Abstract

Stitching interferometry is performed by collecting interferometric data from overlapped sub-apertures and stitching these data together to provide a full surface map. The propagation of the systematic error in the measured subset data is one of the main error sources in stitching interferometry for accurate reconstruction of the surface topography. In this work, we propose, using the redundancy of the captured subset data, two types of two-dimensional (2D) self-calibration stitching algorithms to overcome this issue by in situ estimating the repeatable high-order additive systematic errors, especially for the application of measuring X-ray mirrors. The first algorithm, called CS short for “Calibrate, and then Stitch”, calibrates the high-order terms of the reference by minimizing the de-tilted discrepancies of the overlapped subsets and then stitches the reference-subtracted subsets. The second algorithm, called SC short for “Stitch, and then Calibrate”, stitches a temporarily result and then calibrates the reference from the de-tilted discrepancies of the measured subsets and the temporarily stitched result. In the implementation of 2D scans in $x$- and $y$-directions, step randomization is introduced to generate nonuniformly spaced subsets which can diminish the periodic stitching errors commonly observed in evenly spaced subsets. The regularization on low-order terms enables a highly flexible option to add the curvature and twist acquired by another system. Both numerical simulations and experiments are carried out to verify the proposed method. All the results indicate that 2D high-order repeatable additive systematic errors can be retrieved from the 2D redundant overlapped data in stitching interferometry.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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  5. M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  22. M. Vannoni and G. Molesini, “Absolute planarity with three-flat test: an iterative approach with zernike polynomials,” Opt. Express 16(1), 340–354 (2008).
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  23. L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
    [Crossref]
  24. D. C.-L. Fong and M. Saunders, “Lsmr: An iterative algorithm for sparse least-squares problems,” SIAM J. on Sci. Comput. 33(5), 2950–2971 (2011).
    [Crossref]
  25. D. C.-L. Fong and M. Saunders, “LSMR: Sparse Equations and Least Squares,” http://web.stanford.edu/group/SOL/software/lsmr/ .

2020 (1)

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
[Crossref]

2019 (2)

A. Vivo, R. Barrett, and F. Perrin, “Stitching techniques for measuring x-ray synchrotron mirror topography,” Rev. Sci. Instrum. 90(2), 021710 (2019).
[Crossref]

F. Polack, M. Thomasset, S. Brochet, and D. Dennetiere, “Surface shape determination with a stitching michelson interferometer and accuracy evaluation,” Rev. Sci. Instrum. 90(2), 021708 (2019).
[Crossref]

2018 (3)

2016 (1)

A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instrum. 87(5), 051908 (2016).
[Crossref]

2015 (1)

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

2013 (1)

F. Polack and M. Thomasset, “Determination and compensation of the “reference surface” from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res., Sect. A 710, 67–71 (2013). The 4th international workshop on Metrology for X-ray Optics, Mirror Design, and Fabrication.
[Crossref]

2012 (2)

S. Chen, Y. Dai, S. Li, X. Peng, and J. Wang, “Error reductions for stitching test of large optical flats,” Opt. Laser Technol. 44(5), 1543–1550 (2012).
[Crossref]

S. Chen, W. Liao, Y. Dai, and S. Li, “Self-calibrated subaperture stitching test of hyper-hemispheres using latitude and longitude coordinates,” Appl. Opt. 51(17), 3817–3825 (2012).
[Crossref]

2011 (1)

D. C.-L. Fong and M. Saunders, “Lsmr: An iterative algorithm for sparse least-squares problems,” SIAM J. on Sci. Comput. 33(5), 2950–2971 (2011).
[Crossref]

2010 (2)

P. Su, J. H. Burge, and R. E. Parks, “Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors,” Appl. Opt. 49(1), 21–31 (2010).
[Crossref]

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

2008 (1)

2007 (1)

2005 (2)

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

S. Chen, S. Li, and Y. Dai, “Iterative algorithm for subaperture stitching interferometry for general surfaces,” J. Opt. Soc. Am. A 22(9), 1929–1936 (2005).
[Crossref]

2003 (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

1996 (1)

C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-calibration: Reversal, redundancy, error separation, and ’absolute testing’,” CIRP Ann. 45(2), 617–634 (1996).
[Crossref]

1994 (1)

Baker, R.

A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instrum. 87(5), 051908 (2016).
[Crossref]

Barrett, R.

A. Vivo, R. Barrett, and F. Perrin, “Stitching techniques for measuring x-ray synchrotron mirror topography,” Rev. Sci. Instrum. 90(2), 021710 (2019).
[Crossref]

A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instrum. 87(5), 051908 (2016).
[Crossref]

Bauer, M.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Bray, M.

M. Bray, “Stitching interferometry: the practical side of things,” in Optical Manufacturing and Testing VIII, vol. 7426 (International Society for Optics and Photonics, 2009), p. 74260Q.

Brochet, S.

F. Polack, M. Thomasset, S. Brochet, and D. Dennetiere, “Surface shape determination with a stitching michelson interferometer and accuracy evaluation,” Rev. Sci. Instrum. 90(2), 021708 (2019).
[Crossref]

Burge, J. H.

Campos, J.

Chen, S.

Cocco, D.

Dai, Y.

de Groot, P.

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33(31), 7334–7338 (1994).
[Crossref]

Deck, L.

Dennetiere, D.

F. Polack, M. Thomasset, S. Brochet, and D. Dennetiere, “Surface shape determination with a stitching michelson interferometer and accuracy evaluation,” Rev. Sci. Instrum. 90(2), 021708 (2019).
[Crossref]

DeVries, G.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, vol. 5188 (International Society for Optics and Photonics, 2003), pp. 296–307.

Endo, K.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Estler, W. T.

C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-calibration: Reversal, redundancy, error separation, and ’absolute testing’,” CIRP Ann. 45(2), 617–634 (1996).
[Crossref]

Evans, C. J.

C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-calibration: Reversal, redundancy, error separation, and ’absolute testing’,” CIRP Ann. 45(2), 617–634 (1996).
[Crossref]

Fleig, J.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, vol. 5188 (International Society for Optics and Photonics, 2003), pp. 296–307.

Fong, D. C.-L.

D. C.-L. Fong and M. Saunders, “Lsmr: An iterative algorithm for sparse least-squares problems,” SIAM J. on Sci. Comput. 33(5), 2950–2971 (2011).
[Crossref]

D. C.-L. Fong and M. Saunders, “LSMR: Sparse Equations and Least Squares,” http://web.stanford.edu/group/SOL/software/lsmr/ .

Forbes, G.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” (2005). US Patent 6,956,657.

Forbes, G. W.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, vol. 5188 (International Society for Optics and Photonics, 2003), pp. 296–307.

Gao, B.

Golini, D.

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” (2005). US Patent 6,956,657.

Hocken, R. J.

C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-calibration: Reversal, redundancy, error separation, and ’absolute testing’,” CIRP Ann. 45(2), 617–634 (1996).
[Crossref]

Huang, L.

Idir, M.

Ishikawa, T.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Kuhne, D.

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
[Crossref]

Kulawiec, A.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Lantelme, B.

A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instrum. 87(5), 051908 (2016).
[Crossref]

Li, S.

Liao, W.

Lippmann, E.

Matsuyama, S.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Miladinovich, D.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Mimura, H.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Molesini, G.

Mori, Y.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Murphy, P.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

D. Golini, G. Forbes, and P. Murphy, “Method for self-calibrated sub-aperture stitching for surface figure measurement,” (2005). US Patent 6,956,657.

Murphy, P. E.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, vol. 5188 (International Society for Optics and Photonics, 2003), pp. 296–307.

Ng, M. L.

Nicolas, J.

Nishino, Y.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Parks, R. E.

Pedreira, P.

Peng, X.

S. Chen, Y. Dai, S. Li, X. Peng, and J. Wang, “Error reductions for stitching test of large optical flats,” Opt. Laser Technol. 44(5), 1543–1550 (2012).
[Crossref]

Perrin, F.

A. Vivo, R. Barrett, and F. Perrin, “Stitching techniques for measuring x-ray synchrotron mirror topography,” Rev. Sci. Instrum. 90(2), 021710 (2019).
[Crossref]

Polack, F.

F. Polack, M. Thomasset, S. Brochet, and D. Dennetiere, “Surface shape determination with a stitching michelson interferometer and accuracy evaluation,” Rev. Sci. Instrum. 90(2), 021708 (2019).
[Crossref]

F. Polack and M. Thomasset, “Determination and compensation of the “reference surface” from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res., Sect. A 710, 67–71 (2013). The 4th international workshop on Metrology for X-ray Optics, Mirror Design, and Fabrication.
[Crossref]

Sano, Y.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Saunders, M.

D. C.-L. Fong and M. Saunders, “Lsmr: An iterative algorithm for sparse least-squares problems,” SIAM J. on Sci. Comput. 33(5), 2950–2971 (2011).
[Crossref]

D. C.-L. Fong and M. Saunders, “LSMR: Sparse Equations and Least Squares,” http://web.stanford.edu/group/SOL/software/lsmr/ .

Su, P.

Tamasaku, K.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Tayabaly, K.

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
[Crossref]

L. Huang, M. Idir, C. Zuo, T. Wang, K. Tayabaly, and E. Lippmann, “Two-dimensional stitching interferometry based on tilt measurement,” Opt. Express 26(18), 23278–23286 (2018).
[Crossref]

Thomasset, M.

F. Polack, M. Thomasset, S. Brochet, and D. Dennetiere, “Surface shape determination with a stitching michelson interferometer and accuracy evaluation,” Rev. Sci. Instrum. 90(2), 021708 (2019).
[Crossref]

F. Polack and M. Thomasset, “Determination and compensation of the “reference surface” from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res., Sect. A 710, 67–71 (2013). The 4th international workshop on Metrology for X-ray Optics, Mirror Design, and Fabrication.
[Crossref]

Tricard, M.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Ueno, K.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Vannoni, M.

Vescovi, M.

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
[Crossref]

Vivo, A.

A. Vivo, R. Barrett, and F. Perrin, “Stitching techniques for measuring x-ray synchrotron mirror topography,” Rev. Sci. Instrum. 90(2), 021710 (2019).
[Crossref]

A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instrum. 87(5), 051908 (2016).
[Crossref]

Wang, J.

S. Chen, Y. Dai, S. Li, X. Peng, and J. Wang, “Error reductions for stitching test of large optical flats,” Opt. Laser Technol. 44(5), 1543–1550 (2012).
[Crossref]

Wang, T.

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
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[Crossref]

Xu, W.

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
[Crossref]

L. Huang, T. Wang, K. Tayabaly, D. Kuhne, W. Xu, W. Xu, M. Vescovi, and M. Idir, “Stitching interferometry for synchrotron mirror metrology at National Synchrotron Light Source II (NSLS-II),” Opt. Lasers Eng. 124, 105795 (2020).
[Crossref]

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H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

Yamamura, K.

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

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H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

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H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
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M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
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S. Chen, Y. Dai, S. Li, X. Peng, and J. Wang, “Error reductions for stitching test of large optical flats,” Opt. Laser Technol. 44(5), 1543–1550 (2012).
[Crossref]

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[Crossref]

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[Crossref]

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A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instrum. 87(5), 051908 (2016).
[Crossref]

H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005).
[Crossref]

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Figures (17)

Fig. 1.
Fig. 1. Subsets with $x$- and $y$-shifts are captured in the 2D data stitching.
Fig. 2.
Fig. 2. Two types of self-calibration stitching algorithms (CS and SC) can be applied to both pixel-relation-based stitching and subset-relation-based stitching. (a) The CS-P algorithm can be applied to the pixel-relation-based stitching strategy with no iterations. (b) The CS-S algorithm is for the subset-relation-based stitching strategy with iterative reference compensation. (c) The SC algorithm can work with either pixel-relation-based or subset-relation-based stitching strategy via iterative reference compensations. Note: the $\sigma (\cdot )$ operation is to calculate the RMS value.
Fig. 3.
Fig. 3. In 2D stitching interferometery, uniform steps (a) and randomized nonuniform steps (b) are two kinds of possible SA stepping strategies. We suggest using nonuniform steps (b) to avoid the periodic errors when estimating additive systematic errors.
Fig. 4.
Fig. 4. In our simulation, the SUT is captured in $3\times 23$ unevenly-spaced subsets (a) with a simulated reference $\mathbf {r}$ (b). Note: PTV is the value for Peak-To-Valley.
Fig. 5.
Fig. 5. Stitching results in simulation verifies the CS-P algorithm. (a) Stitching result and (b) Estimated reference. (c) Stitching error (terms up to the second order removed). (d) Reference estimation error (terms up to the second order removed).
Fig. 6.
Fig. 6. Stitching results verifies the CS-S algorithm. (a), (e), and (i) are the estimated $\mathbf {\hat {r}}_i$ after each iteration. (b), (f), and (j) are the updated reference $\mathbf {r}_{i+1}$ , (c), (g), and (k) are the reference estimation errors (terms up to the second order removed), and (d),(h), and (l) are the stitching errors (terms up to the second order removed) after each iteration.
Fig. 7.
Fig. 7. Stitching results with the SC algorithm. (a),(e), and (i) are the stitching errors (terms up to the second order removed) after each iteration. (b), (f), and (j) are the $\mathbf {\hat {r}}_i$ after each iteration. (c), (g), and (k) are the updated reference $\mathbf {r}_{i+1}$, and (d), (h), and (l) are reference estimation errors (terms up to the second order removed) after each iteration.
Fig. 8.
Fig. 8. The 2D self-calibration stitching experiment is performed by using the NSLS-II interferometric stitching platform.
Fig. 9.
Fig. 9. By subtracting a well-calibrated reference (a) from the captured subsets, the stitching result of the SUT (b) will be used as the benchmark for the stitching error evaluation.
Fig. 10.
Fig. 10. The CS-P algorithm directly estimates the reference $\mathbf {r}$ (a) with a reference estimation error (b) and stitch the height $\mathbf {z}$ (c).
Fig. 11.
Fig. 11. With iterative compensations to the reference $\mathbf {r}$ in the SC algorithm, the SUT height $\mathbf {z}$ is stitched with better visualization (less periodic errors).
Fig. 12.
Fig. 12. The data acquisition and the self-calibration stitching algorithm are robust and repeatable in 10 repeating scans. (a) 10 stitched SUT shapes and (b) their discrepancies.
Fig. 13.
Fig. 13. The estimated reference (a) and the discrepancies (b) from their average of these 10 repeating scans indicate the reference estimation is very repeatable.
Fig. 14.
Fig. 14. The comparison of stitching errors (a) without subtracting system error and (b) with the proposed self-calibration stitching method indicates its effectiveness. These two stitching error maps are plotted with surfaces in the same coordinate system with a relative shift in y-direction for better comparison.
Fig. 15.
Fig. 15. Experiments with a $50\%$ overlapping area in common between SA$_1$ and SA$_2$ are designed to study the self-consistency of the 2D self-calibration stitching method.
Fig. 16.
Fig. 16. Two separate stitching results of the same SUT with different SAs on the reference give very similar height maps with a tiny difference of $0.12$ nm RMS. The color range is only within $[-3\sigma , +3\sigma ]$ where $\sigma$ is the RMS value of the stitched height values.
Fig. 17.
Fig. 17. The reference estimations in these two $50\%$ overlapped SAs (a) give self-consistent results in their common area (b) and (c). There is only a tiny height difference in their common area with the tilt terms removed (b) or up to the 2nd order terms removed (c).

Equations (22)

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m n ( x , y ) = z ( x + x n , y + y n ) + r ( x , y ) + t n [ x , y , 1 ] + n a ( x , y ) , n [ 1 , N ] ,
[ G D ] [ t r ] d m i n ,
[ t r ] = ( [ G D ] [ G D ] ) 1 [ G D ] d .
D r ^ i d ^ i m i n .
r ^ i = ( D D ) 1 D d ^ i .
D s r ^ i d s ^ i m i n ,
r ^ i = ( D s D s ) 1 D s d s ^ i .
z ( x , y ) = z ( x , y ) + p ( x ) + q ( y ) ,
r ( x , y ) = r ( x , y ) + p ( x ) + q ( y ) ,
p ( x ) = p ( x k T x ) , k Z ,
q ( y ) = q ( y k T y ) , k Z .
z ( x x n , y y n ) r ( x , y ) + a n x + b n y + c n = z ( x x n , y y n ) + p ( x x n ) + q ( y y n ) r ( x , y ) p ( x ) q ( y ) + a n x + b n y + c n = z ( x x n , y y n ) r ( x , y ) + a n x + b n y + c n = m n ( x , y ) .
z ( x , y ) = z ( x , y ) + C 11 x 2 + C 12 x y + C 22 y 2 ,
r ( x , y ) = r ( x , y ) + C 11 x 2 + C 12 x y + C 22 y 2 ,
a n = a n + 2 C 11 x n + C 12 y n ,
b n = b n + C 12 x n + 2 C 22 y n ,
c n = c n C 11 x n 2 C 12 x n y n C 22 y n 2 .
z ( x x n , y y n ) r ( x , y ) + a n x + b n y + c n = z ( x x n , y y n ) + C 11 ( x x n ) 2 + C 12 ( x x n ) ( y y n ) + C 22 ( y y n ) 2 r ( x , y ) C 11 x 2 C 12 x y C 22 y 2 + ( a n + 2 C 11 x n + C 12 y n ) x + ( b n + C 12 x n + 2 C 22 y n ) y + c n C 11 x n 2 C 12 x n y n C 22 y n 2 = z ( x x n , y y n ) r ( x , y ) + a n x + b n y + c n = m n ( x , y ) .
[ t r ] = ( [ G e x D e x ] [ G e x D e x ] ) 1 [ G e x D e x ] d e x ,
r ^ i = ( D e x D e x ) 1 D e x d e x ^ i ,
r ^ i = ( D e x s D e x s ) 1 D e x s d e x s ^ i ,
z = A [ sin ( 2 π x y p x p y ) + cos ( 2 π ( x p x + y p y ) ) ] ,

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