Abstract

With a view to the next generation of large space telescopes, we investigate guide-star-free, image-based aberration correction using a unimorph deformable mirror in a plane conjugate to the primary mirror. We designed and built a high-resolution imaging testbed to evaluate control algorithms. In this paper we use an algorithm based on the heuristic hill climbing technique and compare the correction in three different domains, namely the voltage domain, the domain of the Zernike modes, and the domain of the singular modes of the deformable mirror. Through our systematic experimental study, we found that successive control in two domains effectively counteracts uncompensated hysteresis of the deformable mirror.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

O. Kazasidis, S. Verpoort, and U. Wittrock, “Algorithm design for image-based wavefront control without wavefront sensing,” Proc. SPIE 10695, 1069502 (2018).

L. Bliek, H. R. G. W. Verstraete, M. Verhaegen, and S. Wahls, “Online Optimization With Costly and Noisy Measurements Using Random Fourier Expansions,” IEEE Transactions on Neural Networks and Learning Systems 29, 167–182 (2018).
[Crossref]

2017 (1)

2016 (5)

J. T. Trauger, D. C. Moody, J. E. Krist, and B. L. Gordon, “Hybrid Lyot coronagraph for WFIRST-AFTA: coronagraph design and performance metrics,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011013 (2016).
[Crossref]

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

D. J. Wahl, Y. Jian, S. Bonora, R. J. Zawadzki, and M. V. Sarunic, “Wavefront sensorless adaptive optics fluorescence biomicroscope for in vivo retinal imaging in mice,” Biomed. Opt. Express 7, 1–12 (2016).
[Crossref] [PubMed]

P. Rausch, S. Verpoort, and U. Wittrock, “Unimorph deformable mirror for space telescopes: environmental testing,” Opt. Express 24, 1528–1542 (2016).
[Crossref] [PubMed]

T. E. Agbana, H. Yang, O. Soloviev, G. Vdovin, and M. Verhaegen, “Sensorless adaptive optics system based on image second moment measurements,” Proc. SPIE 9896, 989609 (2016).
[Crossref]

2015 (5)

2014 (2)

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Y. N. Sulai and A. Dubra, “Non-common path aberration correction in an adaptive optics scanning ophthalmoscope,” Biomed. Opt. Express 5, 3059–3073 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (1)

M. D. Lallo, “Experience with the Hubble Space Telescope: 20 years of an archetype,” Optical Engineering 51, 011011 (2012).
[Crossref]

2011 (2)

A. Give’on, B. D. Kern, and S. Shaklan, “Pair-wise, deformable mirror, image plane-based diversity electric field estimation for high contrast coronagraphy,” Proc. SPIE 8151, 815110 (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)

2008 (1)

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

2007 (1)

2006 (2)

M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14, 1339–1352 (2006).
[Crossref] [PubMed]

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

2005 (2)

M. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterization of deformable membrane mirrors,” Appl. Opt. 44, 5131–5139 (2005).
[Crossref] [PubMed]

L. P. Murray, J. C. Dainty, J. Coignus, and F. Felberer, “Wavefront correction of extended objects through image sharpness maximisation,” Proc. SPIE 6018, 60181A (2005).
[Crossref]

2003 (2)

2001 (1)

P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep,” IEE Proceedings - Control Theory and Applications 148, 185–192 (2001).
[Crossref]

2000 (1)

1974 (1)

Acton, D. S.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Agbana, T. E.

T. E. Agbana, H. Yang, O. Soloviev, G. Vdovin, and M. Verhaegen, “Sensorless adaptive optics system based on image second moment measurements,” Proc. SPIE 9896, 989609 (2016).
[Crossref]

Anderson, R.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Atcheson, P. D.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Balasubramanian, K.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Basinger, S.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Bauer, B.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Bifano, T. G.

Bliek, L.

L. Bliek, H. R. G. W. Verstraete, M. Verhaegen, and S. Wahls, “Online Optimization With Costly and Noisy Measurements Using Random Fourier Expansions,” IEEE Transactions on Neural Networks and Learning Systems 29, 167–182 (2018).
[Crossref]

H. R. G. W. Verstraete, M. Heisler, M. J. Ju, D. Wahl, L. Bliek, J. Kalkman, S. Bonora, Y. Jian, M. Verhaegen, and M. V. Sarunic, “Wavefront sensorless adaptive optics OCT with the DONE algorithm for in vivo human retinal imaging [Invited],” Biomed. Opt. Express 8, 2261–2275 (2017).
[Crossref] [PubMed]

Bonora, S.

Booth, M.

Booth, M. J.

Boreman, G. D.

G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001).
[Crossref]

Bowers, C. W.

Buffington, A.

Burns, L.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Choquet, É.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Coignus, J.

L. P. Murray, J. C. Dainty, J. Coignus, and F. Felberer, “Wavefront correction of extended objects through image sharpness maximisation,” Proc. SPIE 6018, 60181A (2005).
[Crossref]

Contos, A. R.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, Inc., 1992, Vol. 11).

Dainty, J. C.

L. P. Murray, J. C. Dainty, J. Coignus, and F. Felberer, “Wavefront correction of extended objects through image sharpness maximisation,” Proc. SPIE 6018, 60181A (2005).
[Crossref]

C. Paterson, I. Munro, and J. C. Dainty, “A low cost adaptive optics system using a membrane mirror,” Opt. Express 6, 175–185 (2000).
[Crossref] [PubMed]

Dean, B.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Dean, B. H.

Dong, B.

Dubra, A.

Egron, S.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Elliott, E.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Felberer, F.

L. P. Murray, J. C. Dainty, J. Coignus, and F. Felberer, “Wavefront correction of extended objects through image sharpness maximisation,” Proc. SPIE 6018, 60181A (2005).
[Crossref]

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Give’on, A.

A. Give’on, B. D. Kern, and S. Shaklan, “Pair-wise, deformable mirror, image plane-based diversity electric field estimation for high contrast coronagraphy,” Proc. SPIE 8151, 815110 (2011).
[Crossref]

Gleichman, K. W.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Gordon, B. L.

J. T. Trauger, D. C. Moody, J. E. Krist, and B. L. Gordon, “Hybrid Lyot coronagraph for WFIRST-AFTA: coronagraph design and performance metrics,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011013 (2016).
[Crossref]

Hansen, K.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Hartig, G.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Hein, R.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Heisler, M.

Hunt, J. D.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Jian, Y.

Ju, M. J.

Kalkman, J.

Kawata, S.

Kazasidis, O.

O. Kazasidis, S. Verpoort, and U. Wittrock, “Algorithm design for image-based wavefront control without wavefront sensing,” Proc. SPIE 10695, 1069502 (2018).

Kern, B. D.

A. Give’on, B. D. Kern, and S. Shaklan, “Pair-wise, deformable mirror, image plane-based diversity electric field estimation for high contrast coronagraphy,” Proc. SPIE 8151, 815110 (2011).
[Crossref]

Klingemann, L.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Krejci, P.

P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep,” IEE Proceedings - Control Theory and Applications 148, 185–192 (2001).
[Crossref]

Krist, J. E.

J. T. Trauger, D. C. Moody, J. E. Krist, and B. L. Gordon, “Hybrid Lyot coronagraph for WFIRST-AFTA: coronagraph design and performance metrics,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011013 (2016).
[Crossref]

Kuhnen, K.

P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep,” IEE Proceedings - Control Theory and Applications 148, 185–192 (2001).
[Crossref]

K. Kuhnen, Inverse Steuerung piezoelektrischer Aktoren mit Hysterese-, Kriech- und Superpositionsoperatoren (Shaker, 2001).

Lajoie, C.-P.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Lallo, M. D.

M. D. Lallo, “Experience with the Hubble Space Telescope: 20 years of an archetype,” Optical Engineering 51, 011011 (2012).
[Crossref]

Lam, R.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

LeBlanc, J. W.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Leboulleux, L.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Levecq, O.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Linhai, H.

Long, C.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Mertz, J.

Miller, J. J.

Moody, D. C.

J. T. Trauger, D. C. Moody, J. E. Krist, and B. L. Gordon, “Hybrid Lyot coronagraph for WFIRST-AFTA: coronagraph design and performance metrics,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011013 (2016).
[Crossref]

Moore, D.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Moore, J.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Mountain, M.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Muller, R. A.

Munro, I.

Murphy, R. J.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Murray, L. P.

L. P. Murray, J. C. Dainty, J. Coignus, and F. Felberer, “Wavefront correction of extended objects through image sharpness maximisation,” Proc. SPIE 6018, 60181A (2005).
[Crossref]

N’Diaye, M.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Ota, T.

Parker, S. W.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Paterson, C.

Patterson, K.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Paudel, H.

Paxman, R. G.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Perchermeier, J.

Perrin, M. D.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Poberezhskiy, I.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Press, W. H.

W. H. Press, S. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Pueyo, L.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Rao, C.

Rausch, P.

Redding, D.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Sabatke, E.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Sarunic, M. V.

Schwenker, J.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Shaklan, S.

A. Give’on, B. D. Kern, and S. Shaklan, “Pair-wise, deformable mirror, image plane-based diversity electric field estimation for high contrast coronagraphy,” Proc. SPIE 8151, 815110 (2011).
[Crossref]

Shi, F.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Shields, D.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Shields, J.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Sidick, E.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Soloviev, O.

T. E. Agbana, H. Yang, O. Soloviev, G. Vdovin, and M. Verhaegen, “Sensorless adaptive optics system based on image second moment measurements,” Proc. SPIE 9896, 989609 (2016).
[Crossref]

H. Yang, O. Soloviev, and M. Verhaegen, “Model-based wavefront sensorless adaptive optics system for large aberrations and extended objects,” Opt. Express 23, 24587–24601 (2015).
[Crossref] [PubMed]

Soummer, R.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Sulai, Y. N.

Sun, H.-B.

Swensen, J.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Tang, H.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
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W. H. Press, S. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

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M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Towell, T.

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

Trauger, J. T.

J. T. Trauger, D. C. Moody, J. E. Krist, and B. L. Gordon, “Hybrid Lyot coronagraph for WFIRST-AFTA: coronagraph design and performance metrics,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011013 (2016).
[Crossref]

Truong, T.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

van der Marel, R.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Vdovin, G.

T. E. Agbana, H. Yang, O. Soloviev, G. Vdovin, and M. Verhaegen, “Sensorless adaptive optics system based on image second moment measurements,” Proc. SPIE 9896, 989609 (2016).
[Crossref]

Verhaegen, M.

Verpoort, S.

Verstraete, H. R. G. W.

Vetterling, W. T.

W. H. Press, S. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Wahl, D.

Wahl, D. J.

Wahls, S.

L. Bliek, H. R. G. W. Verstraete, M. Verhaegen, and S. Wahls, “Online Optimization With Costly and Noisy Measurements Using Random Fourier Expansions,” IEEE Transactions on Neural Networks and Learning Systems 29, 167–182 (2018).
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H. R. G. W. Verstraete, S. Wahls, J. Kalkman, and M. Verhaegen, “Model-based sensor-less wavefront aberration correction in optical coherence tomography,” Opt. Lett. 40, 5722–5725 (2015).
[Crossref] [PubMed]

Wallace, J. K.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Wang, X.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Warmuth, M. W.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Wilson, A. J.

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

Wilson, D.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Wilson, T.

Wittrock, U.

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, Inc., 1992, Vol. 11).

Yang, H.

T. E. Agbana, H. Yang, O. Soloviev, G. Vdovin, and M. Verhaegen, “Sensorless adaptive optics system based on image second moment measurements,” Proc. SPIE 9896, 989609 (2016).
[Crossref]

H. Yang, O. Soloviev, and M. Verhaegen, “Model-based wavefront sensorless adaptive optics system for large aberrations and extended objects,” Opt. Express 23, 24587–24601 (2015).
[Crossref] [PubMed]

Ygouf, M.

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

Yu, J.

Zawadzki, R. J.

Appl. Opt. (3)

Biomed. Opt. Express (3)

Chin. Opt. Lett. (1)

IEE Proceedings - Control Theory and Applications (1)

P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep,” IEE Proceedings - Control Theory and Applications 148, 185–192 (2001).
[Crossref]

IEEE Transactions on Neural Networks and Learning Systems (1)

L. Bliek, H. R. G. W. Verstraete, M. Verhaegen, and S. Wahls, “Online Optimization With Costly and Noisy Measurements Using Random Fourier Expansions,” IEEE Transactions on Neural Networks and Learning Systems 29, 167–182 (2018).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Journal of Astronomical Telescopes, Instruments, and Systems (2)

J. T. Trauger, D. C. Moody, J. E. Krist, and B. L. Gordon, “Hybrid Lyot coronagraph for WFIRST-AFTA: coronagraph design and performance metrics,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011013 (2016).
[Crossref]

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 2, 011021 (2016).
[Crossref]

Opt. Express (6)

Opt. Lett. (4)

Optical Engineering (1)

M. D. Lallo, “Experience with the Hubble Space Telescope: 20 years of an archetype,” Optical Engineering 51, 011011 (2012).
[Crossref]

Proc. SPIE (7)

O. Kazasidis, S. Verpoort, and U. Wittrock, “Algorithm design for image-based wavefront control without wavefront sensing,” Proc. SPIE 10695, 1069502 (2018).

D. S. Acton, T. Towell, J. Schwenker, J. Swensen, D. Shields, E. Sabatke, L. Klingemann, A. R. Contos, B. Bauer, K. Hansen, P. D. Atcheson, D. Redding, F. Shi, S. Basinger, B. Dean, and L. Burns, “Demonstration of the James Webb Space Telescope commissioning on the JWST testbed telescope,” Proc. SPIE 6265, 62650R (2006).
[Crossref]

M. D. Perrin, R. Soummer, É. Choquet, M. N’Diaye, O. Levecq, C.-P. Lajoie, M. Ygouf, L. Leboulleux, S. Egron, R. Anderson, C. Long, E. Elliott, G. Hartig, L. Pueyo, R. van der Marel, and M. Mountain, “James Webb Space Telescope Optical Simulation Testbed I: overview and first results,” Proc. SPIE 9143, 914309 (2014).
[Crossref]

A. Give’on, B. D. Kern, and S. Shaklan, “Pair-wise, deformable mirror, image plane-based diversity electric field estimation for high contrast coronagraphy,” Proc. SPIE 8151, 815110 (2011).
[Crossref]

M. W. Warmuth, S. W. Parker, A. J. Wilson, K. W. Gleichman, R. G. Paxman, B. J. Thelen, R. J. Murphy, J. D. Hunt, and J. W. LeBlanc, “Operation of phase-diverse adaptive-optics with extended scenes,” Proc. SPIE 7093, 709307 (2008).
[Crossref]

L. P. Murray, J. C. Dainty, J. Coignus, and F. Felberer, “Wavefront correction of extended objects through image sharpness maximisation,” Proc. SPIE 6018, 60181A (2005).
[Crossref]

T. E. Agbana, H. Yang, O. Soloviev, G. Vdovin, and M. Verhaegen, “Sensorless adaptive optics system based on image second moment measurements,” Proc. SPIE 9896, 989609 (2016).
[Crossref]

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J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, Inc., 1992, Vol. 11).

P. Rausch, “Deformierbare Spiegel für Weltraumteleskope und Hochleistungslaser,” Ph.D. thesis, University of Münster (2016).

W. H. Press, S. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

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Figures (12)

Fig. 1
Fig. 1 Sketch of the testbed. The plane of aberration generation (B) is imaged to the plane of aberration correction (B′) with a 4-f telescope and a magnification of −1. The letters A, A′ and A″ indicate the image-forming conjugate planes. The magnification from plane A to plane A′ is −1, and from plane A′ to plane A″ is −3. The angles of incidence on the deformable mirrors are approximately 5°. Due to the non-perpendicular incidence on the deformable mirrors, we place the two silver mirrors around plane A′ to ensure the correct conjugation between the planes B and B′. Lenses focal lengths: f1 = 250 mm, f2 = 750 mm. An image captured with our testbed is shown in Fig. 2.
Fig. 2
Fig. 2 (a) The detected image with the negative 1951 USAF test target. (b) Highlighted zoom of the image showing the designated region of interest of a, with the elements of groups 6 and 7. The white horizontal scale bar in the lower right corner of the image is 31 µm long. (c) The pixel values of the six elements under the red arrow drawn in b. The three bars of the sixth element shown at the end of the graph have 228 lp/mm density and are resolved with about 10 % contrast.
Fig. 3
Fig. 3 Left: The absolute value of the similarity matrix for the first 41 Zernike modes for the deformable mirror used as aberration corrector. The matrix is diagonally dominant. The non-zero off-diagonal elements reveal modal cross-talk. An example is marked with red dashed lines: if we want to generate 1 µm of Z13, we will get a surface with 0.87 µm of Z13 and −0.31 µm of Z22. Right: The generation efficiency for the first 41 Zernike modes. These are the diagonal elements of the similarity matrix shown on the left.
Fig. 4
Fig. 4 Left: The first 11 singular modes of the aberration corrector, ordered with decreasing singular value (i.e., gain) from left to right and from top to bottom. They are very similar to low-order Zernike modes. The singular modes are numerically calculated from the experimentally measured influence matrix. Right: The normalized singular values of the aberration corrector. The dashed red line designates the maximum allowed condition number (200), which leads to truncating the 4 smallest singular values.
Fig. 5
Fig. 5 The control block diagram for the aberration correction. The aberration correction algorithm can be forced to operate in the domain of voltages ( v ), Zernike modes ( z ), or singular modes ( s ). When operating in the Zernike mode domain or in the singular modes domain, the outputs are converted to voltages by matrix multiplication. The control input for the aberration corrector is generated after passing through the open-loop hysteresis compensation (H−1) and the saturation control.
Fig. 6
Fig. 6 The merit function in the voltage domain for the actuators 1, 9, 10, 27, 42, 43, and 44. The plots are centered to the voltage values at the global minimum of the merit function. The solid line corresponds to decreasing amplitude and the dashed line to increasing amplitude. The small difference between the two lines is caused by residual uncompensated hysteresis. The inset at the lower right corner shows the actuator pattern and numbering.
Fig. 7
Fig. 7 The merit function in the Zernike mode domain. Left: The full sensitivity plot for Z7 (coma y). The solid line corresponds to decreasing amplitude and the dashed line to increasing amplitude. The small difference between the two lines is caused by residual uncompensated hysteresis. Right: The MF for decreasing amplitude from 1 µm to −1 µm in grayscale. Each row of the matrix corresponds to one Zernike mode from Z1 to Z32.
Fig. 8
Fig. 8 The merit function in the singular mode domain. Left: The full sensitivity plot for S8. The solid line corresponds to decreasing amplitude and the dashed line to increasing amplitude. The small difference between the two lines is caused by residual uncompensated hysteresis. Right: The MF for decreasing amplitude from 1 µm to −1 µm in grayscale. Each row of the matrix corresponds to one singular mode from S1 to S32. The first 11 singular modes are shown in Fig. 4.
Fig. 9
Fig. 9 Left: Sensitivity plots of the MF for Z9 (trefoil 0°) with (blue solid line) and without (red dashed line) the open-loop hysteresis compensation. The curves start at Z9 = 0 µm, and MF = −1.66 (blue) and MF = −1.63 (red). The horizontal shift of the minimum of the MF is 0.2/4.1=4.9 % (blue double-headed arrow) and 0.7/4.1=17 % (lower red double-headed arrow), respectively. When the open-loop hysteresis compensation is active, the maximum horizontal opening is equal to the horizontal shift of the minimum, i.e., 4.9 % (blue double-headed arrow). When the open-loop hysteresis compensation is inactive, the maximum horizontal opening is 1.0/4.1=24 % (upper red double-headed arrow). Right: Compensation of the residual hysteresis with an algorithm in Z3 to Z5 (blue diamonds) followed by an algorithm in voltages (red circles). The white horizontal scale bar in the lower right corner of the images is 31 µm long.
Fig. 10
Fig. 10 The results of the single Zernike aberration experiments of 1 µm amplitude. Left: The output of the correction process in Zernike modes in color code. Each column of the matrix refers to an experiment when a single Zernike aberration was applied to the aberration corrector. Ideally the matrix would be diagonal with all the diagonal elements equal to −1. To save space we do not show the modes Z15 to Z32, whose maximum absolute value is 0.1 µm. Right: The merit function of the reference images (black circles), the aberrated images (blue diamonds), and the corrected images (red crosses).
Fig. 11
Fig. 11 Left: Progress of the correction for a combination of Zernike modes. The steepest descents correct fast most of the aberrations and the cyclic coordinate descent fine-tunes the image. The aberrated image and the corrected image are shown. The white horizontal scale bar in the lower right corner of the images is 31 µm long. Right: The coefficient of the Zernike modes for the aberration generator (red unfilled bars) and for the aberration corrector (blue filled bars).
Fig. 12
Fig. 12 Left: The voltages set at the aberration generator, randomly chosen in the interval (−25 V, +25 V). Right: Comparison of the aberration correction in the Zernike mode domain and in the singular modes domain. The correction in the Zernike mode domain requires less evaluations for convergence of the MF, but results in a poorer MF than the correction in the singular mode domain. The white horizontal scale bar in the lower right corner of the images is 31 µm long.

Tables (1)

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Table 1 The advantages of each domain for the wavefront-sensorless control of a deformable mirror (DM) with hysteresis in an imaging system.

Equations (4)

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IM = U S W T CM = I M + = W S 1 U T ,
z r e a l = IM v v = CM z d e s ,
s r e a l = S W T v v = W S 1 s d e s ,
M F = n x = 1 N x n y = 1 N y I ( n x , n y ) 2 [ n x = 1 N x n y = 1 N y I ( n x , n y ) ] 2 N x , N y .

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