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. Ju¡skaitis and T.Wilson, "Adaptive aberration correction in a confocal microscope," Proc. Nat. Acad. Sci. 99, 5788-5792 (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. 67, 36-44 (2005).
    [CrossRef]
  4. L. Sherman, J. Y. Ye, O. Albert and 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]
  5. P. N. Marsh, D. Burns and J. M. Girkin, "Practical implementation of adaptive optics in multiphoton microscopy," Opt. Express 11, 1123-1130 (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. 25, 52-54 (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. 236, 145-150 (2004).
    [CrossRef]
  9. A. C. F. Gonte and R. Dandliker, "Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror," Opt. Eng. 41, 1073-1076 (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 19, 356-368 (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. 44, 5131-5139 (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. Z. 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. 61, 665-671 (1939).
    [CrossRef]
  18. E. Vitterbo and J. Boutros, "A universal lattice code decoder for fading channels," IEEE Trans. Inf. Theory 45, 1639-1642 (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 17, 1098-1107 (2000).
    [CrossRef]
  21. R. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-277 (1976).
    [CrossRef]

2005

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. 67, 36-44 (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. 44, 5131-5139 (2005).
[CrossRef] [PubMed]

2004

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

2003

2002

M. 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]

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]

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

L. Sherman, J. Y. Ye, O. Albert and 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]

M. J. Booth, M. A. A. Neil, R. Ju¡skaitis and T.Wilson, "Adaptive aberration correction in a confocal microscope," Proc. Nat. Acad. Sci. 99, 5788-5792 (2002).
[CrossRef] [PubMed]

2000

1999

E. Vitterbo and J. Boutros, "A universal lattice code decoder for fading channels," IEEE Trans. Inf. Theory 45, 1639-1642 (1999).
[CrossRef]

1976

1939

R. Kershner, "The number of circles covering a set," Am. J. Math. 61, 665-671 (1939).
[CrossRef]

Albert, O.

L. Sherman, J. Y. Ye, O. Albert and 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]

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. 25, 52-54 (2000).
[CrossRef]

Arlt, J.

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

Bente, E.

Booth, M. J.

Boutros, J.

E. Vitterbo and J. Boutros, "A universal lattice code decoder for fading channels," IEEE Trans. Inf. Theory 45, 1639-1642 (1999).
[CrossRef]

Burns, D.

Dandliker, R.

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

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. 67, 36-44 (2005).
[CrossRef]

P. N. Marsh, D. Burns and J. M. Girkin, "Practical implementation of adaptive optics in multiphoton microscopy," Opt. Express 11, 1123-1130 (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. 41, 1073-1076 (2002).
[CrossRef]

Hossack, W. J.

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

Kawata, S.

Kershner, R.

R. Kershner, "The number of circles covering a set," Am. J. Math. 61, 665-671 (1939).
[CrossRef]

Lubeigt, W.

Marsh, P. N.

Mourou, G.

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Ju¡skaitis and T.Wilson, "Adaptive aberration correction in a confocal microscope," Proc. Nat. Acad. Sci. 99, 5788-5792 (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 17, 1098-1107 (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 multiphoton scanning microscopy using a deformable mirror," J. Microsc. 206, 65-71 (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. 25, 52-54 (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. 67, 36-44 (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. 67, 36-44 (2005).
[CrossRef]

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert and 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]

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. 25, 52-54 (2000).
[CrossRef]

Sun, H.-B.

Theofanidou, E.

E. Theofanidou, L. Wilson,W. J. Hossack and J. Arlt, "Spherical aberration correction for optical tweezers," Opt. Commun. 236, 145-150 (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. 67, 36-44 (2005).
[CrossRef]

Vdovin, G.

Vitterbo, E.

E. Vitterbo and J. Boutros, "A universal lattice code decoder for fading channels," IEEE Trans. Inf. Theory 45, 1639-1642 (1999).
[CrossRef]

Vorontsov, M.

Wilson, L.

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

Wilson, T.

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. 67, 36-44 (2005).
[CrossRef]

Ye, J. Y.

L. Sherman, J. Y. Ye, O. Albert and 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]

Am. J. Math.

R. Kershner, "The number of circles covering a set," Am. J. Math. 61, 665-671 (1939).
[CrossRef]

Appl. Opt.

IEEE Trans. Inf. Theory

E. Vitterbo and J. Boutros, "A universal lattice code decoder for fading channels," IEEE Trans. Inf. Theory 45, 1639-1642 (1999).
[CrossRef]

J. Microsc.

L. Sherman, J. Y. Ye, O. Albert and 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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microsc. Res. Technol.

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. 67, 36-44 (2005).
[CrossRef]

Opt. Commun.

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

Opt. Eng.

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

Opt. Express

Opt. Lett.

Proc. Nat. Acad. Sci.

M. J. Booth, M. A. A. Neil, R. Ju¡skaitis and T.Wilson, "Adaptive aberration correction in a confocal microscope," Proc. Nat. Acad. Sci. 99, 5788-5792 (2002).
[CrossRef] [PubMed]

Other

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).

Z. 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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