Abstract

Wavefront sensing systems measure the slope or curvature of a surface by calculating the centroid displacement of two focal spot images. Accurately finding the centroid of each focal spot determines the measurement results. This paper studied several widely used centroid-finding techniques and observed that thresholding is the most critical factor affecting the centroid-finding accuracy. Since the focal spot image of a freeform surface usually suffers from various types of image degradation, it is difficult and sometimes impossible to set a best threshold value for the whole image. We propose an adaptive centroid-finding algorithm to tackle this problem and have experimentally proven its effectiveness in measuring freeform surfaces.

© 2013 Optical Society of America

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References

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  1. X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterization—a fresh strategy,” CIRP Ann. 56, 553–556(2007).
    [CrossRef]
  2. R. Henselmans, Non-Contact Measurement Machine for Freeform Optics (Ph.D Thesis, Technische Universiteit Eindhoven, 2009).
  3. B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).
  4. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).
  5. J. C. Wyant, “Advances in interferometric surface measurement,” Proc. SPIE 6024, 602401 (2005).
    [CrossRef]
  6. Spiricon, Inc., “Hartmann wavefront analyzer tutorial,” (2004), http://www.spiricon.com .
  7. R. Schmitt, I. Jakobs, and K. Vielhaber, “Wavefront sensor design based on a micro-mirror array for a high dynamic range measurement at a high lateral resolution,” New Optical Sensors and Measurement Systems (Springer, 2009), pp. 628–633.
  8. S. Rios and D. Lopez, “Interferometric Shack–Hartmann wavefront sensor using an array of superresolution pupil filters,” Appl. Opt. 49, 2334–2338 (2010).
  9. L. Zhao, W. Guo, X. Li, and I-M. Chen, “Reference-free Shack–Hartmann wavefront sensor,” Opt. Lett. 36, 2752–2754 (2011).
    [CrossRef]
  10. Ai-li Xia and C.-w. Ma, “An improved centroid detection method based on higher moment for Shack–Hartmann wavefront sensor,” Proc. SPIE 7850, 78501Q (2010).
    [CrossRef]
  11. A. Vyas, M. B. Roopashree, and B. R. Prasad, “Optimization of existing centroiding algorithms for Shack–Hartmann sensor,” Proceedings of the National Conference on Innovative Computational Intelligence & Security Systems (2009), pp. 400–405.
  12. A. Vyas, M. B. Roopashree, and B. R. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” Proceedings of the International Conference on Advances in Recent Technologies in Communication and Computing (2009), pp. 366–369.
    [CrossRef]
  13. Z. Jiang, S. Gong, and Y. Dai, “Monte Carlo analysis of centroid detected accuracy for wavefront sensor,” Opt. Laser Technol. 37, 541–546 (2005).
    [CrossRef]
  14. X. Yin, L. Zhao, X. Li, and Z. Fang, “Automatic centroid detection and surface measurement with a digital Shack–Hartmann wavefront sensor,” Meas. Sci. Technol. 21, 015304 (2010).
    [CrossRef]
  15. L. P. Zhao, N. Bai, and X. Li, “Asymmetrical optical lenslet array realized by spatial light modulator for measuring toroidal surfaces,” Appl. Opt. 47, 6778–6783 (2008).
    [CrossRef]
  16. Panasonic, http://industrial.panasonic.com/ww/products_e/product_cat2/ADAH000_e/ADAH000_e/index.html .
  17. HOLOEYE Photonics, http://www.holoeye.com/spatial_light_modulator_lc_2002.html .
  18. JAI, http://www.jai.com/en/products/am-1600cl .
  19. W. Guo, L. Zhao, X. Li, and I-M. Chen, “Optimization of scanning strategy of digital Shack–Hartmann wavefront sensing,” Appl. Opt. 51, 121–125 (2012).
    [CrossRef]

2012

2011

2010

Ai-li Xia and C.-w. Ma, “An improved centroid detection method based on higher moment for Shack–Hartmann wavefront sensor,” Proc. SPIE 7850, 78501Q (2010).
[CrossRef]

X. Yin, L. Zhao, X. Li, and Z. Fang, “Automatic centroid detection and surface measurement with a digital Shack–Hartmann wavefront sensor,” Meas. Sci. Technol. 21, 015304 (2010).
[CrossRef]

S. Rios and D. Lopez, “Interferometric Shack–Hartmann wavefront sensor using an array of superresolution pupil filters,” Appl. Opt. 49, 2334–2338 (2010).

2008

2007

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterization—a fresh strategy,” CIRP Ann. 56, 553–556(2007).
[CrossRef]

2005

Z. Jiang, S. Gong, and Y. Dai, “Monte Carlo analysis of centroid detected accuracy for wavefront sensor,” Opt. Laser Technol. 37, 541–546 (2005).
[CrossRef]

J. C. Wyant, “Advances in interferometric surface measurement,” Proc. SPIE 6024, 602401 (2005).
[CrossRef]

2001

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Bai, N.

Chen, I-M.

Dai, Y.

Z. Jiang, S. Gong, and Y. Dai, “Monte Carlo analysis of centroid detected accuracy for wavefront sensor,” Opt. Laser Technol. 37, 541–546 (2005).
[CrossRef]

Fang, Z.

X. Yin, L. Zhao, X. Li, and Z. Fang, “Automatic centroid detection and surface measurement with a digital Shack–Hartmann wavefront sensor,” Meas. Sci. Technol. 21, 015304 (2010).
[CrossRef]

Gong, S.

Z. Jiang, S. Gong, and Y. Dai, “Monte Carlo analysis of centroid detected accuracy for wavefront sensor,” Opt. Laser Technol. 37, 541–546 (2005).
[CrossRef]

Guo, W.

Henselmans, R.

R. Henselmans, Non-Contact Measurement Machine for Freeform Optics (Ph.D Thesis, Technische Universiteit Eindhoven, 2009).

Jakobs, I.

R. Schmitt, I. Jakobs, and K. Vielhaber, “Wavefront sensor design based on a micro-mirror array for a high dynamic range measurement at a high lateral resolution,” New Optical Sensors and Measurement Systems (Springer, 2009), pp. 628–633.

Jiang, X.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterization—a fresh strategy,” CIRP Ann. 56, 553–556(2007).
[CrossRef]

Jiang, Z.

Z. Jiang, S. Gong, and Y. Dai, “Monte Carlo analysis of centroid detected accuracy for wavefront sensor,” Opt. Laser Technol. 37, 541–546 (2005).
[CrossRef]

Li, X.

Lopez, D.

Ma, C.-w.

Ai-li Xia and C.-w. Ma, “An improved centroid detection method based on higher moment for Shack–Hartmann wavefront sensor,” Proc. SPIE 7850, 78501Q (2010).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Prasad, B. R.

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” Proceedings of the International Conference on Advances in Recent Technologies in Communication and Computing (2009), pp. 366–369.
[CrossRef]

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Optimization of existing centroiding algorithms for Shack–Hartmann sensor,” Proceedings of the National Conference on Innovative Computational Intelligence & Security Systems (2009), pp. 400–405.

Rios, S.

Roopashree, M. B.

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Optimization of existing centroiding algorithms for Shack–Hartmann sensor,” Proceedings of the National Conference on Innovative Computational Intelligence & Security Systems (2009), pp. 400–405.

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” Proceedings of the International Conference on Advances in Recent Technologies in Communication and Computing (2009), pp. 366–369.
[CrossRef]

Schmitt, R.

R. Schmitt, I. Jakobs, and K. Vielhaber, “Wavefront sensor design based on a micro-mirror array for a high dynamic range measurement at a high lateral resolution,” New Optical Sensors and Measurement Systems (Springer, 2009), pp. 628–633.

Scott, P.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterization—a fresh strategy,” CIRP Ann. 56, 553–556(2007).
[CrossRef]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Vielhaber, K.

R. Schmitt, I. Jakobs, and K. Vielhaber, “Wavefront sensor design based on a micro-mirror array for a high dynamic range measurement at a high lateral resolution,” New Optical Sensors and Measurement Systems (Springer, 2009), pp. 628–633.

Vyas, A.

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” Proceedings of the International Conference on Advances in Recent Technologies in Communication and Computing (2009), pp. 366–369.
[CrossRef]

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Optimization of existing centroiding algorithms for Shack–Hartmann sensor,” Proceedings of the National Conference on Innovative Computational Intelligence & Security Systems (2009), pp. 400–405.

Whitehouse, D.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterization—a fresh strategy,” CIRP Ann. 56, 553–556(2007).
[CrossRef]

Wyant, J. C.

J. C. Wyant, “Advances in interferometric surface measurement,” Proc. SPIE 6024, 602401 (2005).
[CrossRef]

Xia, Ai-li

Ai-li Xia and C.-w. Ma, “An improved centroid detection method based on higher moment for Shack–Hartmann wavefront sensor,” Proc. SPIE 7850, 78501Q (2010).
[CrossRef]

Yin, X.

X. Yin, L. Zhao, X. Li, and Z. Fang, “Automatic centroid detection and surface measurement with a digital Shack–Hartmann wavefront sensor,” Meas. Sci. Technol. 21, 015304 (2010).
[CrossRef]

Zhao, L.

Zhao, L. P.

Appl. Opt.

CIRP Ann.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterization—a fresh strategy,” CIRP Ann. 56, 553–556(2007).
[CrossRef]

J. Refract. Surg.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Meas. Sci. Technol.

X. Yin, L. Zhao, X. Li, and Z. Fang, “Automatic centroid detection and surface measurement with a digital Shack–Hartmann wavefront sensor,” Meas. Sci. Technol. 21, 015304 (2010).
[CrossRef]

Opt. Laser Technol.

Z. Jiang, S. Gong, and Y. Dai, “Monte Carlo analysis of centroid detected accuracy for wavefront sensor,” Opt. Laser Technol. 37, 541–546 (2005).
[CrossRef]

Opt. Lett.

Proc. SPIE

Ai-li Xia and C.-w. Ma, “An improved centroid detection method based on higher moment for Shack–Hartmann wavefront sensor,” Proc. SPIE 7850, 78501Q (2010).
[CrossRef]

J. C. Wyant, “Advances in interferometric surface measurement,” Proc. SPIE 6024, 602401 (2005).
[CrossRef]

Other

Spiricon, Inc., “Hartmann wavefront analyzer tutorial,” (2004), http://www.spiricon.com .

R. Schmitt, I. Jakobs, and K. Vielhaber, “Wavefront sensor design based on a micro-mirror array for a high dynamic range measurement at a high lateral resolution,” New Optical Sensors and Measurement Systems (Springer, 2009), pp. 628–633.

Panasonic, http://industrial.panasonic.com/ww/products_e/product_cat2/ADAH000_e/ADAH000_e/index.html .

HOLOEYE Photonics, http://www.holoeye.com/spatial_light_modulator_lc_2002.html .

JAI, http://www.jai.com/en/products/am-1600cl .

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Optimization of existing centroiding algorithms for Shack–Hartmann sensor,” Proceedings of the National Conference on Innovative Computational Intelligence & Security Systems (2009), pp. 400–405.

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” Proceedings of the International Conference on Advances in Recent Technologies in Communication and Computing (2009), pp. 366–369.
[CrossRef]

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

R. Henselmans, Non-Contact Measurement Machine for Freeform Optics (Ph.D Thesis, Technische Universiteit Eindhoven, 2009).

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Figures (18)

Fig. 1.
Fig. 1.

Image generated through the digital SHWS system with a mixture of focal spots and noisy signals.

Fig. 2.
Fig. 2.

Image generated through the SHWS system with a distorted focal spot at sample areas with large slope.

Fig. 3.
Fig. 3.

Examples of focal spot image from (a) a freeform sample and (b) a reference plane mirror.

Fig. 4.
Fig. 4.

Flowchart of iteration applied to the centroiding process.

Fig. 5.
Fig. 5.

UA3P measurement results.

Fig. 6.
Fig. 6.

Profile of toroidal samples B and C.

Fig. 7.
Fig. 7.

SHWS setup.

Fig. 8.
Fig. 8.

Experiment setup.

Fig. 9.
Fig. 9.

Focal spot image of sample A with 12×10 windows.

Fig. 10.
Fig. 10.

Focal spot image of sample B with 14×9 windows.

Fig. 11.
Fig. 11.

Focal spot image of sample C with 13×8 windows.

Fig. 12.
Fig. 12.

Measurement results at the best centroiding setting.

Fig. 13.
Fig. 13.

Reconstructed wavefront with a threshold value of 200, (a) power of 1, no subwindowing or iteration and (b) power of 15, subwindowing and iteration.

Fig. 14.
Fig. 14.

Centroid positions found with and without local thresholding at power 1 and no iteration. The threshold value is 0 when global thresholding is conducted.

Fig. 15.
Fig. 15.

Measurement results with the implementation of the adaptive centroid-finding algorithm, at power 15 and no iteration.

Fig. 16.
Fig. 16.

Error of the reconstructed wavefront of sample B obtained through the global threshold centroid-finding algorithm, and adaptive threshold centroid finding.

Fig. 17.
Fig. 17.

Error of the reconstructed wavefront of sample C obtained through global threshold centroid-finding algorithm and adaptive threshold centroid finding.

Fig. 18.
Fig. 18.

Error of the reconstructed wavefront of sample C obtained through the global threshold centroid-finding algorithm, and adaptive threshold centroid finding, in a conventional SHWS system.

Tables (3)

Tables Icon

Table 1. Specifications of Samples B and C

Tables Icon

Table 2. RMSE Values for Various Settings

Tables Icon

Table 3. RMSE Values for Various Settings of Adaptive Centroid Finding

Equations (3)

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

XC=i=1Mj=1NXiI(Xi,Yj)i=1Mj=1NI(Xi,Yj)YC=i=1Mj=1NYjI(Xi,Yj)i=1Mj=1NI(Xi,Yj).
Inew=I(Xi,Yj)α.
{Ithresh=i=1Mj=1NI(Xi,Yj)MNInew(Xi,Yj)={0,I(Xi,Yj)<=IthreshI(Xi,Yj)Ithresh,I(Xi,Yj)>IthreshXC=i=1Mj=1NXiInew(Xi,Yj)i=1Mj=1NInew(Xi,Yj),YC=i=1Mj=1NYjInew(Xi,Yj)i=1Mj=1NInew(Xi,Yj).

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