Abstract

We present an algorithm to extend significantly the dynamic range of a Shack-Hartmann wavefront sensor. With this method, the recorded Shack-Hartmann spots are not constrained to stay in the field of view of their lenslet. The proposed algorithm is computationally effective, robust to a high level of noise on the measured centroid positions and also to missing centroid values. The principle is closely related to the description of wavefronts using Zernike polynomials, which makes optimization for a given sensor and application achievable thanks to numerical simulation. These features make it useful for the measurements of highly aberrated eyes.

© 2009 Optical Society of America

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References

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  1. J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222,81-92 (2003).
    [CrossRef]
  2. A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, "Linearity of the pyramid wavefront sensor," Opt. Express 14,11925-11934 (2006).
    [CrossRef] [PubMed]
  3. D. Malacara, Optical Shop Testing (Wiley-Interscience,2007).
    [CrossRef]
  4. R. Navarro and E. Moreno-Barriuso, "Laser ray-tracing method for optical testing," Opt. Lett. 24,951-953 (1999).
    [CrossRef]
  5. N. Lindlein, J. Pfund, and J. Schwider, "Algorithm for expanding the dynamic range of a Shack-Hartmann sensor by using a spatial light modulator array," Opt. Eng. 40,837-840 (2001).
    [CrossRef]
  6. G. Yoon, S. Pantanelli, and L. J. Nagy, "Large-dynamic-range Shack-Hartmann wavefront sensor for highly aberrated eyes," J. Biomed. Opt. 11,030502-030504 (2006).
    [CrossRef]
  7. L. E. Schmutz and B. M. Levine, "Hartmann sensors detect optical fabrication errors," Laser Focus World 32, (1996).
  8. X. Levecq and S. Bucourt, "Method and device for analysing a highly dynamic wavefront," US patent 6,750,957B1 (2004)
  9. N. Lindlein, J. Pfund, and J. Schwider, "Expansion of the dynamic range of a Shack-Hartmann sensor by using astigmatic microlenses," Opt. Eng. 39,2220-2225 (2000).
    [CrossRef]
  10. N. Lindlein and J. Pfund, "Experimental results for expanding the dynamic range of a Shack-Hartmann sensor using astigmatic microlenses," Opt. Eng. 41,529-533 (2002).
    [CrossRef]
  11. J. Pfund, N. Lindlein, and J. Schwider, "Dynamic range expansion of a Shack-Hartmann sensor by use of a modified unwrapping algorithm," Opt. Lett. 23,995-997 (1998).
    [CrossRef]
  12. S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, "Wave-Front Reconstruction with a Shack-Hartmann Sensor with an Iterative Spline Fitting Method," Appl. Opt. 39,561-567 (2000).
    [CrossRef]
  13. D. G. Smith and J. E. Greivenkamp Schwider, "Generalized method for sorting Shack-Hartmann spot patterns using local similarity," Appl. Opt. 47,4548-4554 (2008).
    [CrossRef] [PubMed]
  14. L. Lundstrm and P. Unsbo, "Unwrapping Hartmann-Shack images from highly aberrated eyes using an iterative B-spline based extrapolation method," Optom. Vis. Sci. 81,383-388 (2004).
    [CrossRef]
  15. N. Lindlein and J. Pfund, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215,285-288 (2003).
    [CrossRef]
  16. M. C. Roggemann and T. J. Schulz, "Algorithm to increase the largest aberration that can be reconstructed from Hartmann sensor measurements," Appl. Opt. 37,4321-4329(1998).
    [CrossRef]
  17. R. K. Tyson, Principles of Adaptive Optics (Academic Press, 1998).
  18. L. Diaz-Santana, G. Walker, and S. Bará, "Sampling geometries for ocular aberrometry: A model for evaluation of performance," J. Opt. Soc. Am. A 13,8801-8818 (2005).
  19. S. Bará, "Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers," J. Opt. Soc. Am. A 24,3700-3707 (2007).
    [CrossRef]
  20. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).
  21. N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
    [CrossRef] [PubMed]
  22. S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
    [CrossRef] [PubMed]

2008 (1)

2007 (2)

S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
[CrossRef] [PubMed]

S. Bará, "Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers," J. Opt. Soc. Am. A 24,3700-3707 (2007).
[CrossRef]

2006 (2)

A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, "Linearity of the pyramid wavefront sensor," Opt. Express 14,11925-11934 (2006).
[CrossRef] [PubMed]

G. Yoon, S. Pantanelli, and L. J. Nagy, "Large-dynamic-range Shack-Hartmann wavefront sensor for highly aberrated eyes," J. Biomed. Opt. 11,030502-030504 (2006).
[CrossRef]

2005 (1)

L. Diaz-Santana, G. Walker, and S. Bará, "Sampling geometries for ocular aberrometry: A model for evaluation of performance," J. Opt. Soc. Am. A 13,8801-8818 (2005).

2004 (1)

L. Lundstrm and P. Unsbo, "Unwrapping Hartmann-Shack images from highly aberrated eyes using an iterative B-spline based extrapolation method," Optom. Vis. Sci. 81,383-388 (2004).
[CrossRef]

2003 (2)

N. Lindlein and J. Pfund, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215,285-288 (2003).
[CrossRef]

J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222,81-92 (2003).
[CrossRef]

2002 (2)

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

N. Lindlein and J. Pfund, "Experimental results for expanding the dynamic range of a Shack-Hartmann sensor using astigmatic microlenses," Opt. Eng. 41,529-533 (2002).
[CrossRef]

2001 (1)

N. Lindlein, J. Pfund, and J. Schwider, "Algorithm for expanding the dynamic range of a Shack-Hartmann sensor by using a spatial light modulator array," Opt. Eng. 40,837-840 (2001).
[CrossRef]

2000 (3)

N. Lindlein, J. Pfund, and J. Schwider, "Expansion of the dynamic range of a Shack-Hartmann sensor by using astigmatic microlenses," Opt. Eng. 39,2220-2225 (2000).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, "Wave-Front Reconstruction with a Shack-Hartmann Sensor with an Iterative Spline Fitting Method," Appl. Opt. 39,561-567 (2000).
[CrossRef]

1999 (1)

1998 (2)

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).

Bará, S.

S. Bará, "Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers," J. Opt. Soc. Am. A 24,3700-3707 (2007).
[CrossRef]

L. Diaz-Santana, G. Walker, and S. Bará, "Sampling geometries for ocular aberrometry: A model for evaluation of performance," J. Opt. Soc. Am. A 13,8801-8818 (2005).

Burvall, A.

Chamot, S. R.

Dainty, C.

Daly, E.

Diaz-Santana, L.

L. Diaz-Santana, G. Walker, and S. Bará, "Sampling geometries for ocular aberrometry: A model for evaluation of performance," J. Opt. Soc. Am. A 13,8801-8818 (2005).

Donner, K.

Fujikado, T.

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

Greivenkamp Schwider, J. E.

Groening, S.

Hirohara, Y.

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

Jeong, T.M.

S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
[CrossRef] [PubMed]

Kuroda, T.

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

Lindlein, N.

N. Lindlein and J. Pfund, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215,285-288 (2003).
[CrossRef]

N. Lindlein and J. Pfund, "Experimental results for expanding the dynamic range of a Shack-Hartmann sensor using astigmatic microlenses," Opt. Eng. 41,529-533 (2002).
[CrossRef]

N. Lindlein, J. Pfund, and J. Schwider, "Algorithm for expanding the dynamic range of a Shack-Hartmann sensor by using a spatial light modulator array," Opt. Eng. 40,837-840 (2001).
[CrossRef]

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, "Wave-Front Reconstruction with a Shack-Hartmann Sensor with an Iterative Spline Fitting Method," Appl. Opt. 39,561-567 (2000).
[CrossRef]

N. Lindlein, J. Pfund, and J. Schwider, "Expansion of the dynamic range of a Shack-Hartmann sensor by using astigmatic microlenses," Opt. Eng. 39,2220-2225 (2000).
[CrossRef]

J. Pfund, N. Lindlein, and J. Schwider, "Dynamic range expansion of a Shack-Hartmann sensor by use of a modified unwrapping algorithm," Opt. Lett. 23,995-997 (1998).
[CrossRef]

Lundstrm, L.

L. Lundstrm and P. Unsbo, "Unwrapping Hartmann-Shack images from highly aberrated eyes using an iterative B-spline based extrapolation method," Optom. Vis. Sci. 81,383-388 (2004).
[CrossRef]

MacRae, S.

S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
[CrossRef] [PubMed]

Maeda, N.

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

Mihashi, T.

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

Moreno-Barriuso, E.

Nagy, L. J.

G. Yoon, S. Pantanelli, and L. J. Nagy, "Large-dynamic-range Shack-Hartmann wavefront sensor for highly aberrated eyes," J. Biomed. Opt. 11,030502-030504 (2006).
[CrossRef]

Navarro, R.

Nishida, K.

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

Pantanelli, S.

S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
[CrossRef] [PubMed]

G. Yoon, S. Pantanelli, and L. J. Nagy, "Large-dynamic-range Shack-Hartmann wavefront sensor for highly aberrated eyes," J. Biomed. Opt. 11,030502-030504 (2006).
[CrossRef]

Pfund, J.

N. Lindlein and J. Pfund, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215,285-288 (2003).
[CrossRef]

N. Lindlein and J. Pfund, "Experimental results for expanding the dynamic range of a Shack-Hartmann sensor using astigmatic microlenses," Opt. Eng. 41,529-533 (2002).
[CrossRef]

N. Lindlein, J. Pfund, and J. Schwider, "Algorithm for expanding the dynamic range of a Shack-Hartmann sensor by using a spatial light modulator array," Opt. Eng. 40,837-840 (2001).
[CrossRef]

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, "Wave-Front Reconstruction with a Shack-Hartmann Sensor with an Iterative Spline Fitting Method," Appl. Opt. 39,561-567 (2000).
[CrossRef]

N. Lindlein, J. Pfund, and J. Schwider, "Expansion of the dynamic range of a Shack-Hartmann sensor by using astigmatic microlenses," Opt. Eng. 39,2220-2225 (2000).
[CrossRef]

J. Pfund, N. Lindlein, and J. Schwider, "Dynamic range expansion of a Shack-Hartmann sensor by use of a modified unwrapping algorithm," Opt. Lett. 23,995-997 (1998).
[CrossRef]

Primot, J.

J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222,81-92 (2003).
[CrossRef]

Roggemann, M. C.

Schulz, T. J.

Schwider, J.

N. Lindlein, J. Pfund, and J. Schwider, "Algorithm for expanding the dynamic range of a Shack-Hartmann sensor by using a spatial light modulator array," Opt. Eng. 40,837-840 (2001).
[CrossRef]

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, "Wave-Front Reconstruction with a Shack-Hartmann Sensor with an Iterative Spline Fitting Method," Appl. Opt. 39,561-567 (2000).
[CrossRef]

N. Lindlein, J. Pfund, and J. Schwider, "Expansion of the dynamic range of a Shack-Hartmann sensor by using astigmatic microlenses," Opt. Eng. 39,2220-2225 (2000).
[CrossRef]

J. Pfund, N. Lindlein, and J. Schwider, "Dynamic range expansion of a Shack-Hartmann sensor by use of a modified unwrapping algorithm," Opt. Lett. 23,995-997 (1998).
[CrossRef]

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).

Sick, B.

Smith, D. G.

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).

Unsbo, P.

L. Lundstrm and P. Unsbo, "Unwrapping Hartmann-Shack images from highly aberrated eyes using an iterative B-spline based extrapolation method," Optom. Vis. Sci. 81,383-388 (2004).
[CrossRef]

Walker, G.

L. Diaz-Santana, G. Walker, and S. Bará, "Sampling geometries for ocular aberrometry: A model for evaluation of performance," J. Opt. Soc. Am. A 13,8801-8818 (2005).

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).

Yoon, G.

S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
[CrossRef] [PubMed]

G. Yoon, S. Pantanelli, and L. J. Nagy, "Large-dynamic-range Shack-Hartmann wavefront sensor for highly aberrated eyes," J. Biomed. Opt. 11,030502-030504 (2006).
[CrossRef]

Appl. Opt. (3)

J. Biomed. Opt. (1)

G. Yoon, S. Pantanelli, and L. J. Nagy, "Large-dynamic-range Shack-Hartmann wavefront sensor for highly aberrated eyes," J. Biomed. Opt. 11,030502-030504 (2006).
[CrossRef]

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

L. Diaz-Santana, G. Walker, and S. Bará, "Sampling geometries for ocular aberrometry: A model for evaluation of performance," J. Opt. Soc. Am. A 13,8801-8818 (2005).

S. Bará, "Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers," J. Opt. Soc. Am. A 24,3700-3707 (2007).
[CrossRef]

J. Ref. Surg. (1)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, "Standards for reporting optical aberrations of eyes," J. Ref. Surg. 18,652-660 (2000).

Ophthalmology (2)

N. Maeda, T. Fujikado, T. Kuroda, T. Mihashi, Y. Hirohara, and K. Nishida, "Wavefront aberrations measured with Hartmann-Shack sensor in patients with keratoconus," Ophthalmology 109,1996-2003 (2002).
[CrossRef] [PubMed]

S. Pantanelli, S. MacRae, T.M. Jeong, G. Yoon, "Characterizing the wave aberration in eyes with keratoconus or penetrating keratoplasty using a high-dynamic range wavefront sensor," Ophthalmology 114,2013-2021 (2007).
[CrossRef] [PubMed]

Opt. Commun. (2)

N. Lindlein and J. Pfund, "Phase retrieval by demodulation of a Hartmann-Shack sensor," Opt. Commun. 215,285-288 (2003).
[CrossRef]

J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222,81-92 (2003).
[CrossRef]

Opt. Eng. (3)

N. Lindlein, J. Pfund, and J. Schwider, "Algorithm for expanding the dynamic range of a Shack-Hartmann sensor by using a spatial light modulator array," Opt. Eng. 40,837-840 (2001).
[CrossRef]

N. Lindlein, J. Pfund, and J. Schwider, "Expansion of the dynamic range of a Shack-Hartmann sensor by using astigmatic microlenses," Opt. Eng. 39,2220-2225 (2000).
[CrossRef]

N. Lindlein and J. Pfund, "Experimental results for expanding the dynamic range of a Shack-Hartmann sensor using astigmatic microlenses," Opt. Eng. 41,529-533 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optom. Vis. Sci. (1)

L. Lundstrm and P. Unsbo, "Unwrapping Hartmann-Shack images from highly aberrated eyes using an iterative B-spline based extrapolation method," Optom. Vis. Sci. 81,383-388 (2004).
[CrossRef]

Other (4)

D. Malacara, Optical Shop Testing (Wiley-Interscience,2007).
[CrossRef]

L. E. Schmutz and B. M. Levine, "Hartmann sensors detect optical fabrication errors," Laser Focus World 32, (1996).

X. Levecq and S. Bucourt, "Method and device for analysing a highly dynamic wavefront," US patent 6,750,957B1 (2004)

R. K. Tyson, Principles of Adaptive Optics (Academic Press, 1998).

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Figures (3)

Fig. 1.
Fig. 1.

Steps for the extrapolation of the centroid positions across the measured pupil.

Fig. 2.
Fig. 2.

Successful pairing of the Shack-Hartmann spots (o) with their reference counterparts (x), for an astigmatic wavefront and a probing beam located 2.25 mm superior to the pupil centre. Left: double-pass measurement with an ophthalmic lens. Right: simulation.

Fig. 3.
Fig. 3.

Dynamic range of the aberrometer of Table 1, with and without our algorithm, for combinations of astigmatism z 2,-2 (vertical axis) and various higher order aberrations (horizontal axis).

Tables (1)

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Table 1. Parameter of the aberrometer

Equations (1)

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X˜=Xref+A×Z0

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