Abstract

A Shack–Hartmann wavefront sensor (SWHS) splits the incident wavefront into many subsections and transfers the distorted wavefront detection into the centroid measurement. The accuracy of the centroid measurement determines the accuracy of the SWHS. Many methods have been presented to improve the accuracy of the wavefront centroid measurement. However, most of these methods are discussed from the point of view of optics, based on the assumption that the spot intensity of the SHWS has a Gaussian distribution, which is not applicable to the digital SHWS. In this paper, we present a centroid measurement algorithm based on the adaptive thresholding and dynamic windowing method by utilizing image processing techniques for practical application of the digital SHWS in surface profile measurement. The method can detect the centroid of each focal spot precisely and robustly by eliminating the influence of various noises, such as diffraction of the digital SHWS, unevenness and instability of the light source, as well as deviation between the centroid of the focal spot and the center of the detection area. The experimental results demonstrate that the algorithm has better precision, repeatability, and stability compared with other commonly used centroid methods, such as the statistical averaging, thresholding, and windowing algorithms.

© 2009 Optical Society of America

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References

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  1. T. J. Kane, B. M. Welsh, and C. S. Gardner, “Wave front detector optimization for laser guided adaptive telescope,” Proc. SPIE 1114, 160-171 (1989).
  2. G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
    [CrossRef]
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    [CrossRef]
  4. A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).
  5. S. K. Park and S. H. Baik, “A study on a fast measuring technique of wavefront using a Shack-Hartmann sensor,” Opt. Laser Technol. 34, 687-684 (2002).
    [CrossRef]
  6. J. F. Ren, C. H. Rao, and Q. M. Li, “An adaptive threshold selection method for Hartmann-Shack wavefront sensor,” Opto-electron. Eng. 29, 1-5 (2002).
  7. S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
    [CrossRef]
  8. P. Arulmozhivarman, L. P. Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack-Hartmann wavefront sensor--a wavelet-based approach and comparison with other methods,” Optik (Jena) 117, 82-87 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. Z. L. Jiang, S. S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol. 38, 614-619 (2006).
    [CrossRef]
  13. L. P. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of SLM as diffractive optical microlens array in digital Shack Hartman wavefront sensor,” Appl. Opt. 45, 90-94 (2006).
    [CrossRef] [PubMed]

2007

S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
[CrossRef]

2006

P. Arulmozhivarman, L. P. Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack-Hartmann wavefront sensor--a wavelet-based approach and comparison with other methods,” Optik (Jena) 117, 82-87 (2006).
[CrossRef]

Z. L. Jiang, S. S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol. 38, 614-619 (2006).
[CrossRef]

L. P. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of SLM as diffractive optical microlens array in digital Shack Hartman wavefront sensor,” Appl. Opt. 45, 90-94 (2006).
[CrossRef] [PubMed]

2004

J. Ares and J. Arines, “Influence of thresholding on centroid statistics: full analytical description,” Appl. Opt. 43, 5796-5805 (2004).
[CrossRef] [PubMed]

A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).

2002

S. K. Park and S. H. Baik, “A study on a fast measuring technique of wavefront using a Shack-Hartmann sensor,” Opt. Laser Technol. 34, 687-684 (2002).
[CrossRef]

J. F. Ren, C. H. Rao, and Q. M. Li, “An adaptive threshold selection method for Hartmann-Shack wavefront sensor,” Opto-electron. Eng. 29, 1-5 (2002).

2000

1999

1997

1994

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

1989

T. J. Kane, B. M. Welsh, and C. S. Gardner, “Wave front detector optimization for laser guided adaptive telescope,” Proc. SPIE 1114, 160-171 (1989).

Ares, J.

Arines, J.

Arulmozhivarman, P.

P. Arulmozhivarman, L. P. Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack-Hartmann wavefront sensor--a wavelet-based approach and comparison with other methods,” Optik (Jena) 117, 82-87 (2006).
[CrossRef]

Asundi, A. K.

Bai, N.

Baik, S. H.

S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
[CrossRef]

S. K. Park and S. H. Baik, “A study on a fast measuring technique of wavefront using a Shack-Hartmann sensor,” Opt. Laser Technol. 34, 687-684 (2002).
[CrossRef]

Cao, G.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

Cha, B.

S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
[CrossRef]

Dai, Y.

Z. L. Jiang, S. S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol. 38, 614-619 (2006).
[CrossRef]

Fang, Z. P.

Ganesan, A. R.

P. Arulmozhivarman, L. P. Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack-Hartmann wavefront sensor--a wavelet-based approach and comparison with other methods,” Optik (Jena) 117, 82-87 (2006).
[CrossRef]

Gardner, C. S.

T. J. Kane, B. M. Welsh, and C. S. Gardner, “Wave front detector optimization for laser guided adaptive telescope,” Proc. SPIE 1114, 160-171 (1989).

Gong, S. S.

Z. L. Jiang, S. S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol. 38, 614-619 (2006).
[CrossRef]

Irwan, R.

Jiang, W.

A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).

Jiang, Z. L.

Z. L. Jiang, S. S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol. 38, 614-619 (2006).
[CrossRef]

Kane, T. J.

T. J. Kane, B. M. Welsh, and C. S. Gardner, “Wave front detector optimization for laser guided adaptive telescope,” Proc. SPIE 1114, 160-171 (1989).

Kim, C. J.

S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
[CrossRef]

Kumar, L. P.

P. Arulmozhivarman, L. P. Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack-Hartmann wavefront sensor--a wavelet-based approach and comparison with other methods,” Optik (Jena) 117, 82-87 (2006).
[CrossRef]

Lane, R. G.

Li, Q. M.

J. F. Ren, C. H. Rao, and Q. M. Li, “An adaptive threshold selection method for Hartmann-Shack wavefront sensor,” Opto-electron. Eng. 29, 1-5 (2002).

Li, X.

Ong, L. S.

Park, S. K.

S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
[CrossRef]

S. K. Park and S. H. Baik, “A study on a fast measuring technique of wavefront using a Shack-Hartmann sensor,” Opt. Laser Technol. 34, 687-684 (2002).
[CrossRef]

Rao, C.

A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).

Rao, C. H.

J. F. Ren, C. H. Rao, and Q. M. Li, “An adaptive threshold selection method for Hartmann-Shack wavefront sensor,” Opto-electron. Eng. 29, 1-5 (2002).

Ren, J. F.

J. F. Ren, C. H. Rao, and Q. M. Li, “An adaptive threshold selection method for Hartmann-Shack wavefront sensor,” Opto-electron. Eng. 29, 1-5 (2002).

Roggemann, M. C.

Sallberg, S. A.

van Dam, M. A.

Welsh, B. M.

S. A. Sallberg, B. M. Welsh, and M. C. Roggemann, “Maximum a posteriori estimation of wave-front slopes using a Shack-Hartmann wave-front sensor,” J. Opt. Soc. Am. A 14, 1347-1354 (1997).
[CrossRef]

T. J. Kane, B. M. Welsh, and C. S. Gardner, “Wave front detector optimization for laser guided adaptive telescope,” Proc. SPIE 1114, 160-171 (1989).

Yu, X.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

Zhang, A.

A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).

Zhang, Y.

A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).

Zhao, L. P.

Appl. Opt.

J. Mod. Opt.

A. Zhang, C. Rao, Y. Zhang, and W. Jiang, “Sampling error analysis of Shack-Hartmann wavefront sensor with variable subaperture pixels,” J. Mod. Opt. 51, 2267-2278 (2004).

J. Opt. Soc. Am. A

Opt. Eng.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

Opt. Laser Technol.

S. H. Baik, S. K. Park, C. J. Kim, and B. Cha, “A center detection algorithm for Shack-Hartmann sensor,” Opt. Laser Technol. 39, 262-267 (2007).
[CrossRef]

S. K. Park and S. H. Baik, “A study on a fast measuring technique of wavefront using a Shack-Hartmann sensor,” Opt. Laser Technol. 34, 687-684 (2002).
[CrossRef]

Z. L. Jiang, S. S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol. 38, 614-619 (2006).
[CrossRef]

Optik (Jena)

P. Arulmozhivarman, L. P. Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack-Hartmann wavefront sensor--a wavelet-based approach and comparison with other methods,” Optik (Jena) 117, 82-87 (2006).
[CrossRef]

Opto-electron. Eng.

J. F. Ren, C. H. Rao, and Q. M. Li, “An adaptive threshold selection method for Hartmann-Shack wavefront sensor,” Opto-electron. Eng. 29, 1-5 (2002).

Proc. SPIE

T. J. Kane, B. M. Welsh, and C. S. Gardner, “Wave front detector optimization for laser guided adaptive telescope,” Proc. SPIE 1114, 160-171 (1989).

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Figures (14)

Fig. 1
Fig. 1

Lenslet array generated with a spatial light modulator.

Fig. 2
Fig. 2

Focal spot array detected by a CCD camera with undesired diffraction spots surrounding designed focal spots.

Fig. 3
Fig. 3

Defined focal spot subareas, detected focal spot centers, and identified detection windows in the image.

Fig. 4
Fig. 4

(a) Detected centroids of focal spots. (b) Enlarged picture of one centroid.

Fig. 5
Fig. 5

Influence of the intensity level of the threshold on the precision of centroid detection.

Fig. 6
Fig. 6

Differences between the centroids detected with adjacent threshold levels.

Fig. 7
Fig. 7

Influence of the size of the detection window on the precision of centroid detection.

Fig. 8
Fig. 8

Differences between the centroids detected in detection windows with adjacent window sizes.

Fig. 9
Fig. 9

(a) Detected centroids by the averaging algorithm. (b) Enlarged picture of one centroid.

Fig. 10
Fig. 10

(a) Detected centroids by the thresholding algorithm. (b) Enlarged picture of one centroid.

Fig. 11
Fig. 11

(a) Detected centroids by the windowing algorithm. (b) Enlarged picture of one centroid.

Fig. 12
Fig. 12

Centroid shifts between the first image and subsequent images.

Fig. 13
Fig. 13

Detected centroids by the autocentroid algorithm from a spot image generated by a Hartmann wavefront sensor.

Fig. 14
Fig. 14

Detected centroids by the autocentroid algorithm from a spot image generated by the digital Shack– Hartmann wavefront sensor with strong diffraction noise.

Tables (2)

Tables Icon

Table 1 Coordinates of the Centroids Detected by Autocentroid, Averaging, Thresholding, and Windowing Algorithms Before and After the Focal Spot Subareas Shift 20 × 20 Pixels, and Their Differences

Tables Icon

Table 2 Mean and Standard Deviation of the Coordinates of Centroids detected by Autocentroid, Averaging, Thresholding, and Windowing Algorithms from 36 Images

Equations (7)

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

x c = i = 1 U j = 1 V ( I ( i , j ) I n ) H ( i , j ) · x i j i = 1 U j = 1 V ( I ( i , j ) I n ) H ( i , j ) ,
y c = i = 1 U j = 1 V ( I ( i , j ) I n ) H ( i , j ) · y i j i = 1 U j = 1 V ( I ( i , j ) I n ) H ( i , j ) .
H ( i , j ) = { 1 , I ( i , j ) I n 0 , I ( i , j ) < I n ,
Δ = a b s ( C n C n 1 ) .
Δ = a b s ( C u C u 1 ) .
S i j = a b s ( C i j C i 1 ) ,
S j = 1 N i = 1 N a b s ( C i j C i 1 ) ,

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