Abstract

The use of a Shack–Hartmann wave-front sensor as a position-sensing device is proposed and demonstrated. The coordinates of a pointlike object are determined from the modal Zernike coefficients of the wave fronts emitted by the object and detected by the sensor. The position of the luminous centroid of a moderately extended incoherent flat object can also be measured with this device. Experimental results with off-the-shelf CCD cameras and conventional relay optics as well as inexpensive diffractive microlens arrays show that axial positioning accuracies of 74 µm rms at 300 mm and angular accuracies of 4.3 µrad rms can easily be achieved.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. J. Pfund, N. Lindlein, J. Schwider, R. Burow, Th. Blümel, K.-E. Elssner, “Absolute sphericity measurement: a comparative study of the use of interferometry and a Shack–Hartmann sensor,” Opt. Lett. 23, 742–744 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. Ref. 20, p. 757.

1999

1998

G. Artzner, “Aspherical wavefront measurements: Shack–Hartmann numerical and practical experiments,” Pure Appl. Opt. 7, 435–448 (1998).
[CrossRef]

J. Pfund, N. Lindlein, J. Schwider, R. Burow, Th. Blümel, K.-E. Elssner, “Absolute sphericity measurement: a comparative study of the use of interferometry and a Shack–Hartmann sensor,” Opt. Lett. 23, 742–744 (1998).
[CrossRef]

V. V. Voitsekhovich, S. Bará, S. Ríos, E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

J. Pfund, N. Lindlein, J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37, 22–27 (1998).
[CrossRef]

1997

H. J. Tiziani, J. H. Chen, “Shack–Hartmann sensor for fast infrared wave-front testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

1996

1995

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

1994

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

C. Castellini, F. Francini, B. Tiribilli, “Hartmann test modification for measuring ophtalmic progressive lenses,” Appl. Opt. 33, 4120–4124 (1994).
[CrossRef] [PubMed]

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 wavefront sensor,” J. Opt. Soc. Am. A. 11, 1949–1957 (1994).
[CrossRef]

T. Kohno, S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

1993

G. Roblin, D. Horville, “Study of the aberration induced by a microlens array,” J. Opt. 24, 77–87 (1993).
[CrossRef]

1992

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451–466 (1992).
[CrossRef]

1990

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A. 7, 1589–1608 (1990).
[CrossRef]

1988

1986

1983

1982

1980

1975

1904

J. Hartmann, “Objectivuntersuchungen,” Z. Instrum. XXIV, 1–21, 3–47, 98–117 (1904).

Acosta, E.

V. V. Voitsekhovich, S. Bará, S. Ríos, E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Artzner, G.

G. Artzner, “Aspherical wavefront measurements: Shack–Hartmann numerical and practical experiments,” Pure Appl. Opt. 7, 435–448 (1998).
[CrossRef]

Bakut, P. A.

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Bará, S.

V. V. Voitsekhovich, S. Bará, S. Ríos, E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Bille, J. F.

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 wavefront sensor,” J. Opt. Soc. Am. A. 11, 1949–1957 (1994).
[CrossRef]

Blümel, Th.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), pp. 464–466, 767–772.

Burow, R.

Castellini, C.

Chen, J. H.

H. J. Tiziani, J. H. Chen, “Shack–Hartmann sensor for fast infrared wave-front testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

Cho, K. H.

Dainty, J. C.

L. Diaz, J. C. Dainty, “Single-pass measurements of the wave-front aberrations of the human eye by use of retinal lipofuscin autofluorescence,” Opt. Lett. 24, 61–63 (1999).
[CrossRef]

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

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

DeVore, S. L.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed, D. Malacara, ed. (Wiley, New York, 1992), Chap. 13, pp. 455–499.

Diaz, L.

Doyle, S. M.

N. S. Prasad, S. M. Doyle, M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Elssner, K.-E.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A. 7, 1589–1608 (1990).
[CrossRef]

Francini, F.

Giles, M. K.

N. S. Prasad, S. M. Doyle, M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Goelz, S.

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 wavefront sensor,” J. Opt. Soc. Am. A. 11, 1949–1957 (1994).
[CrossRef]

Grimm, B.

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 wavefront sensor,” J. Opt. Soc. Am. A. 11, 1949–1957 (1994).
[CrossRef]

Harbers, G.

Hartmann, J.

J. Hartmann, “Objectivuntersuchungen,” Z. Instrum. XXIV, 1–21, 3–47, 98–117 (1904).

Häusler, G.

Herrmann, J.

Horville, D.

G. Roblin, D. Horville, “Study of the aberration induced by a microlens array,” J. Opt. 24, 77–87 (1993).
[CrossRef]

Hutfless, J.

Jitsuno, T.

Kirakoshyants, V. E.

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Kohno, T.

T. Kohno, S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

Kunst, P. J.

Leibbrandt, G. W. R.

Liang, J.

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 wavefront sensor,” J. Opt. Soc. Am. A. 11, 1949–1957 (1994).
[CrossRef]

Liebelt, P. B.

P. B. Liebelt, An Introduction to Optimal Estimation (Addison-Wesley, Reading, Mass., 1967), pp. 135–172.

Lindlein, N.

Loginov, V. A.

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Malacara, D.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed, D. Malacara, ed. (Wiley, New York, 1992), Chap. 13, pp. 455–499.

Maul, M.

Merkle, F.

F. Merkle, “Adaptive optics,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), Chap. 26, pp. 375–390.
[CrossRef]

Moreno-Barriuso, E.

Nakai, S.

Nakatsuka, M.

Navarro, R.

Petersen, D. P.

Pfund, J.

Prasad, N. S.

N. S. Prasad, S. M. Doyle, M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A. 7, 1589–1608 (1990).
[CrossRef]

Rimmer, M. P.

Ríos, S.

V. V. Voitsekhovich, S. Bará, S. Ríos, E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Roblin, G.

G. Roblin, D. Horville, “Study of the aberration induced by a microlens array,” J. Opt. 24, 77–87 (1993).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451–466 (1992).
[CrossRef]

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A. 7, 1589–1608 (1990).
[CrossRef]

Schneider, G.

Schwider, J.

Silva, D. E.

Solomon, C. J.

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

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Southwell, W. H.

Tanaka, S.

T. Kohno, S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

Teague, M. R.

Tiribilli, B.

Tiziani, H. J.

H. J. Tiziani, J. H. Chen, “Shack–Hartmann sensor for fast infrared wave-front testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, Mass., 1991).

Voitsekhovich, V. V.

V. V. Voitsekhovich, S. Bará, S. Ríos, E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Wallner, E. P.

Wang, J. Y.

Weissmann, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), pp. 464–466, 767–772.

Wooder, N.

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

Wyant, J. C.

Yoon, G. Y.

Appl. Opt.

Comput. Electr. Eng.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451–466 (1992).
[CrossRef]

J. Mod. Opt.

H. J. Tiziani, J. H. Chen, “Shack–Hartmann sensor for fast infrared wave-front testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

J. Opt.

G. Roblin, D. Horville, “Study of the aberration induced by a microlens array,” J. Opt. 24, 77–87 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

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 wavefront sensor,” J. Opt. Soc. Am. A. 11, 1949–1957 (1994).
[CrossRef]

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A. 7, 1589–1608 (1990).
[CrossRef]

Opt. Commun.

V. V. Voitsekhovich, S. Bará, S. Ríos, E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

P. A. Bakut, V. E. Kirakoshyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Opt. Eng.

N. S. Prasad, S. M. Doyle, M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Opt. Lett.

Opt. Rev.

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

T. Kohno, S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

Pure Appl. Opt.

G. Artzner, “Aspherical wavefront measurements: Shack–Hartmann numerical and practical experiments,” Pure Appl. Opt. 7, 435–448 (1998).
[CrossRef]

Z. Instrum.

J. Hartmann, “Objectivuntersuchungen,” Z. Instrum. XXIV, 1–21, 3–47, 98–117 (1904).

Other

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, Mass., 1991).

F. Merkle, “Adaptive optics,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), Chap. 26, pp. 375–390.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), pp. 464–466, 767–772.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed, D. Malacara, ed. (Wiley, New York, 1992), Chap. 13, pp. 455–499.

P. B. Liebelt, An Introduction to Optimal Estimation (Addison-Wesley, Reading, Mass., 1967), pp. 135–172.

Ref. 20, p. 757.

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Figures (6)

Fig. 1
Fig. 1

Basic schematic of a SH setup.

Fig. 2
Fig. 2

Experimental setup: (a) sensor configuration, (b) diffractive microlens array.

Fig. 3
Fig. 3

Emitting region of the LED used as the object.

Fig. 4
Fig. 4

(a) Typical spot field (after thresholding) of the microlens array illuminated by a LED located at a z 0 = 300 mm. (b) Enlarged image of an individual focus.

Fig. 5
Fig. 5

Undersampling and undermodeling error ∊ i versus actual modal coefficient for i = 4 (defocus). The range of values of a i corresponds to a set of pointlike objects located on the optical axis from z = 1000 mm to z = 50 mm away from the sensor.

Fig. 6
Fig. 6

Calibration curves. (a) Filled circles, actual z 0 versus estimated (before calibration) z 0′ for a z displacement of 5 mm in increments of 250 µm; solid line, fit of Eq. (19a) to the experimental data. (c) Filled circles, actual x 0 versus estimated (before calibration) x 0′ for an x displacement of 1.05 mm in increments of 250 µm at z = 300 mm; solid line, fit of Eq. (19b) to the experimental data. (e) Same as (c) but for an x displacement of 120 µm in increments of 5 µm. (b), (d), (f) Residuals of the fits for (a), (c), and (d), respectively.

Equations (30)

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

xnIn-1CxIxd2x=zIn-1L A2rWr+Drd2r,
xn/z=Sn-1L Wrd2r,
Wr=i=1 aiZir/R
aˆ=ξm,
σ2=1SS Wr-Wˆr2d2r.
ξ=ATA-1AT,
z0=R2/43a4, x0=-2z0a2/R, y0=-2z0a3/R.
Ix=S0 Ix; r0d2r0,
xn=zIn-1LS0 A2r; r0Wr; r0+Drd2rd2r0,
xn/z=SL-1L Wr; rcod2r,
rco=S0r0Ir0d2r0S0 Ir0d2r0.
ai=1πR2S WrZir/Rd2r.
aˆi=n=12N ξinmn
aˆi=n=12N ξinmn,
ξin=αRξin
mn=xn-xnr/z+μn=αs/αzxn-xnr/z+μn=αs/αzxn-xnp/z-αs/αzxnr-xnp/z+μn
aˆi=αsαR/αzaˆi-airˆ,
aˆi=αsαR/αz1-κai-airAai+B.
z0=z0rz0γz0r+z0,
x0=σz0/z0x0+z0x0r/z0r,
y0=σz0/z0y0+z0y0r/z0r,
Uir=ArexpikWr
tLr=exp-ik/2fxL2+yL2,
Ux=1iλzexpikzL tLrUirexpik2z|x-r|2d2r,
Ix=UxU*x=λz-2LL Fr1, r2exp-ik/z×x·r1-r2d2r1d2r2,
Fr1, r2=Ar1Ar2expikWr1-Wr2×expikD/2|r1-r2|2;
CxIxd2x=λz-2LL Fr1, r2Cx exp-ik/z×x·r1-r2d2xd2r1d2r2.
C x expik/zxL1-xL2xdx×C expik/zyL1-yL2ydy,
 xn exp-icxdx=2π-i-nδnc
 fxδncx-x0dx=-1nc-n-1nxn fx0.

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