Abstract

We propose a simple and powerful algorithm to extend the dynamic range of a Shack–Hartmann wave-front sensor. In a conventional Shack–Hartmann wave-front sensor the dynamic range is limited by the f-number of a lenslet, because the focal spot is required to remain in the area confined by the single lenslet. The sorting method proposed here eliminates such a limitation and extends the dynamic range by tagging each spot in a special sequence. Since the sorting method is a simple algorithm that does not change the measurement configuration, there is no requirement for extra hardware, multiple measurements, or complicated algorithms. We not only present the theory and a calculation example of the sorting method but also actually implement measurement of a highly aberrated wave front from nonrotational symmetric optics.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. C. Platt, R. Shack, “History and principles of Shack–Hartmann wave-front sensing,” J. Refract. Surg. 17, s573–s577 (2001).
    [PubMed]
  2. I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed., Wiley Series in Pure and Applied Optics (Wiley, 1992), pp. 367–396.
  3. J. W. Hardy, “Adaptive optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  4. C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensor,” Opt. Rev. 2, 217–220 (1995).
    [CrossRef]
  5. J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  6. J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
    [CrossRef]
  7. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  8. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  9. W. H. Southwell, “Wave-front estimation from wave-front slope measurement,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  10. J. Hermann, “Least-squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  11. G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989), Chap. 2, pp. 48–86.
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C++, 2nd ed. (Cambridge U. Press, 1993), Chap. 2, pp. 35–107.
  13. Å. Bjørk, Numerical Methods for Least Squares Problems, 1st ed. (Society for Industrial and Applied Mathematics, 1996), Chap. 1, pp. 1–36.
    [CrossRef]
  14. H. H. Barrett, K. J. Myers, Foundations of Image Science, 1st ed. (Wiley, 2003), Chap. 1, pp. 1–62.
  15. J. Pfund, N. Lindlein, J. Schwider, “Dynamic range expansion of a Shack–Hartmann sensor by using a modified unwrapping algorithm,” Opt. Lett. 23, 995–997 (1998).
    [CrossRef]
  16. T. L. Bruno, A. Wirth, A. J. Jankevics, “Applying Hartmann wavefront-sensing technology to precision optical testing of the HST correctors,” in Active and Adaptive Optical Components and Systems II,M. A. Ealey, ed., Proc. SPIE1920, 328–336 (1993).
    [CrossRef]
  17. M. C. Roggermann, T. J. Schulz, “Algorithm to increase the largest aberration that can be reconstructed from Hartmann sensor measurements,” Appl. Opt. 37, 4321–4329 (1998).
    [CrossRef]
  18. S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, J. Schwider, “Wave-front reconstruction with a Shack–Hartmann sensor with an iterative spline fitting method,” Appl. Opt. 39, 561–567 (2000).
    [CrossRef]
  19. N. Lindlein, J. Pfund, J. Schwider, “Expansion of the dynamic range of a Shack–Hartmann sensor by using astigmatic microlenses,” Opt. Eng. 39, 2220–2225 (2000).
    [CrossRef]
  20. N. Lindlein, J. Pfund, 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]
  21. G. Rousset, “Wave-front sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., Vol. 423 of NATO Advanced Science Institutes Series (Kluwer Academic, 1994), pp. 115–138.
    [CrossRef]
  22. J. C. Wyant, “Zernike polynomial,” Optical Science Center, University of Arizona, Tucson, Arizona, 1999, http://www.optics.arizona.edu/jcwyant/Zernikes/ZernikePolynomials.htm .
  23. G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak, D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 108–112.
  24. CDM Optics, Inc., Suite 2110, 4001 Discovery Drive, Boulder, Colo. 80303, http://www.cdm-optics.com .
  25. E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]

2001 (2)

B. C. Platt, R. Shack, “History and principles of Shack–Hartmann wave-front sensing,” J. Refract. Surg. 17, s573–s577 (2001).
[PubMed]

N. Lindlein, J. Pfund, 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 (2)

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, 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, J. Schwider, “Expansion of the dynamic range of a Shack–Hartmann sensor by using astigmatic microlenses,” Opt. Eng. 39, 2220–2225 (2000).
[CrossRef]

1998 (2)

1997 (1)

1995 (2)

E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef] [PubMed]

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensor,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

1994 (1)

1980 (2)

1978 (2)

R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
[CrossRef]

J. W. Hardy, “Adaptive optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977 (1)

Barrett, H. H.

H. H. Barrett, K. J. Myers, Foundations of Image Science, 1st ed. (Wiley, 2003), Chap. 1, pp. 1–62.

Bille, J. F.

Bjørk, Å.

Å. Bjørk, Numerical Methods for Least Squares Problems, 1st ed. (Society for Industrial and Applied Mathematics, 1996), Chap. 1, pp. 1–36.
[CrossRef]

Bruno, T. L.

T. L. Bruno, A. Wirth, A. J. Jankevics, “Applying Hartmann wavefront-sensing technology to precision optical testing of the HST correctors,” in Active and Adaptive Optical Components and Systems II,M. A. Ealey, ed., Proc. SPIE1920, 328–336 (1993).
[CrossRef]

Cathey, W. T.

Dainty, J. C.

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensor,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Donner, K.

Dowski, E. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C++, 2nd ed. (Cambridge U. Press, 1993), Chap. 2, pp. 35–107.

Fried, D. L.

Ghozeil, I.

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed., Wiley Series in Pure and Applied Optics (Wiley, 1992), pp. 367–396.

Goelz, S.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989), Chap. 2, pp. 48–86.

Grimm, B.

Groening, S.

Hardy, J. W.

J. W. Hardy, “Adaptive optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Hermann, J.

Jankevics, A. J.

T. L. Bruno, A. Wirth, A. J. Jankevics, “Applying Hartmann wavefront-sensing technology to precision optical testing of the HST correctors,” in Active and Adaptive Optical Components and Systems II,M. A. Ealey, ed., Proc. SPIE1920, 328–336 (1993).
[CrossRef]

Liang, J.

Lindlein, N.

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

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, J. Schwider, “Wave-front reconstruction with a Shack–Hartmann sensor with an iterative spline fitting method,” Appl. Opt. 39, 561–567 (2000).
[CrossRef]

J. Pfund, N. Lindlein, J. Schwider, “Dynamic range expansion of a Shack–Hartmann sensor by using a modified unwrapping algorithm,” Opt. Lett. 23, 995–997 (1998).
[CrossRef]

Myers, K. J.

H. H. Barrett, K. J. Myers, Foundations of Image Science, 1st ed. (Wiley, 2003), Chap. 1, pp. 1–62.

Noll, R. J.

Pfund, J.

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

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, J. Schwider, “Wave-front reconstruction with a Shack–Hartmann sensor with an iterative spline fitting method,” Appl. Opt. 39, 561–567 (2000).
[CrossRef]

J. Pfund, N. Lindlein, J. Schwider, “Dynamic range expansion of a Shack–Hartmann sensor by using a modified unwrapping algorithm,” Opt. Lett. 23, 995–997 (1998).
[CrossRef]

Platt, B. C.

B. C. Platt, R. Shack, “History and principles of Shack–Hartmann wave-front sensing,” J. Refract. Surg. 17, s573–s577 (2001).
[PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C++, 2nd ed. (Cambridge U. Press, 1993), Chap. 2, pp. 35–107.

Roggermann, M. C.

Rousset, G.

G. Rousset, “Wave-front sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., Vol. 423 of NATO Advanced Science Institutes Series (Kluwer Academic, 1994), pp. 115–138.
[CrossRef]

Schulz, T. J.

Schwider, J.

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

S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, J. Schwider, “Wave-front reconstruction with a Shack–Hartmann sensor with an iterative spline fitting method,” Appl. Opt. 39, 561–567 (2000).
[CrossRef]

J. Pfund, N. Lindlein, J. Schwider, “Dynamic range expansion of a Shack–Hartmann sensor by using a modified unwrapping algorithm,” Opt. Lett. 23, 995–997 (1998).
[CrossRef]

Shack, R.

B. C. Platt, R. Shack, “History and principles of Shack–Hartmann wave-front sensing,” J. Refract. Surg. 17, s573–s577 (2001).
[PubMed]

Sick, B.

Solomon, C. J.

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensor,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Sommargren, G. E.

G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak, D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 108–112.

Southwell, W. H.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C++, 2nd ed. (Cambridge U. Press, 1993), Chap. 2, pp. 35–107.

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989), Chap. 2, pp. 48–86.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C++, 2nd ed. (Cambridge U. Press, 1993), Chap. 2, pp. 35–107.

Williams, D. R.

Wirth, A.

T. L. Bruno, A. Wirth, A. J. Jankevics, “Applying Hartmann wavefront-sensing technology to precision optical testing of the HST correctors,” in Active and Adaptive Optical Components and Systems II,M. A. Ealey, ed., Proc. SPIE1920, 328–336 (1993).
[CrossRef]

Wooder, N. J.

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensor,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

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

J. Refract. Surg. (1)

B. C. Platt, R. Shack, “History and principles of Shack–Hartmann wave-front sensing,” J. Refract. Surg. 17, s573–s577 (2001).
[PubMed]

Opt. Eng. (2)

N. Lindlein, J. Pfund, 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, J. Pfund, 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]

Opt. Lett. (1)

Opt. Rev. (1)

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensor,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Proc. IEEE (1)

J. W. Hardy, “Adaptive optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other (10)

G. H. Golub, C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989), Chap. 2, pp. 48–86.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C++, 2nd ed. (Cambridge U. Press, 1993), Chap. 2, pp. 35–107.

Å. Bjørk, Numerical Methods for Least Squares Problems, 1st ed. (Society for Industrial and Applied Mathematics, 1996), Chap. 1, pp. 1–36.
[CrossRef]

H. H. Barrett, K. J. Myers, Foundations of Image Science, 1st ed. (Wiley, 2003), Chap. 1, pp. 1–62.

T. L. Bruno, A. Wirth, A. J. Jankevics, “Applying Hartmann wavefront-sensing technology to precision optical testing of the HST correctors,” in Active and Adaptive Optical Components and Systems II,M. A. Ealey, ed., Proc. SPIE1920, 328–336 (1993).
[CrossRef]

G. Rousset, “Wave-front sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., Vol. 423 of NATO Advanced Science Institutes Series (Kluwer Academic, 1994), pp. 115–138.
[CrossRef]

J. C. Wyant, “Zernike polynomial,” Optical Science Center, University of Arizona, Tucson, Arizona, 1999, http://www.optics.arizona.edu/jcwyant/Zernikes/ZernikePolynomials.htm .

G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak, D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 108–112.

CDM Optics, Inc., Suite 2110, 4001 Discovery Drive, Boulder, Colo. 80303, http://www.cdm-optics.com .

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed., Wiley Series in Pure and Applied Optics (Wiley, 1992), pp. 367–396.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Plot of the focal spots before sorting is performed: top, reference focal spots; bottom, aberrated focal spots in the same configuration as in the top figure. The rectangular grid indicates the subaperture bounded by the lenslet array.

Fig. 2
Fig. 2

Plot of aberrated focal spots after sorting is performed in the y position.

Fig. 3
Fig. 3

Plot of aberrated focal spots after sorting is performed in the y and x positions sequentially.

Fig. 4
Fig. 4

Spot crossing. Two adjacent focal spots cross each other.

Fig. 5
Fig. 5

Row crossing. Horizontal lines pass on the spots that are aligned on the y axis: top, spot displacement when the wave front has 24λ of spherical aberration; bottom, crossing of two adjacent rows when the wave front has 27λ of a spherical aberration.

Fig. 6
Fig. 6

Actual Shack–Hartmann wave-front sensor. The pinhole is moved out of sight. The system total track is 15 cm, which is the distance from the optics being tested to the CCD chip.

Fig. 7
Fig. 7

Spot displacements and the reconstructed wave front of a cubic phase plate. The patterns show the unique characteristics of the cubic phase plate.

Tables (1)

Tables Icon

Table 1 Calculation of the Maximum Zernike Coefficient in Wavesa

Equations (13)

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

Δ x j = - f A A j W ( x , y ) x d x d y , Δ y j = - f A A j W ( x , y ) y d x d y ,
W ( x , y ) = i = 1 i max c i B i ( x , y ) ,
Δ x j = - f l A j A j [ i = 1 i max c i B i ( x , y ) ] x d x d y = i = 1 i max [ - f l A j A j B i ( x , y ) x d x d y ] c i , Δ y j = i = 1 i max [ - f l A j A j B i ( x , y ) y d x d y ] c i .
Δ = S · c ,
c ^ = S + · Δ ,
| 1 A A j [ W ( x , y ) x ] d x d y | = w 2 f , | 1 A A j [ W ( x , y ) y ] d x d y | = w 2 f ,
( p a ) T = { α 1 , 1 , β 1 , 2 , χ 1 , 3 , δ 1 , 4 , ɛ 1 , 5 } ,
( p a ) T = { α 1 , 1 , β 1 , 2 , χ 1 , 3 ,     , φ 2 , 1 , γ 2 , 2 , η 2 , 3 , } .
( p r ) T = { A 1 , 1 , B 1 , 2 , X 1 , 3 ,     , Φ 2 , 1 , Γ 2 , 2 , H 2 , 3 , } .
p a - p r = Δ .
1 w · A { A j [ W ( x , y ) x ] d x d y - A j ± 1 [ W ( x , y ) x ] d x d y } = 1 f .
max { 1 w · A | A j [ W ( x , y ) y ] d x d y - A k [ W ( x , y ) y ] d x d y | ] = 1 f ,
P ( x , y ) 2 π λ ( x 3 + y 3 ) .

Metrics