Abstract

An algorithm for expanding the dynamic range of Shack–Hartmann sensors is proposed. The distribution of the spot dislocations is treated with a modified unwrapping algorithm that is widely used in interferometry. The algorithm unwraps the spot dislocations and assigns the spots to their original subapertures, leading to a huge expansion of the dynamic range. For the proposed algorithm there remains a limitation on the maximum wave-front curvature instead of on the maximum wave-front slope. Examples are given that show spot fields that were wrapped four times; the measured wave front had a peak-to-valley value of 116λ.

© 1998 Optical Society of America

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References

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  1. I. Ghozeil, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 323–349.
  2. H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
    [CrossRef]
  3. T. L. Bruno, A. Wirth, and A. J. Jankevics, Proc. SPIE 1920, 328 (1993).
    [CrossRef]
  4. H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).
  5. N. H. Ching, D. Rosenfeld, and M. Braun, IEEE Trans. Image Process. 1, 355 (1992).
    [CrossRef]

1994 (1)

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
[CrossRef]

1993 (2)

T. L. Bruno, A. Wirth, and A. J. Jankevics, Proc. SPIE 1920, 328 (1993).
[CrossRef]

H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).

1992 (1)

N. H. Ching, D. Rosenfeld, and M. Braun, IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Braun, M.

N. H. Ching, D. Rosenfeld, and M. Braun, IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Bruno, T. L.

T. L. Bruno, A. Wirth, and A. J. Jankevics, Proc. SPIE 1920, 328 (1993).
[CrossRef]

Ching, N. H.

N. H. Ching, D. Rosenfeld, and M. Braun, IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
[CrossRef]

H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).

Ghozeil, I.

I. Ghozeil, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 323–349.

Jankevics, A. J.

T. L. Bruno, A. Wirth, and A. J. Jankevics, Proc. SPIE 1920, 328 (1993).
[CrossRef]

Lindlein, N.

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
[CrossRef]

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, and M. Braun, IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Schreiber, H.

H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).

Schwider, J.

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
[CrossRef]

H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
[CrossRef]

Wirth, A.

T. L. Bruno, A. Wirth, and A. J. Jankevics, Proc. SPIE 1920, 328 (1993).
[CrossRef]

Zöller, A.

H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).

IEEE Trans. Image Process. (1)

N. H. Ching, D. Rosenfeld, and M. Braun, IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Opt. Eng. (1)

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, Opt. Eng. 33, 2680 (1994).
[CrossRef]

Proc. SPIE (2)

T. L. Bruno, A. Wirth, and A. J. Jankevics, Proc. SPIE 1920, 328 (1993).
[CrossRef]

H. Schreiber, O. Falkenstörfer, J. Schwider, and A. Zöller, Proc. SPIE 1983, 716 (1993).

Other (1)

I. Ghozeil, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 323–349.

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Figures (4)

Fig. 1
Fig. 1

a, Mean wave-front slopes across the microlenses translated into lateral spot dislocations; b, spot distribution wrapped modulo P; c, numbers ni,j;s calculated by the unwrapping procedure; d, unwrapped spot distribution.

Fig. 2
Fig. 2

Wrapped spot dislocations in the x direction (σ); PVσ = 12.76 pixels.

Fig. 3
Fig. 3

Unwrapped spot dislocations in the x direction (Σ); PVΣ = 41.75 pixels.

Fig. 4
Fig. 4

Plot of the xy polynomial calculated out of the unwrapped spot dislocations; PVwf = 115.9λ; distance of the contours, ΔW = 10λ.

Equations (10)

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Wi,j=σi,j/f,
W=l=0Gm=0lalmxmyl-m.
σi,j;s,τi,j;s-P2,P2
np,1;1=0p,1;1=σp,1;1, Xp,1;1=xp,1;1.
Δσi,1;s=σi,1;s-σi1,1;s,
-P/2Δσi,1;sP/2,
ni,1;s=ni1,1 sif-P/2Δσi,1;sP/2,ni1,1 s-signΔσi,1;selse
i,1;s=σi,1;s+ni,1;s×P, Xi,1;s=xi,1;s-ni,1;s×P,
i,j;s=σi,j;s+P×ni,j;s, Ri,j;s=ri,j;s-P×ni,j;s.
W/xi,j-W/xi1,j<P/2f.

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