Abstract

We describe the conditions required for a set of displaced sub-aperture measurements to contain sufficient information to reconstruct the stitched mirror profile removing all additive systematic errors of the measuring instrument, independent of the reference surface and of the guidance error of the linear stage used for the translation. We show that even-spaced stitching must be avoided and that the pitch error of the linear stage or the curvature of the reference must be measured, to avoid periodic errors and curvature errors in the reconstructed profile. We show that once these uncertainties are solved, the 1D profile can be reconstructed free of any additive systematic error. The theory is supported by computer simulations and by experimental results using two different instruments.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, “Improvements in the accuracy and the repeatability of long trace profiler measurements,” Appl. Opt. 38(25), 5468–5479 (1999).
    [Crossref] [PubMed]
  2. S. Qian, W. Jark, and P. Z. Takacs, “The penta‐prism LTP: a long‐trace‐profiler with stationary optical head and moving penta prism,” Rev. Sci. Instrum. 66(3), 2562–2569 (1995).
    [Crossref]
  3. A. Rommeveaux, M. Thomasset, and D. Cocco, “The long trace profilers” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed. (Springer, 2008) pp. 175–184.
  4. F. Siewert, H. Lammert, and T. Zeschke, “The nanometer optical component measuring machine” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed. (Springer, 2008) pp. 185–192.
  5. S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
    [Crossref]
  6. J. Nicolas and J. C. Martínez, “Characterization of the error budget of Alba-NOM,” Nucl. Instrum. Methods Phys. Res. A 710, 24–30 (2013).
    [Crossref]
  7. T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
    [Crossref]
  8. M. Idir, K. Kaznatcheev, G. Dovillaire, J. Legrand, and R. Rungsawang, “A 2 D high accuracy slope measuring system based on a Stitching Shack Hartmann Optical Head,” Opt. Express 22(3), 2770–2781 (2014).
    [Crossref] [PubMed]
  9. H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
    [Crossref] [PubMed]
  10. 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] [PubMed]
  11. J. Xue, L. Huang, B. Gao, K. Kaznatcheev, and M. Idir, “One-dimensional stitching interferometry assisted by a triple-beam interferometer,” Opt. Express 25(8), 9393–9405 (2017).
    [Crossref] [PubMed]
  12. J. Vidal, J. Nicolas, and J. Campos, “New method to improve the accuracy in a sequential lateral shearing interferometer,” Opt. Eng. 50(11), 115601 (2011).
    [Crossref]
  13. F. Polack and M. Thomasset, “Determination and compensation of the reference surface from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res. A 710, 67–71 (2013).
    [Crossref]
  14. J. Nicolas and J. Campos, “Error compensation for the calibration of mechanical mirror benders,” Proc. SPIE 8848, 88480K (2013).
    [Crossref]
  15. J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).
  16. J. Vidal, J. Nicolás, and J. Campos, “Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer,” Proc. SPIE 7390, 73900N (2009).
    [Crossref]
  17. M. Vannoni and I. F. Martìn, “Absolute interferometric characterization of an x-ray mirror surface profile,” Metrologia 53(1), 1–6 (2016).
    [Crossref]
  18. M. Vannoni, “Calibration of absolute planarity flats: generalized iterative approach,” Opt. Eng. 51(8), 081510 (2012).
    [Crossref]

2017 (1)

2016 (4)

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

M. Vannoni and I. F. Martìn, “Absolute interferometric characterization of an x-ray mirror surface profile,” Metrologia 53(1), 1–6 (2016).
[Crossref]

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

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] [PubMed]

2014 (1)

2013 (3)

J. Nicolas and J. C. Martínez, “Characterization of the error budget of Alba-NOM,” Nucl. Instrum. Methods Phys. Res. A 710, 24–30 (2013).
[Crossref]

F. Polack and M. Thomasset, “Determination and compensation of the reference surface from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res. A 710, 67–71 (2013).
[Crossref]

J. Nicolas and J. Campos, “Error compensation for the calibration of mechanical mirror benders,” Proc. SPIE 8848, 88480K (2013).
[Crossref]

2012 (1)

M. Vannoni, “Calibration of absolute planarity flats: generalized iterative approach,” Opt. Eng. 51(8), 081510 (2012).
[Crossref]

2011 (1)

J. Vidal, J. Nicolas, and J. Campos, “New method to improve the accuracy in a sequential lateral shearing interferometer,” Opt. Eng. 50(11), 115601 (2011).
[Crossref]

2010 (2)

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

2009 (1)

J. Vidal, J. Nicolás, and J. Campos, “Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer,” Proc. SPIE 7390, 73900N (2009).
[Crossref]

1999 (1)

1995 (1)

S. Qian, W. Jark, and P. Z. Takacs, “The penta‐prism LTP: a long‐trace‐profiler with stationary optical head and moving penta prism,” Rev. Sci. Instrum. 66(3), 2562–2569 (1995).
[Crossref]

Alcock, S. G.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Assoufid, L.

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] [PubMed]

Barrett, 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] [PubMed]

Bresloff, C. J.

Campos, J.

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

J. Nicolas and J. Campos, “Error compensation for the calibration of mechanical mirror benders,” Proc. SPIE 8848, 88480K (2013).
[Crossref]

J. Vidal, J. Nicolas, and J. Campos, “New method to improve the accuracy in a sequential lateral shearing interferometer,” Opt. Eng. 50(11), 115601 (2011).
[Crossref]

J. Vidal, J. Nicolás, and J. Campos, “Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer,” Proc. SPIE 7390, 73900N (2009).
[Crossref]

Church, E. L.

Dovillaire, G.

Gao, B.

Goto, S.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Huang, L.

Idir, M.

Ishikawa, T.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Jark, W.

S. Qian, W. Jark, and P. Z. Takacs, “The penta‐prism LTP: a long‐trace‐profiler with stationary optical head and moving penta prism,” Rev. Sci. Instrum. 66(3), 2562–2569 (1995).
[Crossref]

Kaznatcheev, K.

Kimura, T.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Koyama, T.

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

Lammert, H.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (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] [PubMed]

Legrand, J.

Martìn, I. F.

M. Vannoni and I. F. Martìn, “Absolute interferometric characterization of an x-ray mirror surface profile,” Metrologia 53(1), 1–6 (2016).
[Crossref]

Martínez, J. C.

J. Nicolas and J. C. Martínez, “Characterization of the error budget of Alba-NOM,” Nucl. Instrum. Methods Phys. Res. A 710, 24–30 (2013).
[Crossref]

Masunaga, T.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Matsuyama, S.

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Mimura, H.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Nicolas, J.

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

J. Nicolas and J. Campos, “Error compensation for the calibration of mechanical mirror benders,” Proc. SPIE 8848, 88480K (2013).
[Crossref]

J. Nicolas and J. C. Martínez, “Characterization of the error budget of Alba-NOM,” Nucl. Instrum. Methods Phys. Res. A 710, 24–30 (2013).
[Crossref]

J. Vidal, J. Nicolas, and J. Campos, “New method to improve the accuracy in a sequential lateral shearing interferometer,” Opt. Eng. 50(11), 115601 (2011).
[Crossref]

Nicolás, J.

J. Vidal, J. Nicolás, and J. Campos, “Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer,” Proc. SPIE 7390, 73900N (2009).
[Crossref]

Noll, T.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Ohashi, H.

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Okada, H.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Pedersen, U.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Pedreira, P.

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

Polack, F.

F. Polack and M. Thomasset, “Determination and compensation of the reference surface from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res. A 710, 67–71 (2013).
[Crossref]

Qian, S.

S. Qian, W. Jark, and P. Z. Takacs, “The penta‐prism LTP: a long‐trace‐profiler with stationary optical head and moving penta prism,” Rev. Sci. Instrum. 66(3), 2562–2569 (1995).
[Crossref]

Ramírez, C.

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

Rungsawang, R.

Sawhney, K. J. S.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Scott, S.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Senba, Y.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Senf, F.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Sics, I.

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

Siewert, F.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Takacs, P. Z.

P. Z. Takacs, E. L. Church, C. J. Bresloff, and L. Assoufid, “Improvements in the accuracy and the repeatability of long trace profiler measurements,” Appl. Opt. 38(25), 5468–5479 (1999).
[Crossref] [PubMed]

S. Qian, W. Jark, and P. Z. Takacs, “The penta‐prism LTP: a long‐trace‐profiler with stationary optical head and moving penta prism,” Rev. Sci. Instrum. 66(3), 2562–2569 (1995).
[Crossref]

Thomasset, M.

F. Polack and M. Thomasset, “Determination and compensation of the reference surface from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res. A 710, 67–71 (2013).
[Crossref]

Tsumura, T.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Vannoni, M.

M. Vannoni and I. F. Martìn, “Absolute interferometric characterization of an x-ray mirror surface profile,” Metrologia 53(1), 1–6 (2016).
[Crossref]

M. Vannoni, “Calibration of absolute planarity flats: generalized iterative approach,” Opt. Eng. 51(8), 081510 (2012).
[Crossref]

Vidal, J.

J. Vidal, J. Nicolas, and J. Campos, “New method to improve the accuracy in a sequential lateral shearing interferometer,” Opt. Eng. 50(11), 115601 (2011).
[Crossref]

J. Vidal, J. Nicolás, and J. Campos, “Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer,” Proc. SPIE 7390, 73900N (2009).
[Crossref]

Vivo, A.

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] [PubMed]

Walton, R.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Xue, J.

Yamakawa, D.

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Yamauchi, K.

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Yumoto, H.

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

Zeschke, T.

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

Appl. Opt. (1)

Metrologia (1)

M. Vannoni and I. F. Martìn, “Absolute interferometric characterization of an x-ray mirror surface profile,” Metrologia 53(1), 1–6 (2016).
[Crossref]

Nucl. Instrum. Methods Phys. Res. A (4)

S. G. Alcock, K. J. S. Sawhney, S. Scott, U. Pedersen, R. Walton, F. Siewert, T. Zeschke, F. Senf, T. Noll, and H. Lammert, “The Diamond-NOM: a non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 224–228 (2010).
[Crossref]

J. Nicolas and J. C. Martínez, “Characterization of the error budget of Alba-NOM,” Nucl. Instrum. Methods Phys. Res. A 710, 24–30 (2013).
[Crossref]

T. Kimura, H. Ohashi, H. Mimura, D. Yamakawa, H. Yumoto, S. Matsuyama, T. Tsumura, H. Okada, T. Masunaga, Y. Senba, S. Goto, T. Ishikawa, and K. Yamauchi, “A stitching figure profiler of large X-ray mirrors using RADSI for subaperture data adquisition,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 229–232 (2010).
[Crossref]

F. Polack and M. Thomasset, “Determination and compensation of the reference surface from redundant sets of surface measurements,” Nucl. Instrum. Methods Phys. Res. A 710, 67–71 (2013).
[Crossref]

Opt. Eng. (2)

J. Vidal, J. Nicolas, and J. Campos, “New method to improve the accuracy in a sequential lateral shearing interferometer,” Opt. Eng. 50(11), 115601 (2011).
[Crossref]

M. Vannoni, “Calibration of absolute planarity flats: generalized iterative approach,” Opt. Eng. 51(8), 081510 (2012).
[Crossref]

Opt. Express (2)

Proc. SPIE (3)

J. Nicolas and J. Campos, “Error compensation for the calibration of mechanical mirror benders,” Proc. SPIE 8848, 88480K (2013).
[Crossref]

J. Nicolas, P. Pedreira, I. Sics, C. Ramírez, and J. Campos, “Nanometer accuracy with continuous scans at the ALBA-NOM,” Proc. SPIE 9962, 995203 (2016).

J. Vidal, J. Nicolás, and J. Campos, “Analysis of the positioning error on lateral shearing surface reconstruction with a Fizeau interferometer,” Proc. SPIE 7390, 73900N (2009).
[Crossref]

Rev. Sci. Instrum. (3)

S. Qian, W. Jark, and P. Z. Takacs, “The penta‐prism LTP: a long‐trace‐profiler with stationary optical head and moving penta prism,” Rev. Sci. Instrum. 66(3), 2562–2569 (1995).
[Crossref]

H. Yumoto, T. Koyama, S. Matsuyama, K. Yamauchi, and H. Ohashi, “Stitching interferometry for ellipsoidal x-ray mirrors,” Rev. Sci. Instrum. 87(5), 051905 (2016).
[Crossref] [PubMed]

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] [PubMed]

Other (2)

A. Rommeveaux, M. Thomasset, and D. Cocco, “The long trace profilers” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed. (Springer, 2008) pp. 175–184.

F. Siewert, H. Lammert, and T. Zeschke, “The nanometer optical component measuring machine” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed. (Springer, 2008) pp. 185–192.

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Figures (7)

Fig. 1
Fig. 1 Diagram representing the measurement process. The interferometer measures the distance at every point between the surface under test and a reference surface. The instrument (including the reference surface, have to be scanned along the SUT. Guidance error of such motion introduces a pitch and piston at every measurement.
Fig. 2
Fig. 2 Reconstruction of the mirror profile by stitching sub-aperture measurements, for (a) evenly spaced sub-aperture measurements, and (b) irregularly spaced sub-apertures. The mean curvature has been removed from the represented mirror profiles.
Fig. 3
Fig. 3 Curvature of the reconstruction error. (a) Reconstructed and actual profile of the surface under test, differences are caused by the straightness error of the scanning stage. (b) Reconstruction error. (c) and (d) correspond to the case when pitch data is included in the stitching algorithm.
Fig. 4
Fig. 4 Periodic error enhanced by evenly spaced sub-aperture stitching. (a) reconstructed profile using even and un-even displacements. (b) Power Spectral Density of the reconstructed profiles, and their differences.
Fig. 5
Fig. 5 Stitching using two different interference-microscope objectives. (a) reconstructed profiles (spatially filtered to match the same spatial frequency). (b) reconstructed reference error.
Fig. 6
Fig. 6 Reconstructed profile of the surface under test. The red and blue profiles correspond to the profile reconstructed using different parts of the field of view of the Fizeau interferometer. The black line corresponds to a measurement obtained by the NOM.
Fig. 7
Fig. 7 Reconstructed reference error for the two regions of the Fizeau interferometer reference flat. The dashed line indicates the best fit curvature for each region.

Equations (13)

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m k ( x )=s( x X k )r( x )+ T k + U k x
m k [ x n ]= i=1 N i s i B i [ x n X k ] j=1 N j r j B j [ x n ]+ T k + U k x n .
s * ( x )=s( x )+q( x ) r * ( x )=r( x )+q( x )
m k * ( x )= s * ( xkΔx ) r * ( x )+ T k + U k x
m k * ( x )=s( xkΔx )+q( xkΔx )r( x )q( x )+ T k + U k x
s * ( x )=s( x )+ x 2 2R r * ( x )=r( x )+ x 2 2R
T k * = T k X k 2 2R   U k * = U k + X k R
m k * ( x )=s( x X k )+ 1 2R ( x X k ) 2 r( x ) x 2 2R + T k X k 2 2R +( U k + 1 R X k )x
k=1 N k ( X k X k ) ( U k meas U k meas )= k=1 N k ( X k X k ) ( U k U k )
δs( x X k )δr( x )+δ T k +δ U k x=0
δ s '' ( x X k )=δ r '' ( x ),  x,k
δ s '' ( x X k )=δ s '' ( x X j ),  x, k,j
δ s '' ( x )=δ s '' ( xnΔ x 1 ) δ s '' ( x )=δ s '' ( xmΔ x 2 )

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