## Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method

Optics Express, Vol. 15, Issue 20, pp. 12890-12899 (2007)

http://dx.doi.org/10.1364/OE.15.012890

Acrobat PDF (755 KB)

### Abstract

Based on our previously reported annular subaperture reconstruction algorithm with Zernike annular polynomials and matrix method, we provide an experimental demonstration by testing a parabolic mirror with the complementary annular subaperture interferometric method. By comparing the results of annular subaperture method with that of the classical auto-collimation method, it is shown that the reconstruction results are in good agreement with the auto-collimation measurement. In addition, we discuss some limitations of characterizing annular subaperture measurement data with finite Zernike coefficients in our algorithm, and also show the possibility of characterizing higher spatial frequency information with adequate Zernike coefficients. It is believable that the reported method can be extended to test the surface shape of some large concave aspheric mirrors with acceptable accuracy.

© 2007 Optical Society of America

## 1. Introduction

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

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

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

08. M. Bray, “Stitching interferometer for large plano optics using a standard interferometer,” Proc. SPIE **3134**,
39–50 (1997). [CrossRef]

10. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News **14**, 38–43 (2003). [CrossRef]

11. P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, “High precision metrology of demes and aspheric optics,” Proc. SPIE **5786**, 112–121 (2005). [CrossRef]

^{TM}[12] adds asphere metrology to the SSI’s capabilities, in which the subaperture configuration is shown in Fig. 1(b). Except for the above circle subaperture method, the annular subaperture interferometric method [14–16

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

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

*et al*. in 1988, which can calculate full-aperture Zernike coefficients from the subaperture Zernike coefficients. Liu

*et al*. indicated that the overlapping areas or gaps may exist in the subaperture configuration, and overlapping will contribute to the reconstruction accuracy. As is shown in Fig. 1(c), another reduction method with successive overlapping phase maps was presented by Melozzi

*et al*. [15] in 1993. Starting from the inner phase map and subtracting the misalignment errors from its adjacent annular map, the latter is brought to coincidence to the former. This process is then repeated until the requested diameter, obtaining the total phase map of aspheric surfaces under test. F.Granados-Agustín

*et al*. [16

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

10. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News **14**, 38–43 (2003). [CrossRef]

13. S. Chen, S. Li, Y. Dai, and Z. Zheng, “Testing of large optical surfaces with subaperture stitching,” Appl. Opt. **46**, 3504–3509 (2007). [CrossRef] [PubMed]

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

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

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

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

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

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

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

*f*/2 parabolic mirror, and we compare the reconstruction results with that of the classical auto-collimation null measurement and full-aperture non-null measurement. This paper is organized as follows. In Section 2, the basic theory of annular subaperture interferometric method is described. In Section 3, we describe the experimental setup and results. In Section 4, we discuss some limitations of characterizing annular subaperture measurement data with finite Zernike coefficients, and also show the possibility of characterizing higher spatial frequency information with adequate Zernike coefficients. The conclusion is given in Sec.5.

## 2. Theory

*f*is the focal distance of the transmission sphere,

*R*

_{0}is the vertex curvature radius of aspheric surface, and

*d*is the moving distance relative to the vertex curvature measurement. When

*d*= 0, the aspheric surface will be tested at its vertex center of curvature. When

*d*> 0 , the aspheric surface will be tested with some defocus from its vertex center of curvature.

*d*, a set of simulated interferograms are shown in Fig. 3. Note that, every over-sampled interferogram can not accurately represent the difference between the aspheric surface and the reference spherical wavefront owing to the higher fringe density involved. If a series of interferograms are taken at different longitudinal position of the aspheric surface, the resolvable subaperture interferograms can be extracted from those over-sampled interferograms containing the full-aperture surface information with the “mask” method, which is used to define the interesting sub-areas within the test part. The next critical step is to combine all subaperture data together with a suitable algorithm to get a complete surface map.

*W*(P,Θ,ε

_{0}) , can be segmented into the global surface information and the local subaperture misalignment information in terms of the Zernike annular polynomials,

*ρ*, θ) is the normalized local pixel coordinate system for the

_{k}*k*th subaperture, and (P, Θ) is the normalized global coordinate system for the full-aperture.

*K*is the number of subapertures,

*L*is the number of Zernike annular polynomials,

*b*is the misalignment coefficient for the

_{ki}*k*th subaperture and

*i*th Zernike mode, and

*B*is the full-aperture coefficient for the

_{i}*i*th Zernike mode. The obscuration ratio of full-aperture and the

*k*th subaperture is

*ε*

_{0}and ε

*, respectively. The*

_{k}*Z*(

_{ki}*ρ*,

_{k}*θ*,

*ε*) is the

_{k}*i*th Zernike annular polynomial of the

*k*th subaperture, and the

*Z*(P, Θ,

_{i}*ε*

_{0}) is the

*i*th Zernike annular polynomial of the full-aperture. The Zernike polynomials are ordered according to Ref. 17 and are also adopted in the commercial interferogram reduction software MetroPro [21

21. MetroPro Manual, Version 7.4.2,2001, Zygo corporation,http://www.zygo.com

*a*is the subaperture coefficient for the

_{ki}*k*th subaperture and

*i*th Zernike mode. But since the aspheric surface itself has not changed, Eq. (1) must equal Eq. (2), giving

*B*can be calculated, which was described in detail in a published paper [20

_{i}20. X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data using Zernike annular polynomials and matrix method for testing large aspheric surfaces,” Appl. Opt. **45**, 3442–3455 (2006). [CrossRef] [PubMed]

## 3. Experimental results

*f*/2 and 130 mm parabolic mirror under test is mounted with a 5-axis adjustor. The aspheric departure from the best-fit sphere is about four microns. The annular subaperture data can be accurately extracted by the fiducial mark and “Mask” function of MetroPro software [21

21. MetroPro Manual, Version 7.4.2,2001, Zygo corporation,http://www.zygo.com

*λ*and 0.061

*λ*, respectively. The maps based on the first 36-term Zernike fitting results are relatively smooth, and mainly represent the surface shape. By comparison the Figs. 6(a) and 6(b) with Figs. 7(c) and 7(d), the surface map is highly similar, and the difference of PV and rms value is approximate 0.02

*λ*and 0.001

*λ*, respectively. In summary, the reconstruction results with annular subaperture method have good agreement with the classical auto-collimation measurement.

## 4. Discussion

*et al*. [22

22. D. Dutton, A. Cornejo, and M. Latta, “A semiautomatic method for interpreting shearing interferograms,” Appl. Opt. **7**, 125–132 (1968). [CrossRef] [PubMed]

23. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. **14**, 142–150 (1975). [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

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

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

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

04. | J. E. Negro, “Subaperture optical system testing,” Appl. Opt. |

05. | C. R. De Hainaut and A. Erteza, “Numerical processing of dynamic subaperture testing measurements,” Appl. Opt. |

06. | G. N. Lawrence and R. D. Day, “Interferometric characterization of full spheres: Data reduction techniques,” Appl. Opt. |

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

08. | M. Bray, “Stitching interferometer for large plano optics using a standard interferometer,” Proc. SPIE |

09. | T. Hänsel, A. Nickel, and A. Schindler, “Stitching interferometry of aspherical surfaces,” Proc. SPIE |

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

11. | P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, “High precision metrology of demes and aspheric optics,” Proc. SPIE |

12. | M. Tricard and P. E. Murphy, “Subaperture stitching for large aspheric surfaces,” in |

13. | S. Chen, S. Li, Y. Dai, and Z. Zheng, “Testing of large optical surfaces with subaperture stitching,” Appl. Opt. |

14. | Y. M. Liu, G. Lawrence, and C. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl.Opt. |

15. | M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using mulitiple annular interferograms,” OptEng. |

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

17. | J. C. Wyant and K. Creath, |

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

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

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

21. | MetroPro Manual, Version 7.4.2,2001, Zygo corporation,http://www.zygo.com |

22. | D. Dutton, A. Cornejo, and M. Latta, “A semiautomatic method for interpreting shearing interferograms,” Appl. Opt. |

23. | M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” 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: July 24, 2007

Revised Manuscript: August 30, 2007

Manuscript Accepted: September 2, 2007

Published: September 21, 2007

**Citation**

Xi Hou, Fan Wu, Li Yang, and Qiang Chen, "Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method," Opt. Express **15**, 12890-12899 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12890

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### References

- C. J. Kim, "Polynomial fit of interferograms," Appl. Opt. 21, 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, 468-470 (1983). [CrossRef] [PubMed]
- J. E. Negro, "Subaperture optical system testing," Appl. Opt. 23, 1921-1930 (1984). [CrossRef] [PubMed]
- C. R. De Hainaut and A. Erteza, "Numerical processing of dynamic subaperture testing measurements," Appl. Opt. 25, 503-509 (1986). [CrossRef] [PubMed]
- G. N. Lawrence and R. D. Day, "Interferometric characterization of full spheres: Data reduction techniques," Appl. Opt. 26, 4875-4882 (1987). [CrossRef] [PubMed]
- M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994). [CrossRef]
- M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," Proc. SPIE 3134, 39-50 (1997). [CrossRef]
- T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," Proc. SPIE 4449, 265-273 (2001). [CrossRef]
- P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photonics News 14, 38-43 (2003). [CrossRef]
- P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," Proc. SPIE 5786,112-121 (2005). [CrossRef]
- M. Tricard and P. E. Murphy, "Subaperture stitching for large aspheric surfaces," in Talk for NASA Tech Day 2006.
- S. Chen, S. Li, Y. Dai, and Z. Zheng, "Testing of large optical surfaces with subaperture stitching," Appl. Opt. 46, 3504-3509 (2007). [CrossRef] [PubMed]
- Y. M. Liu, G. Lawrence, and C. Koliopoulos, "Subaperture testing of aspheres with annular zones," Appl. Opt. 27, 4504-4513 (1988). [CrossRef] [PubMed]
- M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using mulitiple annular interferograms," Opt. Eng. 32, 1073-1079 (1993).
- F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004). [CrossRef]
- J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, of Applied Optics and Optical Engineering Series (Academic, 1992), Vol. 11 p. 28.
- X. Hou, F. Wu, L. Yang, and Q. Chen, "Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials," Appl. Opt. 45, 8893-8901 (2006). [CrossRef] [PubMed]
- V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75 (1981). [CrossRef]
- X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, "Full-aperture wavefront reconstruction from annular subaperture interferometric data using Zernike annular polynomials and matrix method for testing large aspheric surfaces," Appl. Opt. 45, 3442-3455 (2006). [CrossRef] [PubMed]
- MetroPro Manual, Version 7.4.2,2001, Zygo corporation,http://www.zygo.com
- D. Dutton, A. Cornejo, and M. Latta, "A semiautomatic method for interpreting shearing interferograms," Appl. Opt. 7, 125-132 (1968). [CrossRef] [PubMed]
- M. P. Rimmer and J. C. Wyant, "Evaluation of large aberrations using a lateral-shear interferometer having variable shear," Appl. Opt. 14, 142-150 (1975). [PubMed]

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