## Algorithm and experiment of whole-aperture wavefront reconstruction from annular subaperture Hartmann–Shack gradient data

Optics Express, Vol. 18, Issue 13, pp. 13431-13443 (2010)

http://dx.doi.org/10.1364/OE.18.013431

Acrobat PDF (1343 KB)

### Abstract

Abstract: A new method is proposed for testing a rotationally symmetric aspheric surface with several annular subapertures based on a Hartmann–Shack sensor. In consideration of the limited sampling of Hartmann–Shack subapertures in the matching annular subaperture, a new algorithm for whole-aperture wavefront reconstruction from annular subaperture Hartmann–Shack gradient data is established. The algorithm separates the tip, tilt, and defocus misalignments for each annular subaperture by introducing annular Zernike polynomials. The performance of the algorithm is evaluated for different annular subaperture configurations, and the sensitivity of the algorithm to the detector error of the wavefront gradient is analyzed. The algorithm is verified by the experimental results.

© 2010 OSA

## 1. Introduction

1. C. J. Kim, “Polynomial fit of interferograms,” Appl. Opt. **21**(24), 4521–4525 (1982). [CrossRef] [PubMed]

15. S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, and S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express **16**(7), 4760–4765 (2008). [CrossRef] [PubMed]

5. Y.-M. Liu, G. N. Lawrance, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. **27**(21), 4504–4513 (1988). [CrossRef] [PubMed]

5. Y.-M. Liu, G. N. Lawrance, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. **27**(21), 4504–4513 (1988). [CrossRef] [PubMed]

11. F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. **11**, 82–86 (2004). [CrossRef]

*et al.*[5

5. Y.-M. Liu, G. N. Lawrance, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. **27**(21), 4504–4513 (1988). [CrossRef] [PubMed]

*et al.*[5

**27**(21), 4504–4513 (1988). [CrossRef] [PubMed]

*et al.*[6

6. X. Hou, F. Wu, L. Yang, S. B. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. **45**(15), 3442–3455 (2006). [CrossRef] [PubMed]

7. X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express **15**(20), 12890–12899 (2007). [CrossRef] [PubMed]

*et al.*[8

8. M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. **32**(5), 1073–1079 (1993). [CrossRef]

*et al.,*Granados-Agustín

*et al.*[11

11. F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. **11**, 82–86 (2004). [CrossRef]

12. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. **33**(2), 608–613 (1994). [CrossRef]

*et al.*[16] presented a rectangle subaperture stitching method using a Hartmann–Shack sensor for measuring flat surfaces, and the method was successfully applied to the measurement of nanotopographic features on silicon wafers with high speed.

*et al.*[5

**27**(21), 4504–4513 (1988). [CrossRef] [PubMed]

*et al.*[17

17. T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE **4809**, 208–216 (2002). [CrossRef]

## 2. The whole-aperture wavefront reconstruction algorithm

*K*is the number of the annular subapertures,

18. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. **71**(1), 75–85 (1981). [CrossRef]

19. X. Hou, F. Wu, L. Yang, and Q. Chen, “Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials,” Appl. Opt. **45**(35), 8893–8901 (2006). [CrossRef] [PubMed]

*W*including annular subaperture misalignments can be represented as a linear combination of annular Zernike polynomials as shown in Eq. (5),where

*L*is the number of the annular Zernike modes,

*M*is the number of the annular subapertures,

*σ*is the sum area of the Hartmann–Shack subaperture in a global normalized coordinate system, and

*S*is the sum of the valid Hartmann–Shack subapertures in all matching annular subapertures as shown in Eq. (9), and

*G*can be shown as Eq. (10) and Eq. (11).

## 3. Simulation analysis

*τ*is the noise-signal ratio,

*δ*is the standard deviation of the added noise gradient data, and

*S*is the standard deviation of the signal gradient data.

- 1. The original whole-aperture wavefront with a central obstruction of 0.2 is generated with a series of Zernike polynomials.
- 2. The whole-aperture wavefront is divided into three concentric annular subapertures, as shown in Fig. 2.
- 3. Each annular aperture is measured with different misalignments of piston, tip, tilt, and defocus, and then the gradient data of all the annular subapertures is added with noise of different levels.
- 4. The whole-aperture wavefront is reconstructed from the gradient data by the algorithm established in Section 2. Whole-aperture annular Zernike coefficients can be obtained.
- 5. The whole-aperture wavefront is generated with whole-aperture annular Zernike coefficients.
- 6. The reconstructed whole-aperture wavefront is compared with the original whole-aperture wavefront, and the residual wavefront between them is calculated.

## 4. Experiment verification

### 4.1 Results of measuring spherical surface

### 4.2 Results of testing parabolic surface

## 5. Conclusion

## Acknowledgements

## References and links

1. | C. J. Kim, “Polynomial fit of interferograms,” Appl. Opt. |

2. | J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE |

3. | W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. |

4. | T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE |

5. | Y.-M. Liu, G. N. Lawrance, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. |

6. | X. Hou, F. Wu, L. Yang, S. B. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. |

7. | X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express |

8. | M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. |

9. | D. Malacara, M. Servin, A. Morales, and Z. Maracara, “Aspherical wavefront testing with several defusing steps,” in International Conference on Optical Fabrication and Testing, T. Kasai,ed., Proc.SPIE 2576,190–192 (1995). |

10. | M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, “Method of testing aspherical optical surfaces with an interferometer,” US Patent 5416586 (May.16, 1995). |

11. | F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. |

12. | M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. |

13. | P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News |

14. | P. Murphy, G. Devries, C. Brophy, and G. Forbes, “Stitching of near-nulled subaperture measurements,” US Patent 2009/0251702 A1 (October 8, 2009). |

15. | S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, and S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express |

16. | D. R. Neal, R. R. Rammage, D. J. Armstrong, W. T. Turner, and J. D. Mansell, “Apparatus and method for evaluating a target larger than a measuring aperture of a sensor,” US Patent 6184974 B1 (February 6, 2001). |

17. | T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE |

18. | V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. |

19. | X. Hou, F. Wu, L. Yang, and Q. Chen, “Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials,” Appl. Opt. |

**OCIS Codes**

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(220.1250) Optical design and fabrication : Aspherics

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 8, 2010

Revised Manuscript: May 19, 2010

Manuscript Accepted: May 23, 2010

Published: June 8, 2010

**Citation**

Hongyan Xu, Hao Xian, and Yudong Zhang, "Algorithm and experiment of whole-aperture wavefront reconstruction from annular subaperture Hartmann–Shack gradient data," Opt. Express **18**, 13431-13443 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13431

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### References

- C. J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21(24), 4521–4525 (1982). [CrossRef] [PubMed]
- J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).
- W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8(9), 468–470 (1983). [CrossRef] [PubMed]
- T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).
- Y.-M. Liu, G. N. Lawrance, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27(21), 4504–4513 (1988). [CrossRef] [PubMed]
- X. Hou, F. Wu, L. Yang, S. B. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. 45(15), 3442–3455 (2006). [CrossRef] [PubMed]
- X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007). [CrossRef] [PubMed]
- M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993). [CrossRef]
- D. Malacara, M. Servin, A. Morales, and Z. Maracara, “Aspherical wavefront testing with several defusing steps,” in International Conference on Optical Fabrication and Testing, T. Kasai,ed., Proc.SPIE 2576,190–192 (1995).
- M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, “Method of testing aspherical optical surfaces with an interferometer,” US Patent 5416586 (May.16, 1995).
- F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004). [CrossRef]
- M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994). [CrossRef]
- P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003). [CrossRef]
- P. Murphy, G. Devries, C. Brophy, and G. Forbes, “Stitching of near-nulled subaperture measurements,” US Patent 2009/0251702 A1 (October 8, 2009).
- S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, and S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express 16(7), 4760–4765 (2008). [CrossRef] [PubMed]
- D. R. Neal, R. R. Rammage, D. J. Armstrong, W. T. Turner, and J. D. Mansell, “Apparatus and method for evaluating a target larger than a measuring aperture of a sensor,” US Patent 6184974 B1 (February 6, 2001).
- T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002). [CrossRef]
- V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981). [CrossRef]
- X. Hou, F. Wu, L. Yang, and Q. Chen, “Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials,” Appl. Opt. 45(35), 8893–8901 (2006). [CrossRef] [PubMed]

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