Abstract

The use of Fourier methods in wave-front reconstruction can significantly reduce the computation time for large telescopes with a high number of degrees of freedom. However, Fourier algorithms for discrete data require a rectangular data set which conform to specific boundary requirements, whereas wave-front sensor data is typically defined over a circular domain (the telescope pupil). Here we present an iterative Gerchberg routine modified for the purposes of discrete wave-front reconstruction which adapts the measurement data (wave-front sensor slopes) for Fourier analysis, fulfilling the requirements of the fast Fourier transform (FFT) and providing accurate reconstruction. The routine is used in the adaptation step only and can be coupled to any other Wiener-like or least-squares method. We compare simulations using this method with previous Fourier methods and show an increase in performance in terms of Strehl ratio and a reduction in noise propagation for a 40×40 SPHERE-like adaptive optics system. For closed loop operation with minimal iterations the Gerchberg method provides an improvement in Strehl, from 95.4% to 96.9% in K-band. This corresponds to ~ 40 nm improvement in rms, and avoids the high spatial frequency errors present in other methods, providing an increase in contrast towards the edge of the correctable band.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2014 (2)

2008 (1)

C. Correia, C. Kulcsar, J.-M. Conan, and H.-F. Raynaud, “Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in adaptive optics,” Proc. SPIE 7015, 701551 (2008).
[Crossref]

2007 (1)

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” The Computer Journal 52, 80–89 (2007).
[Crossref]

2006 (3)

2005 (1)

2003 (1)

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

2002 (1)

1998 (1)

F. Rigaut, J.-P. Véran, and O. Lai, “An analytical model for Shack-Hartmann based adaptive optics systems,” Proc. SPIE 3353, 1038 (1998).
[Crossref]

1993 (1)

K. R. Freischlad, “Wave-front integration from difference data,” Proc. SPIE 1755, 212 (1993).
[Crossref]

1991 (1)

1986 (2)

K. R. Freischlad and C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 0551, 74 (1986).
[Crossref]

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]

1979 (1)

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Optica Acta 21, 709–720 (1974).
[Crossref]

Baudoz, P.

Baxter, B. S.

Béchet, C.

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Beuzit, J.-L.

Brase, J. M.

Carmon, Y.

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

Charton, J.

Chatterjee, P.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” The Computer Journal 52, 80–89 (2007).
[Crossref]

Chaudhuri, S.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” The Computer Journal 52, 80–89 (2007).
[Crossref]

Conan, J.-M.

C. Correia, C. Kulcsar, J.-M. Conan, and H.-F. Raynaud, “Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in adaptive optics,” Proc. SPIE 7015, 701551 (2008).
[Crossref]

Conan, R.

R. Conan and C. Correia, “Object-oriented Matlab adaptive optics toolbox,” Proc. SPIE 9148, 91486C (2014).
[Crossref]

Correia, C.

R. Conan and C. Correia, “Object-oriented Matlab adaptive optics toolbox,” Proc. SPIE 9148, 91486C (2014).
[Crossref]

C. Correia, C. Kulcsar, J.-M. Conan, and H.-F. Raynaud, “Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in adaptive optics,” Proc. SPIE 7015, 701551 (2008).
[Crossref]

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Correia, C. M.

Dohlen, K.

Freischlad, K. R.

K. R. Freischlad, “Wave-front integration from difference data,” Proc. SPIE 1755, 212 (1993).
[Crossref]

K. R. Freischlad and C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 0551, 74 (1986).
[Crossref]

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]

Frost, R. L.

Fusco, T.

Gavel, D. T.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Optica Acta 21, 709–720 (1974).
[Crossref]

Glazer, O.

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

Hardy, J. W.

J. W. Hardy, “Adaptive Optics for Astronomical Telescopes” (Oxford University Press, 1998).

Kasper, M.

Koliopoulos, C. L.

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]

K. R. Freischlad and C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 0551, 74 (1986).
[Crossref]

Kulcsar, C.

C. Correia, C. Kulcsar, J.-M. Conan, and H.-F. Raynaud, “Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in adaptive optics,” Proc. SPIE 7015, 701551 (2008).
[Crossref]

Lai, O.

F. Rigaut, J.-P. Véran, and O. Lai, “An analytical model for Shack-Hartmann based adaptive optics systems,” Proc. SPIE 3353, 1038 (1998).
[Crossref]

LeLouarn, M.

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Montilla, I.

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Mouillet, D.

Mukherjee, S.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” The Computer Journal 52, 80–89 (2007).
[Crossref]

Nicolle, M.

Niranjan, A.

A. Niranjan, “HARMONI: the first light integral field spectrograph for the E-ELT,” in “Proc. SPIE 9147”, Ground-based and Airborne Instrumentation for Astronomy V, 914725, (2014).

Petit, C.

Poyneer, L. A.

Puget, P.

Raynaud, H.-F.

C. Correia, C. Kulcsar, J.-M. Conan, and H.-F. Raynaud, “Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in adaptive optics,” Proc. SPIE 7015, 701551 (2008).
[Crossref]

Reyes, M.

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Ribak, E. N.

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

Rigaut, F.

F. Rigaut, J.-P. Véran, and O. Lai, “An analytical model for Shack-Hartmann based adaptive optics systems,” Proc. SPIE 3353, 1038 (1998).
[Crossref]

Roddier, C.

Roddier, F.

Rolland, J. P.

Rousset, G.

Rushforth, C. K.

Sauvage, J.-F.

Seetharaman, G.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” The Computer Journal 52, 80–89 (2007).
[Crossref]

Srour, O.

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

Tallon, M.

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Talmi, A.

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

Teixeira, J.

Thiébut, E.

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

Véran, J.-P.

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier-transform wavefront control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005).
[Crossref]

F. Rigaut, J.-P. Véran, and O. Lai, “An analytical model for Shack-Hartmann based adaptive optics systems,” Proc. SPIE 3353, 1038 (1998).
[Crossref]

Zon, N.

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

Zou, W.

Appl. Opt. (2)

J. Opt. Soc. Am. A (5)

Opt. Express (1)

Optica Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Optica Acta 21, 709–720 (1974).
[Crossref]

Proc. SPIE (7)

R. Conan and C. Correia, “Object-oriented Matlab adaptive optics toolbox,” Proc. SPIE 9148, 91486C (2014).
[Crossref]

K. R. Freischlad, “Wave-front integration from difference data,” Proc. SPIE 1755, 212 (1993).
[Crossref]

K. R. Freischlad and C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 0551, 74 (1986).
[Crossref]

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

E. N. Ribak, Y. Carmon, A. Talmi, O. Glazer, O. Srour, and N. Zon, “Full wave front reconstruction in the Fourier domain,” Proc. SPIE 6272, 627254 (2006).
[Crossref]

C. Correia, C. Kulcsar, J.-M. Conan, and H.-F. Raynaud, “Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in adaptive optics,” Proc. SPIE 7015, 701551 (2008).
[Crossref]

F. Rigaut, J.-P. Véran, and O. Lai, “An analytical model for Shack-Hartmann based adaptive optics systems,” Proc. SPIE 3353, 1038 (1998).
[Crossref]

The Computer Journal (1)

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis-Gerchberg method in image super-resolution and inpainting,” The Computer Journal 52, 80–89 (2007).
[Crossref]

Other (6)

T. de Zeeuw, R. Tamai, and J. Liske, “Constructing the E-ELT,” http://www.eso.org/sci/facilities/eelt/docs/ , (2014).

The TMT Observatory Corporation, “Thirty meter telescope construction proposal,” http://www.tmt.org/news/TMT-ConstructionProposal-Public.pdf , (2007).

GMTO Corporation, “Giant magellan telescope scientific promise and opportunities,” http://www.gmto.org/Resources/GMT-SCI-REF-00482_2_GMT_Science_Book.pdf , (2012).

I. Montilla, C. Béchet, M. LeLouarn, C. Correia, M. Tallon, M. Reyes, and E. Thiébut, “Comparison of reconstruction and control algorithms on the ESO end-to-end simulator OCTOPUS,” in “AO4ELT-I”, 03002, (2010).

J. W. Hardy, “Adaptive Optics for Astronomical Telescopes” (Oxford University Press, 1998).

A. Niranjan, “HARMONI: the first light integral field spectrograph for the E-ELT,” in “Proc. SPIE 9147”, Ground-based and Airborne Instrumentation for Astronomy V, 914725, (2014).

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Figures (9)

Fig. 1
Fig. 1 Examples of different extension methods. Left: Original x slopes. Middle: x slopes extended using Hudgin extension method. Right: Extended x slopes using the Gerchberg method with 3 iterations. The examples are shown here before/without the imposition of the periodic condition.
Fig. 2
Fig. 2 Plot illustrating the number of operations required for wave-front reconstruction of an N mode system, both in direct space (N × N) and for possible FFT reconstruction (see Eq. (8)). For FFT reconstruction two cases are shown, one without the Gerchberg extension (niter. = 0) and one with 10 Gerchberg iterations (niter. = 10). The state for current systems and ELTs are highlighted, showing the potential for a 1 – 2 order of magnitude reduction in the number of operations using Fourier methods.
Fig. 3
Fig. 3 Plots of the K-band Strehl ratio versus Gerchberg iteration for open loop end-to-end simulations. The results for 2 different extension windows (n = 3 and n = 6) are shown for 3 extension methods: 1) Gerchberg extension with enforced periodicity; 2) Gerchberg extension without enforced periodicity; and 3) the Hudgin extension.
Fig. 4
Fig. 4 PSDs (power spectral densities) of the turbulent phase and residual phase for different open loop simulations. In all cases the phase is reconstructed in the Fourier domain using different extension methods: 1) no extension; 2) Hudgin extension; 3) Gerchberg extension (3 iterations); and 4) Gerchberg extension with periodicity enforced (3 iterations). In the 3 extension cases (2 – 4) an extension window of n = 3 is used.
Fig. 5
Fig. 5 Error propagation coefficient versus Gerchberg iteration for different extension methods and extension windows (n).
Fig. 6
Fig. 6 Plots of Strehl ratio versus Gerchberg iteration for closed loop simulations. The results shown are for two different extension windows (n) and 3 different extension methods: 1) the Gerchberg extension; 2) the Gerchberg extension with periodicity enforced; and 3) the Hudgin extension.
Fig. 7
Fig. 7 Error propagation coefficient versus Gerchberg iteration for different extension methods and extension windows (n) in end-to-end closed loop simulations.
Fig. 8
Fig. 8 Short exposure Strehl ratio vs. closed loop time step for 3 different extension methods: 1) Hudgin extension; 2) edge correction extension [13]; and 3) Gerchberg extension (2 iterations). In all cases the local and global waffle modes have been filtered out.
Fig. 9
Fig. 9 Closed loop PSFs for Fourier reconstruction using different extension methods. Left: Hudgin extension. Right: Gerchberg extension with niter. = 2. Only the corrected region is shown and the same colour scheme is used for each image.

Tables (2)

Tables Icon

Algorithm 1. Modified Gerchberg algorithm in which slope measurements are extended beyond the aperture. The gradients obtained are periodic over the domain and the rotational is as close to zero as possible on account of the measurement noise.

Tables Icon

Table 1 Summary of simulation parameters used in end-to-end simulations of a full AO loop. The atmospheric properties (the Fried parameter r0 and outer scale L0) are both given, as well as the telescope properties and wavelengths for the wave-front sensing and science star. The parameters correspond to a SPHERE-like system with the omission of the spatial filter before the WFS [18].

Equations (8)

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

s ˜ ( κ ) = G ˜ ϕ ˜ ( κ ) + η ˜
ϕ ^ ˜ = R ˜ x s ˜ x + R ˜ y s ˜ y
R ˜ = G ˜ * | G ˜ | 2 + γ W η W ϕ
G = | | | ( x d ) × [ Π ( x d ) ]
s x ( m , N ) = n = 1 N 1 s x ( m , n ) 1 m N s y ( N , n ) = m = 1 N 1 s y ( m , n ) 1 n N
ϕ ˜ ^ = G ˜ x * s ˜ x + G ˜ y * s ˜ y | G ˜ x | 2 + | G ˜ y | 2
s ˜ ^ = G ˜ ϕ ˜ ^ s ^ = F 1 ( s ˜ ^ )
N op . = n iter . ( 2 N log ( N ) + 3 N ) Gerchberg algorithm + 2 N log ( N ) + N Final reconstruction

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