Abstract

We demonstrate two methods for the characterization of deformable membrane mirrors and the training of adaptive optics systems that employ these mirrors. Neither method employs a wave-front sensor. In one case, aberrations produced by a wave-front generator are corrected by the deformable mirror by use of a rapidly converging iterative algorithm based on orthogonal deformation modes of the mirror. In the other case, a simple interferometer is used with fringe analysis and phase-unwrapping algorithms. We discuss how the choice of singular values can be used to control the pseudoinversion of the control matrix.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2004

E. Theofanidou, L. Wilson, W. J. Hossack, J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236, 145–150 (2004).
[CrossRef]

M. Schwertner, M. J. Booth, M. A. A. Neil, T. Wilson, “Measurement of specimen induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[CrossRef]

2003

2002

D. Dayton, J. Gonglewski, S. Restaino, J. Martin, J. Phillips, M. Hartman, S. Browne, P. Kervin, J. Snodgrass, N. Heimann, M. Shilko, R. Pohle, B. Carrion, C. Smith, D. Thiel, “Demonstration of new technology MEMS and liquid crystal adaptive optics on bright astronomical objects and satellites,” Opt. Express 10, 1508–1519 (2002).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

F. Gonte, A. Courteville, R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41, 1073–1076 (2002).
[CrossRef]

M. A. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A 19, 356–368 (2002).
[CrossRef]

2001

2000

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

M. A. A. Neil, M. J. Booth, T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A 17, 1098–1107 (2000).
[CrossRef]

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

1999

G. V. Vdovin, P. M. Sarro, S. Middelhoek, “Technology and applications of micromachined adaptive mirrors,” J. Micromech. Microeng. 9, R8–R20 (1999).
[CrossRef]

L. Zhu, P. C. Sun, D. U. Bartsch, W. R. Freeman, Y. Fainman, “Wave-front generation of Zernike polynomial modes with a micromachined membrane deformable mirror,” Appl. Opt. 38, 6019–6026 (1999).
[CrossRef]

1998

1995

1987

1986

1983

1982

1977

1976

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Arlt, J.

E. Theofanidou, L. Wilson, W. J. Hossack, J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236, 145–150 (2004).
[CrossRef]

Artal, P.

Bareket, N.

Bartsch, D. U.

Booth, M. J.

M. Schwertner, M. J. Booth, M. A. A. Neil, T. Wilson, “Measurement of specimen induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[CrossRef]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

M. A. A. Neil, M. J. Booth, T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A 17, 1098–1107 (2000).
[CrossRef]

M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
[CrossRef]

M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wave front sensor,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 79–90 (2003).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1975).

Browne, S.

D. Dayton, J. Gonglewski, S. Restaino, J. Martin, J. Phillips, M. Hartman, S. Browne, P. Kervin, J. Snodgrass, N. Heimann, M. Shilko, R. Pohle, B. Carrion, C. Smith, D. Thiel, “Demonstration of new technology MEMS and liquid crystal adaptive optics on bright astronomical objects and satellites,” Opt. Express 10, 1508–1519 (2002).
[CrossRef] [PubMed]

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Burns, D.

Carrion, B.

Claflin, E. S.

Courteville, A.

F. Gonte, A. Courteville, R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41, 1073–1076 (2002).
[CrossRef]

Dainty, J. C.

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

Dandliker, R.

F. Gonte, A. Courteville, R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41, 1073–1076 (2002).
[CrossRef]

Dayton, D.

D. Dayton, J. Gonglewski, S. Restaino, J. Martin, J. Phillips, M. Hartman, S. Browne, P. Kervin, J. Snodgrass, N. Heimann, M. Shilko, R. Pohle, B. Carrion, C. Smith, D. Thiel, “Demonstration of new technology MEMS and liquid crystal adaptive optics on bright astronomical objects and satellites,” Opt. Express 10, 1508–1519 (2002).
[CrossRef] [PubMed]

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Fainman, Y.

Fernández, E. J.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Frazier, B. W.

Freeman, W. R.

Gallegos, J.

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Girkin, J. M.

Gonglewski, J.

D. Dayton, J. Gonglewski, S. Restaino, J. Martin, J. Phillips, M. Hartman, S. Browne, P. Kervin, J. Snodgrass, N. Heimann, M. Shilko, R. Pohle, B. Carrion, C. Smith, D. Thiel, “Demonstration of new technology MEMS and liquid crystal adaptive optics on bright astronomical objects and satellites,” Opt. Express 10, 1508–1519 (2002).
[CrossRef] [PubMed]

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Gonte, F.

F. Gonte, A. Courteville, R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41, 1073–1076 (2002).
[CrossRef]

Grosso, R. P.

Hartman, M.

Heimann, N.

Hossack, W. J.

E. Theofanidou, L. Wilson, W. J. Hossack, J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236, 145–150 (2004).
[CrossRef]

Ina, H.

Juškaitis, R.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Kawata, S.

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Kervin, P.

Kobayashi, S.

Macy, W. M.

Marsh, P. N.

Martin, J.

McDer-mott, S.

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Middelhoek, S.

G. V. Vdovin, P. M. Sarro, S. Middelhoek, “Technology and applications of micromachined adaptive mirrors,” J. Micromech. Microeng. 9, R8–R20 (1999).
[CrossRef]

Munro, I.

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

Neil, M. A. A.

M. Schwertner, M. J. Booth, M. A. A. Neil, T. Wilson, “Measurement of specimen induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[CrossRef]

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

M. A. A. Neil, M. J. Booth, T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A 17, 1098–1107 (2000).
[CrossRef]

M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23, 1849–1851 (1998).
[CrossRef]

Noll, R. J.

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Ota, T.

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Paterson, C.

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

Phillips, J.

Pohle, R.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Restaino, S.

D. Dayton, J. Gonglewski, S. Restaino, J. Martin, J. Phillips, M. Hartman, S. Browne, P. Kervin, J. Snodgrass, N. Heimann, M. Shilko, R. Pohle, B. Carrion, C. Smith, D. Thiel, “Demonstration of new technology MEMS and liquid crystal adaptive optics on bright astronomical objects and satellites,” Opt. Express 10, 1508–1519 (2002).
[CrossRef] [PubMed]

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Roddier, C.

Roddier, F.

Rogers, S.

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Sarro, P. M.

G. V. Vdovin, P. M. Sarro, S. Middelhoek, “Technology and applications of micromachined adaptive mirrors,” J. Micromech. Microeng. 9, R8–R20 (1999).
[CrossRef]

G. V. Vdovin, P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
[CrossRef] [PubMed]

Schwertner, M.

M. Schwertner, M. J. Booth, M. A. A. Neil, T. Wilson, “Measurement of specimen induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[CrossRef]

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Shilko, M.

D. Dayton, J. Gonglewski, S. Restaino, J. Martin, J. Phillips, M. Hartman, S. Browne, P. Kervin, J. Snodgrass, N. Heimann, M. Shilko, R. Pohle, B. Carrion, C. Smith, D. Thiel, “Demonstration of new technology MEMS and liquid crystal adaptive optics on bright astronomical objects and satellites,” Opt. Express 10, 1508–1519 (2002).
[CrossRef] [PubMed]

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Smith, C.

Snodgrass, J.

Sugiura, T.

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Sun, P. C.

Takeda, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Theofanidou, E.

E. Theofanidou, L. Wilson, W. J. Hossack, J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236, 145–150 (2004).
[CrossRef]

Thiel, D.

Tyson, R. K.

Vaidyanathan, M.

D. Dayton, S. Restaino, J. Gonglewski, J. Gallegos, S. McDer-mott, S. Browne, S. Rogers, M. Vaidyanathan, M. Shilko, “Laboratory and field demonstration of a low cost membrane mirror adaptive optics system,” Opt. Commun. 176, 339–345 (2000).
[CrossRef]

Vdovin, G. V.

G. V. Vdovin, P. M. Sarro, S. Middelhoek, “Technology and applications of micromachined adaptive mirrors,” J. Micromech. Microeng. 9, R8–R20 (1999).
[CrossRef]

G. V. Vdovin, P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Vorontsov, M. A.

Wilson, L.

E. Theofanidou, L. Wilson, W. J. Hossack, J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236, 145–150 (2004).
[CrossRef]

Wilson, T.

M. Schwertner, M. J. Booth, M. A. A. Neil, T. Wilson, “Measurement of specimen induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[CrossRef]

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

T. Ota, S. Kawata, T. Sugiura, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Dynamic axial-position control of a laser-trapped particle by wave-front modification,” Opt. Lett. 28, 465–467 (2003).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

M. A. A. Neil, M. J. Booth, T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A 17, 1098–1107 (2000).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

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[CrossRef]

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J. Opt. Soc. Am. A

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T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

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E. Theofanidou, L. Wilson, W. J. Hossack, J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236, 145–150 (2004).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Proc. Natl. Acad. Sci. USA

M. J. Booth, M. A. A. Neil, R. Juškaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
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Figures (7)

Fig. 1
Fig. 1

Experimental setup for deformable mirror training by the wave-front generator method. PBS: polarizing beam splitter; SF, spatial filter.

Fig. 2
Fig. 2

(a) Photodetector responses for a selection of mirror deformation modes. (b) Output signals from modal sensor measurements based on the mirror modes. Solid curves, mode 0; dashed curves, mode 6; long-dashed curves, mode 12; dotted–dashed curves, mode 18. Mode 0 is the mode with the largest corresponding singular value. Other modes are ordered according to descending singular values.

Fig. 3
Fig. 3

Range of each Zernike mode generated by the deformable mirror for a maximum wave-front error variance of 0.05 rad2.

Fig. 4
Fig. 4

Experimental setup for the interferometer method.

Fig. 5
Fig. 5

Graphic representations of similarity matrices M obtained when N singular values are removed during the SVD inversion. The gray-scale shading represents the absolute value of the matrix element. The Zernike modes represented range from mode 2 (top left) to 37 (bottom right). The histograms at the right show the values of the first 20 diagonal elements, corresponding to Zernike modes 2–21.

Fig. 6
Fig. 6

Saturation likelihood Q after N singular values had been removed during the SVD matrix inversion.

Fig. 7
Fig. 7

Experimentally determined similarity matrix after 20 singular values had been removed during the matrix pseudoinversion. The scale represents the absolute value of the matrix element.

Tables (1)

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Table 1 Zernike Polynomials

Equations (20)

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Φ ( r , θ ) = i = 1 a i Z i ( r , θ ) ,
a = Bc .
c = B a .
W = V | P exp [ i Ψ ( r , θ ) + i Φ ( r , θ ) ] × exp [ i ν r cos ( θ - ϕ ) ] r d r d θ | 2 ν d ν d ϕ ,
c n + 1 = c n + γ S i - 1 ( W 1 - W 2 ) ξ i ,
ζ i = 1 a c .
W = W 0 [ 1 - var ( Δ Φ ) ] ,
I ( x , y ) = F { M ( u , v ) } { A r A 0 exp [ i ( ϕ + Δ τ ) ] } ,
ψ ( x , y ) arg [ I ( x , y ) ] - Δ τ ( x , y ) ,
z ( x , y ) = k = 1 K c ψ k ( x , y ) .
ψ k ( x , y ) = 1 c p = 1 P β k p Z p ( x , y ) .
β k p = 1 π x 2 + y 2 1 ψ k ( x , y ) Z p ( x , y ) d x d y .
a = B B a .
Q = max ( B ) .
Z n m ( r , θ ) = { m < 0 , 2 R n - m ( r ) sin ( - m θ ) m = 0 , 0 m > 0 , 2 R n m ( r ) cos ( m θ ) ; R n m ( r ) = n + 1 s = 0 ( n - m ) / 2 × ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] ! r n - 2 s ,
2 ϕ = 2 ψ .
Δ i , j x = W ( ψ ˜ i + 1 , j - ψ ˜ i , j ) , Δ i , j y = W ( ψ ˜ i , j + 1 - ψ ˜ i , j ) .
ρ ˜ i , j = ( Δ i , j x - Δ i - 1 , j x ) + ( Δ i , j y - Δ i , j - 1 y ) .
( ϕ ˜ i + 1 , j - 2 ϕ ˜ i , j + ϕ ˜ i - 1 , j ) + ( ϕ ˜ i , j + 1 - 2 ϕ ˜ i , j + ϕ ˜ i , j - 1 ) = ρ ˜ i , j ,
Φ m , n = P m , n 2 cos ( π m / N ) + 2 cos ( π n / N ) - 4 ,

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