 (2)

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

C. Béchet, M. Tallon, and E. Thiébaut, “Frim: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

2005 (1)

2003 (1)

2002 (3)

2001 (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

1986 (1)

1979 (1)

Béchet, C.

Brase, J. M.

Ellerbroek, B. L.

Freischlad, K. R.

Fried, D. L.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Gavel, D. T.

Hubin, N.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

Hunt, B. R.

Koliopoulos, C. L.

Korkiakoski, V.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

Lall, S.

Le Louarn, M.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

M. Le Louarn, “Multi-conjugate adaptive optics with laser guide stars: performance in the infrared and visible,” Mon. Not. R. Astron. Soc. 334, 865–874 (2002).
[CrossRef]

Lessard, L.

MacMartin, D. G.

MacMynowski, D.

Marchetti, E.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

Neubauer, A.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
[CrossRef]

Poyneer, L. A.

Ramlau, R.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
[CrossRef]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Rosensteiner, M.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
[CrossRef]

M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for Extremely Large Telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
[CrossRef]

Tallon, M.

Thiébaut, E.

C. Béchet, M. Tallon, and E. Thiébaut, “Frim: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Thiébaut, É.

Véran, J.-P.

Verinaud, C.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

West, M.

Zhariy, M.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
[CrossRef]

Inverse Probl. Imaging (1)

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartmann sensor,” Inverse Probl. Imaging 5, 893–913 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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

M. Le Louarn, “Multi-conjugate adaptive optics with laser guide stars: performance in the infrared and visible,” Mon. Not. R. Astron. Soc. 334, 865–874 (2002).
[CrossRef]

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Proc. SPIE (2)

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, U1048–U1056 (2006).

C. Béchet, M. Tallon, and E. Thiébaut, “Frim: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Other (1)

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

Geometry of a subaperture.

Fig. 2.
Fig. 2.

Possible decomposition of the domain ΩT.

Fig. 3.
Fig. 3.

Residual wavefront error for reconstructing open-loop SH-wfs data (seeing 0.2 m) for annular aperture versus size of telescope, subaperture size fixed d = 0.2 m for different levels of subdivision in the CuReD algorithm.

Fig. 4.
Fig. 4.

Comparison of noise propagation for circular aperture versus size of aperture for several reconstruction methods showing the average amplification of white noise in the measurements.

Fig. 5.
Fig. 5.

Noise propagation for different levels of subdivisions with constant sizes of subdomains.

Fig. 6.
Fig. 6.

Comparison of CuReD versus MVM at OCTOPUS end-to-end simulator for different levels of noise in closed loop, r0=0.129m.

Fig. 7.
Fig. 7.

Reconstruction times for 84×84 Shack–Hartmann wavefront sensor data with CuReD on a multicore machine.

Equations (45)

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sx[i,j]=Γxϕ[i,j]1|Ωij|Ωijϕx(x,y)d(x,y),
sx[i,j]ϕ[i,j12]ϕ[i1,j12],
sy[i,j]ϕ[i12,j]ϕ[i12,j1].
lx[i+1,j12]=lx[i,j12]+sx[i+1,j],
ly[i12,j+1]=ly[i12,j]+sy[i,j+1],
lx0[i,j12]=lx[i,j12]mx[j12],
tx[i]=1Nj=1Nlx0[i,j].
tx[i+12]=12(tx[i]+tx[i+1]).
ϕx[i,j12]=lx0[i,j12]+ty[j12].
ϕ[i,j]=14(ϕ1[i,j]+ϕ2[i,j]+ϕ3[i,j]+ϕ4[i,j]),
ϕ1[i,j]ϕ[i12,j]+14(sx[i,j]+sx[i,j+1]).
ΩT=iΩpi,
ΩpiΩpj=ΩpiΩpj,ij.
d1,2=mean(ϕ1|Ωp1Ωp2ϕ2|Ωp1Ωp2),
ϵ=d1,2+d2,4d1,3d3,4,
ϕ¯2=ϕ2+d1,2,
ϕ¯3=ϕ3+d1,3,
ϕ¯4=ϕ4+d1,2+d2,4.
ϕx^=ϕx,ϕy^=ϕy,ΩTϕ^(x,y)d(x,y)=0,
ϕi^=ϕ|Ωpi+ci,
d1,2=ϕ1^|Ωp1Ωp2ϕ2^|Ωp1Ωp2=ϕ|Ωp1Ωp2+c1ϕ|Ωp1Ωp2+c2=c1c2,
ϕ¯2=ϕ2^+d1,2=ϕ|Ωp2+c1,
ϕ¯3=ϕ3^+d1,3=ϕ|Ωp3+c1,
ϕ¯4=ϕ4^+d1,2+d2,4=ϕ|Ωp4+c1.
sx[i,j]12(ϕ[i,j]+ϕ[i,j1]ϕ[i1,j]ϕ[i1,j1]),
sy[i,j]12(ϕ[i,j]+ϕ[i1,j]ϕ[i,j1]ϕ[i1,j1]),
mse=trace(MMT)n,
mse=0.24+0.63lnn0.119(lnn)2+0.00825(lnn)3.
lx[i+1,j12]=lx[i,j12]+sx[i+1,j]
lx0[i,j12]=lx[i,j12]mx[j12],
mx[j12]=1N(k=1N1lx[k,j12]+12lx[0,j12]+12lx[N,j12]).
tx[i]=1Nj=1Nlx0[i,j],
tx[i+12]=12(tx[i]+tx[i+1]).
ϕx[i,j12]=lx0[i,j12]+ty[j12].
ϕm[a+1,b]ϕm[a,b]=14(ϕm[a+12,b]+ϕm[a+32,b]+ϕm[a+1,b12]+ϕm[a+1,b+12](ϕm[a12,b]+ϕm[a+12,b]+ϕm[a,b12]+ϕm[a,b+12])+R1),
14(ly0[a+32,b]+tx[a+32]ly0[a12,b]tx[a12]+lx0[a+1,b12]+ty[b12]+lx0[a+1,b+12]+ty[b+12]lx0[a,b12]ty[b12]lx0[a,b+12]ty[b+12]+R1).
ly[a+32,b]my[a+23]ly[a12,b]my[a12]+12(tx[a+2]+tx[a+1])12(tx[a]+tx[a1])+lx[a+1,b12]+lx[a+1,b+12]lx[a,b12]lx[a,b+12]+R1.
lx[a+1,b12]+lx[a+1,b+12]lx[a,b12]lx[a,b+12]=sx[a+1,b]+sx[a+1,b+1],
ly[a+32,b]ly[a12,b]1N(12ly[a+32,0]+k=1N1ly[a+32,k]+12ly[a+32,N](12ly[a+12,0]+k=1N1ly[a+12,k]+12ly[a+12,N]))12k=1N(l0x[a+2,k12]+l0x[a+1,k12]l0x[a,k12]l0x[a1,k12])+R2,
l=1b(sy[a+2,l]sy[a,l])12N(2k=1N1l=1k(sy[a+2,l]sy[a,l])+l=1N(sy[a+2,l]sy[a,l])k=1N(sx[a+2,k]+2sx[a+1,k]+sx[a,k]))+R2,
l=1b(ϕ¯[l]ϕ¯[l1])12N(2k=1N1l=1k(ϕ¯[l]ϕ¯[l1])+l=1N(ϕ¯[l]ϕ¯[l1])k=1N(ϕ¯[k]+ϕ¯[k1]))+R2,
(ϕ¯[b]ϕ¯[0])12N(2k=1N1(ϕ¯[k]ϕ¯[0])+(ϕ¯[N]ϕ¯[0])2k=1N1ϕ¯[k]ϕ¯[N]ϕ¯[0])+R2.
ϕ1[i,N2]=ϕ2[i,N2]+c.
14(ϕ1[i12,N2]+14(sx[i,N2]+sx[i,N2+1])+ϕ1[i+12,N2]14(sx[i+1,N2]+sx[i+1,N2+1])+ϕ1[i,N212]+14(sy[i,N2]+sy[i+1,N2])+ϕ1[i,N2+12]14(sy[i,N2+1]+sy[i+1,N2+1]))=14(ϕ2[i12,N2]+14(sx[i,N2]+sx[i,N2+1])+ϕ2[i+12,N2]14(sx[i+1,N2]+sx[i+1,N2+1])+ϕ2[i,12]+14(sy[i,N2]+sy[i+1,N2])+ϕ2[i,12]14(sy[i,N2+1]+sy[i+1,N2+1]))+c.
14(ϕ1[i12,N2]+12(sx[i,N2])+ϕ1[i+12,N2]12(sx[i+1,N2])+2(ϕ1[i,N212]+14(sy[i,N2]+sy[i+1,N2])))=14(ϕ2[i12,0]+12(sx[i,N2+1])+ϕ2[i+12,0]12(sx[i+1,N2+1])+2(ϕ2[i,12]14(sy[i,N2+1]+sy[i+1,N2+1])))+c.

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