Abstract

Certain adaptive optics systems do not employ a wave front sensor but rather maximise a photodetector signal by appropriate control of an adaptive element. The maximisation procedure must be optimised if the system is to work efficiently. Such optimisation is often implemented empirically, but further insight can be obtained by using an appropriate mathematical model. In many practical systems aberrations can be accurately represented by a small number of modes of an orthogonal basis, such as the Zernike polynomials. By heuristic reasoning we develop a model for the operation of such systems and demonstrate a link with the geometrical problems of sphere packings and coverings. This approach aids the optimisation of control algorithms and is illustrated by application to direct search and hill climbing algorithms. We develop an efficient scheme using a direct maximisation calculation that permits the measurement of N Zernike modes with only N +1 intensity measurements.

© 2006 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford University Press, 1998).
  2. M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. 995788–5792 (2002).
    [Crossref] [PubMed]
  3. A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
    [Crossref]
  4. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
    [Crossref] [PubMed]
  5. P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 111123–1130 (2003).
    [Crossref] [PubMed]
  6. O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 2552–54 (2000).
    [Crossref]
  7. W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express 10, 550–555 (2002).
    [PubMed]
  8. E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
    [Crossref]
  9. A. C. F. Gonte and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 411073–1076 (2002).
    [Crossref]
  10. M. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A 19356–368 (2002).
    [Crossref]
  11. M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. 445131–5139 (2005).
    [Crossref] [PubMed]
  12. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, (Academic Press, London, 1984).
  13. M. Born and E. Wolf, Principles of Optics, 6th Edition, (Pergamon Press, 1983).
  14. Michalewicz and D. B. Fogel, How to Solve It: Modern Heuristics, (Springer, Berlin, 2000).
  15. J. H. Conway and N. J. A. Sloan, Sphere Packings, Lattices and Groups, 3rd Edition, (Springer-Verlag, 1998).
  16. T. C. Hales, “An overview of the Kepler conjecture,” http://xxx.lanl.gov/ math.MG/9811071 (1999).
  17. R. Kershner, “The number of circles covering a set,” Am. J. Math. 61665–671 (1939).
    [Crossref]
  18. E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory 451639–1642 (1999).
    [Crossref]
  19. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).
  20. M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A 171098–1107 (2000).
    [Crossref]
  21. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66207–277 (1976).
    [Crossref]

2005 (2)

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. 445131–5139 (2005).
[Crossref] [PubMed]

2004 (1)

E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
[Crossref]

2003 (1)

2002 (5)

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
[Crossref] [PubMed]

A. C. F. Gonte and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 411073–1076 (2002).
[Crossref]

M. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A 19356–368 (2002).
[Crossref]

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. 995788–5792 (2002).
[Crossref] [PubMed]

W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express 10, 550–555 (2002).
[PubMed]

2000 (2)

1999 (1)

E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory 451639–1642 (1999).
[Crossref]

1976 (1)

1939 (1)

R. Kershner, “The number of circles covering a set,” Am. J. Math. 61665–671 (1939).
[Crossref]

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
[Crossref] [PubMed]

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 2552–54 (2000).
[Crossref]

Arlt, J.

E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
[Crossref]

Bente, E.

Booth, M. J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th Edition, (Pergamon Press, 1983).

Boutros, J.

E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory 451639–1642 (1999).
[Crossref]

Burns, D.

Conway, J. H.

J. H. Conway and N. J. A. Sloan, Sphere Packings, Lattices and Groups, 3rd Edition, (Springer-Verlag, 1998).

Dandliker, R.

A. C. F. Gonte and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 411073–1076 (2002).
[Crossref]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).

Fogel, D. B.

Michalewicz and D. B. Fogel, How to Solve It: Modern Heuristics, (Springer, Berlin, 2000).

Girkin, J.

Girkin, J. M.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 111123–1130 (2003).
[Crossref] [PubMed]

Gonte, A. C. F.

A. C. F. Gonte and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 411073–1076 (2002).
[Crossref]

Hales, T. C.

T. C. Hales, “An overview of the Kepler conjecture,” http://xxx.lanl.gov/ math.MG/9811071 (1999).

Hardy, J. W.

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

Hossack, W. J.

E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
[Crossref]

Juškaitis, R.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. 995788–5792 (2002).
[Crossref] [PubMed]

Kawata, S.

Kershner, R.

R. Kershner, “The number of circles covering a set,” Am. J. Math. 61665–671 (1939).
[Crossref]

Lubeigt, W.

Marsh, P. N.

Michalewicz,

Michalewicz and D. B. Fogel, How to Solve It: Modern Heuristics, (Springer, Berlin, 2000).

Mourou, G.

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. 995788–5792 (2002).
[Crossref] [PubMed]

M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A 171098–1107 (2000).
[Crossref]

Noll, R.

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
[Crossref] [PubMed]

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 2552–54 (2000).
[Crossref]

Ota, T.

Patterson, B. A.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

Poland, S. P.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).

Sheppard, C. J. R.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, (Academic Press, London, 1984).

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
[Crossref] [PubMed]

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 2552–54 (2000).
[Crossref]

Sloan, N. J. A.

J. H. Conway and N. J. A. Sloan, Sphere Packings, Lattices and Groups, 3rd Edition, (Springer-Verlag, 1998).

Sun, H.-B.

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).

Theofanidou, E.

E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
[Crossref]

Valentine, G.

Valentine, G. J.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

Vdovin, G.

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).

Vitterbo, E.

E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory 451639–1642 (1999).
[Crossref]

Vorontsov, M.

Wilson, L.

E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
[Crossref]

Wilson, T.

M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. 445131–5139 (2005).
[Crossref] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. 995788–5792 (2002).
[Crossref] [PubMed]

M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A 171098–1107 (2000).
[Crossref]

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, (Academic Press, London, 1984).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th Edition, (Pergamon Press, 1983).

Wright, A. J.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

Ye, J. Y.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
[Crossref] [PubMed]

Am. J. Math. (1)

R. Kershner, “The number of circles covering a set,” Am. J. Math. 61665–671 (1939).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Inf. Theory (1)

E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory 451639–1642 (1999).
[Crossref]

J. Microsc. (1)

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. 20665–71 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

Microsc. Res. Technol. (1)

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. 6736–44 (2005).
[Crossref]

Opt. Commun. (1)

E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236145–150 (2004).
[Crossref]

Opt. Eng. (1)

A. C. F. Gonte and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 411073–1076 (2002).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Proc. Nat. Acad. Sci. (1)

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. 995788–5792 (2002).
[Crossref] [PubMed]

Other (7)

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

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, (Academic Press, London, 1984).

M. Born and E. Wolf, Principles of Optics, 6th Edition, (Pergamon Press, 1983).

Michalewicz and D. B. Fogel, How to Solve It: Modern Heuristics, (Springer, Berlin, 2000).

J. H. Conway and N. J. A. Sloan, Sphere Packings, Lattices and Groups, 3rd Edition, (Springer-Verlag, 1998).

T. C. Hales, “An overview of the Kepler conjecture,” http://xxx.lanl.gov/ math.MG/9811071 (1999).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the adaptive system. The input wave front is incident from the left whereupon it passes through the pupil plane of the lens. A phase element in this plane adds a chosen phase aberration to the input wave front. The wave front is then focussed by the lens onto pinhole positioned at the nominal focus where the intensity signal is measured by the photodetector.

Fig. 2.
Fig. 2.

The photodetector signal calculated as a function of aberration magnitude for random combinations of the six Zernike aberration modes i = 4 to i = 9. Each data point shows the mean and the 10th and 90th percentiles of a collection of 100 random samples.

Fig. 3.
Fig. 3.

Sphere packings in two dimensions: (a) incomplete covering based upon an integer lattice with lattice constant 2ε; (b) complete covering based upon an integer lattice with lattice constant 2 ε ; (c) thinnest possible covering based upon hexagonal lattice. The points at the centre of each circle represent the candidate solutions b. Part (d) illustrates the body centred cubic arrangement, the optimal three dimensional covering.

Fig. 4.
Fig. 4.

Base 10 logarithmic plots of the covering thickness for different packings in N dimensions, corresponding to the efficiency of measurement of N modes: (i) Incomplete integer lattice covering (long dashed line), Θ1; (ii) Complete integer lattice covering (short dashed line), Θ2; (iii) Optimum known lattice covering (solid line), Θ2.

Fig. 5.
Fig. 5.

Base 10 logarithmic plots of the number of evalutions, K, required for the exaustive search in N dimensions, based upon (i) Complete integer lattice (diamonds); (ii) the AN* lattice (stars); (iii) three-level branch and bound exhaustive search using the AAN* lattice (squares).

Fig. 6.
Fig. 6.

Linear plots of K, the mean number of evalutions required for the hill climbing algorithm in N dimensions, using (i) a simplex with fixed orientation (diamonds); (ii) a simplex with reversals at each step (stars).

Fig. 7.
Fig. 7.

Aberration measurement accuracy for direct estimation using a simplex arrangement of abcissæ for N = 6 and ∣b n ∣ = 0.5. The data points show the mean and standard deviation from a sample of 1000 input aberrations. The solid line is an even polynomial fit. The dashed line shows the measurement tolerance of E= 0.1 .

Tables (1)

Tables Icon

Table 1. Zernike mode definitions

Equations (25)

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

F = I 0 1 π θ = 0 2 π r = 0 1 exp { r θ r θ } r d r d θ 2
Φ r θ = n = 1 N a n Z n r θ
Ψ r θ = n = 1 N b n Z n r θ
F ( c ) = I 0 f ( c ) ,
f ( c ) = 1 π θ = 0 2 π r = 0 1 exp { j n = 1 N c n Z n r θ } r d r d θ 2 ,
f ( c ) 1 c 2 ,
V N = π N 2 Γ ( N 2 + 1 ) ,
Θ 1 = V N 2 N .
Θ 2 = V N N N 2 2 N .
Θ 3 = V N N + 1 ( N ( N + 2 ) 12 ( N + 1 ) ) N 2 .
W = b F ( a b ) d V F ( a b ) d V a ,
W = m = 1 M γ m b m F ( a b m ) m = 1 M γ M F ( a b m )
a S 1 W ,
S ik = W i a k a = 0 .
E = S 1 W a .
n = 1 N + 1 b n = 0 .
n = 2 N + 1 ( b n ) 1 = 1 ,
( b n ) 1 = 1 N n > 1 .
( b 2 ) 2 = 1 ( b 2 ) 1 2 = 1 ( 1 N ) 2 .
( b n ) m = ( b m ) m N m + 1 m < n
= 1 p = 1 m ( b m ) p 2 m = n
= 0 m > n
s b 2 ε b 1 2 = s 2 2 εs N + ε 2 ε 2 .
s ( s 2 ε N ) 0 ,
0 s 2 ε N .

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