Abstract

One limitation of the conventional Shack–Hartmann sensor is that the spots of each microlens have to remain in their respective subapertures. We present an algorithm that assigns the spots to their reference points unequivocally even if they are situated far outside their subaperture. For this assignment a spline function is extrapolated in successive steps of the iterative algorithm. The proposed method works in a single-shot technique and does not need any aid from mechanical devices. The reconstruction of a simulated steep aspherical wave front (∼100λ/mm slope) is described as well as experimental results of the measurement of a spherical wave front with a huge peak-to-valley value (∼400λ). The performance of the method is compared with the unwrapping method, which has been published before.

© 2000 Optical Society of America

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References

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  1. I. Ghozeil, “Hartmann and other Screen Tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 10, pp. 323–349.
  2. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  3. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  4. 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]
  5. M. C. Roggemann, T. J. Schulz, “Algorithm to increase the largest aberration that can be reconstructed from Hartmann sensor measurements,” Appl. Opt. 37, 4321–4329 (1998).
    [CrossRef]
  6. 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]
  7. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
    [CrossRef] [PubMed]
  8. Å. Bjørk, Numerical Methods for Least Squares Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1996).
    [CrossRef]
  9. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).
  10. P. Dierckx, Curve and Surface Fitting with Splines (Clarendon, Oxford, UK, 1995).
  11. H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
    [CrossRef]

1998

1994

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

1987

1980

1978

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

Bjørk, Å.

Å. Bjørk, Numerical Methods for Least Squares Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1996).
[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]

Dierckx, P.

P. Dierckx, Curve and Surface Fitting with Splines (Clarendon, Oxford, UK, 1995).

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Ghozeil, I.

I. Ghozeil, “Hartmann and other Screen Tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 10, pp. 323–349.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

Greivenkamp, J. E.

Hardy, J. W.

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

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]

Lindlein, N.

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]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Pfund, J.

Roggemann, M. C.

Schulz, T. J.

Schwider, J.

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]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Southwell, W. H.

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

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]

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Opt. Lett.

Proc. IEEE

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

Other

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]

Å. Bjørk, Numerical Methods for Least Squares Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1996).
[CrossRef]

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

P. Dierckx, Curve and Surface Fitting with Splines (Clarendon, Oxford, UK, 1995).

I. Ghozeil, “Hartmann and other Screen Tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 10, pp. 323–349.

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Figures (11)

Fig. 1
Fig. 1

Scheme of the Shack–Hartmann sensor.

Fig. 2
Fig. 2

Scheme of the iterative spline fitting algorithm working on a small-example spot field of 25 spots.

Fig. 3
Fig. 3

Starting heuristic of the iterative spline fitting algorithm.

Fig. 4
Fig. 4

Spot distribution of the simulated aspherical wave front: microlens design, f = 4 mm; P = 0.15 mm.

Fig. 5
Fig. 5

Contour plot of the simulated aspherical wave front: distance between the contours, 5λ; PV value of the wavefront, W PV = 32.37λ.

Fig. 6
Fig. 6

Contour plot of the difference between the wave front reconstructed from the simulated spot distribution and the simulated aspherical wave front: distance between the contours, 0.005λ; PV value of the difference, ΔW PV = 0.073λ.

Fig. 7
Fig. 7

Contour plot of the wave front reconstructed without an extended spot assignment: distance between the contours, 2λ; PV value of the wavefront, W PV = 24.50λ.

Fig. 8
Fig. 8

Spot distribution of the diverging spherical wave front with a radius of curvature of r ≅ 9 mm: microlens design, f = 4 mm; P = 0.15 mm.

Fig. 9
Fig. 9

Contour plot of the wave front reconstructed with the iterative spline fitting method from the spot distribution shown in Fig. 9: distance between the contours, 50λ; PV value of the wave front, W PV = 406.54λ.

Fig. 10
Fig. 10

Contour plot of the difference between the wave fronts reconstructed with the iterative spline fitting method and the unwrapping method: distance between the contours, 0.01λ; PV value of the difference, ΔW PV = 0.096λ.

Fig. 11
Fig. 11

Contour plot of the wave front (Fig. 9) reconstructed without an extended spot assignment: distance between the contours 1λ; PV value of the wave front, W PV = 10.43λ.

Equations (11)

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iW=σif2+Π1σi2+Π2σi21/2=σif2+σi12+σi221/2,
A :=φ1x1φnx1φ1xmφnxm,
b1 :=Π1y1Π1ym,  b2 :=Π2y1Π2ym.
minξRnAξ-b1,  minvRnAv-b2,
Φ :=i=1n ξiφii=1n viφi,
ξ, vj=1mi=1n ξiφixj-Π1yj2+j=1mi=1n viφixj-Π2yj2.
Ni,k+1x :=x-τiτi+k-τi Ni,kx+τi+k+1-xτi+k+1-τi+1 Ni+1,kx,
Ni,1x :=1,if x[τi, τi+1)0,if x  [τi, τi+1).
F=D2λf
εerr=WPV,differenceWPV,ideal0.0028.
εerr=WPV,differenceWPV,iterative0.00025.

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