Abstract

We present a method to enhance the achievable lateral resolution of a multi-sensor scanning profile measurement method. The relationship between the profile measurement method considered and established shearing techniques is illustrated. Simulation and measurement results show that non-equidistant sensor spacing can improve the lateral resolution significantly.

© 2010 OSA

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References

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  1. H. Tanaka and H. Sato, “Extensive analysis and development of straightness measurement by sequential-two-points method,” Trans. ASME J. Eng. Ind 108, 167–182 (1986).
    [Crossref]
  2. W. Gao, and S. Kiyono, “On-Machine Profile Measurement of Machined Surface using the Combined Three-Point Method,” JSME Int. J. Ser. C 40 (2), 253–259 (1997). http://ci.nii.ac.jp/naid/110004164205/en/ .
  3. W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).
  4. I. Weingärtner and C. Elster, “System of four distance sensors for high-accuracy measurement of topography,” Precis. Eng. 28(2), 164–170 (2004).
    [Crossref]
  5. C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Precis. Eng. 30(1), 32–38 (2006).
    [Crossref]
  6. 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), doi:.
    [Crossref]
  7. H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
    [Crossref]
  8. J. Flügge, R. Köning, and C. Weichert, “W. Häßler-Grohne, R. D. Geckeler, A. Wiegmann, M. Schulz, C. Elster and H. Bosse, “Development of a 1.5D reference comparator for position and straightness metrology on photomasks,” Proc. SPIE 7122, 71222Y (2008), doi:, http://link.aip.org/link/?PSI/7122/71222Y/1 .
    [Crossref]
  9. W. H. Press, P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, (Cambridge University Press, 1992).
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  11. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, CRC Press (1998), Chapt. 1.6.4.
  12. T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41(10), 1954–1961 (2002).
    [Crossref] [PubMed]
  13. C. Elster and I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38(23), 5024–5031 (1999).
    [Crossref]
  14. C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comput. Appl. Math. 110(1), 177–180 (1999).
    [Crossref]
  15. A. Wiegmann, M. Schulz, and C. Elster, “Suppression of aliasing in multi-sensor scanning absolute profile measurement,” Opt. Express 17(13), 11098–11106 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-13-11098 .
    [Crossref] [PubMed]
  16. B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE 5878, 587806 (2005).
    [Crossref]
  17. R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete,” Tech. Mess. 74, 572–576 (2007).
  18. C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
    [Crossref]

2010 (1)

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

2009 (2)

A. Wiegmann, M. Schulz, and C. Elster, “Suppression of aliasing in multi-sensor scanning absolute profile measurement,” Opt. Express 17(13), 11098–11106 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-13-11098 .
[Crossref] [PubMed]

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

2008 (1)

J. Flügge, R. Köning, and C. Weichert, “W. Häßler-Grohne, R. D. Geckeler, A. Wiegmann, M. Schulz, C. Elster and H. Bosse, “Development of a 1.5D reference comparator for position and straightness metrology on photomasks,” Proc. SPIE 7122, 71222Y (2008), doi:, http://link.aip.org/link/?PSI/7122/71222Y/1 .
[Crossref]

2007 (1)

R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete,” Tech. Mess. 74, 572–576 (2007).

2006 (1)

C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Precis. Eng. 30(1), 32–38 (2006).
[Crossref]

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), doi:.
[Crossref]

B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE 5878, 587806 (2005).
[Crossref]

2004 (1)

I. Weingärtner and C. Elster, “System of four distance sensors for high-accuracy measurement of topography,” Precis. Eng. 28(2), 164–170 (2004).
[Crossref]

2002 (2)

W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).

T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41(10), 1954–1961 (2002).
[Crossref] [PubMed]

1999 (2)

C. Elster and I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38(23), 5024–5031 (1999).
[Crossref]

C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comput. Appl. Math. 110(1), 177–180 (1999).
[Crossref]

1986 (1)

H. Tanaka and H. Sato, “Extensive analysis and development of straightness measurement by sequential-two-points method,” Trans. ASME J. Eng. Ind 108, 167–182 (1986).
[Crossref]

Bakucz, P.

R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete,” Tech. Mess. 74, 572–576 (2007).

Bremer, H.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

Doerband, B.

B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE 5878, 587806 (2005).
[Crossref]

Elster, C.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

A. Wiegmann, M. Schulz, and C. Elster, “Suppression of aliasing in multi-sensor scanning absolute profile measurement,” Opt. Express 17(13), 11098–11106 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-13-11098 .
[Crossref] [PubMed]

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Precis. Eng. 30(1), 32–38 (2006).
[Crossref]

I. Weingärtner and C. Elster, “System of four distance sensors for high-accuracy measurement of topography,” Precis. Eng. 28(2), 164–170 (2004).
[Crossref]

C. Elster and I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38(23), 5024–5031 (1999).
[Crossref]

C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comput. Appl. Math. 110(1), 177–180 (1999).
[Crossref]

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), doi:.
[Crossref]

Flügge, J.

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

J. Flügge, R. Köning, and C. Weichert, “W. Häßler-Grohne, R. D. Geckeler, A. Wiegmann, M. Schulz, C. Elster and H. Bosse, “Development of a 1.5D reference comparator for position and straightness metrology on photomasks,” Proc. SPIE 7122, 71222Y (2008), doi:, http://link.aip.org/link/?PSI/7122/71222Y/1 .
[Crossref]

Gao, W.

W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).

Hetzler, J.

B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE 5878, 587806 (2005).
[Crossref]

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), doi:.
[Crossref]

Jung, L.

R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete,” Tech. Mess. 74, 572–576 (2007).

Kamiya, K.

Kiyono, S.

W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).

Kojima, H.

W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).

Köning, R.

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

J. Flügge, R. Köning, and C. Weichert, “W. Häßler-Grohne, R. D. Geckeler, A. Wiegmann, M. Schulz, C. Elster and H. Bosse, “Development of a 1.5D reference comparator for position and straightness metrology on photomasks,” Proc. SPIE 7122, 71222Y (2008), doi:, http://link.aip.org/link/?PSI/7122/71222Y/1 .
[Crossref]

Krey, S.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

Krueger-Sehm, R.

R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete,” Tech. Mess. 74, 572–576 (2007).

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), doi:.
[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), doi:.
[Crossref]

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), doi:.
[Crossref]

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), doi:.
[Crossref]

Nomura, T.

Okuda, S.

Ruprecht, A.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[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), doi:.
[Crossref]

Sato, H.

H. Tanaka and H. Sato, “Extensive analysis and development of straightness measurement by sequential-two-points method,” Trans. ASME J. Eng. Ind 108, 167–182 (1986).
[Crossref]

Schmahling, F.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

Schulz, M.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

A. Wiegmann, M. Schulz, and C. Elster, “Suppression of aliasing in multi-sensor scanning absolute profile measurement,” Opt. Express 17(13), 11098–11106 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-13-11098 .
[Crossref] [PubMed]

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Precis. Eng. 30(1), 32–38 (2006).
[Crossref]

Stavridis, M.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

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), doi:.
[Crossref]

Tanaka, H.

H. Tanaka and H. Sato, “Extensive analysis and development of straightness measurement by sequential-two-points method,” Trans. ASME J. Eng. Ind 108, 167–182 (1986).
[Crossref]

Tashiro, H.

Tutsch, R.

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[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), doi:.
[Crossref]

Walzel, M.

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

Weichert, C.

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

J. Flügge, R. Köning, and C. Weichert, “W. Häßler-Grohne, R. D. Geckeler, A. Wiegmann, M. Schulz, C. Elster and H. Bosse, “Development of a 1.5D reference comparator for position and straightness metrology on photomasks,” Proc. SPIE 7122, 71222Y (2008), doi:, http://link.aip.org/link/?PSI/7122/71222Y/1 .
[Crossref]

Weingärtner, I.

C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Precis. Eng. 30(1), 32–38 (2006).
[Crossref]

I. Weingärtner and C. Elster, “System of four distance sensors for high-accuracy measurement of topography,” Precis. Eng. 28(2), 164–170 (2004).
[Crossref]

C. Elster and I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38(23), 5024–5031 (1999).
[Crossref]

Wiegmann, A.

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

A. Wiegmann, M. Schulz, and C. Elster, “Suppression of aliasing in multi-sensor scanning absolute profile measurement,” Opt. Express 17(13), 11098–11106 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-13-11098 .
[Crossref] [PubMed]

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
[Crossref]

Wilhelms, H.

R. Krueger-Sehm, P. Bakucz, L. Jung, and H. Wilhelms, “Chirp-Kalibriernormale fuer Obeflaechenmessgeraete,” Tech. Mess. 74, 572–576 (2007).

Yabashi, M.

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), doi:.
[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), doi:.
[Crossref]

Yamauchi, 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), doi:.
[Crossref]

Yokoyama, J.

W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).

Yoshikawa, K.

Yumoto, 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), doi:.
[Crossref]

Appl. Opt. (2)

J. Comput. Appl. Math. (1)

C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comput. Appl. Math. 110(1), 177–180 (1999).
[Crossref]

Opt. Express (1)

Precis. Eng. (3)

W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precis. Eng. 26, 279–288 (2002).

I. Weingärtner and C. Elster, “System of four distance sensors for high-accuracy measurement of topography,” Precis. Eng. 28(2), 164–170 (2004).
[Crossref]

C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Precis. Eng. 30(1), 32–38 (2006).
[Crossref]

Proc. SPIE (4)

H. Bremer, F. Schmahling, C. Elster, S. Krey, A. Ruprecht, M. Schulz, M. Stavridis, and A. Wiegmann, “Simple methods for alignment of line distance sensor arrays,” Proc. SPIE 7718, 77181M (2010), doi:.
[Crossref]

J. Flügge, R. Köning, and C. Weichert, “W. Häßler-Grohne, R. D. Geckeler, A. Wiegmann, M. Schulz, C. Elster and H. Bosse, “Development of a 1.5D reference comparator for position and straightness metrology on photomasks,” Proc. SPIE 7122, 71222Y (2008), doi:, http://link.aip.org/link/?PSI/7122/71222Y/1 .
[Crossref]

B. Doerband and J. Hetzler, “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE 5878, 587806 (2005).
[Crossref]

C. Weichert, M. Stavridis, M. Walzel, C. Elster, A. Wiegmann, M. Schulz, R. Köning, J. Flügge, and R. Tutsch, “A model based approach to reference-free straightness measurement at the Nanometer Comparator,” Proc. SPIE 7390, 73900O (2009), doi:.
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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), doi:.
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Figures (14)

Fig. 1
Fig. 1

Diagram of the absolute profile measurement method used. The thick red horizontal line represents the array of 3 distance sensors. The vertical blue lines represent the systematic sensor errors εj.

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Fig. 2
Fig. 2

Illustration of the indistinguishability between scanning stage errors and topography for wavelength λ = σ/N (N = 1,2,…). (a): scanning stage without height errors and a sinusoidal topography with wavelength λ = σ. (b): scanning stage with sinusoidal height errors (λ = σ) and an ideal plane topography. (c): measurement results of the three distance sensors for (a) and (b).

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Fig. 3
Fig. 3

Transfer functions for the sensor distances σ1 = 1 and σ2 = 1.5.

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Fig. 4
Fig. 4

Rms reconstruction error for equidistant (σ2 = σ1, left) and non-equidistant sensors (σ2 = 1.5σ1, right) in dependence on reconstruction distance ds and topography wavelength λ.

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Fig. 5
Fig. 5

Rms reconstruction error against relative sensor spacing (σ2/σ1) and spatial frequency of the topography ( f t o p o ). The reconstruction distance ds was set to ds = σ1 on the left and ds = 0.5σ1 on the right.

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Fig. 6
Fig. 6

Rms reconstruction error against relative sensor spacing (σ2/σ1) and spatial frequency of the topography ( f t o p o ). The reconstruction distance ds was set to ds = 0.3σ1 on the left and ds = 0.25σ1 on the right.

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Fig. 7
Fig. 7

Rms reconstruction error against relative sensor spacing (σ2/σ1) and reconstruction frequency (fs = 1/ds).

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Fig. 8
Fig. 8

Local wavelength λ of the manufactured chirp specimen as a function of the distance to the center of the specimen.

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Fig. 9
Fig. 9

Reconstructed topography for an array of three distance sensors with σ2/σ1 = 1.028 (blue curve at the top) in comparison to the high-resolution scan (orange) and the difference between both topographies on the bottom. The vertical green lines mark the positions where the chirp specimen has a local wavelength corresponding to σ1/N (N = 1,2,3,…).

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Fig. 10
Fig. 10

Transfer functions for the two sensor distances σ1 and σ2 (top) and spectrum (bottom) of the difference between both reconstructed topographies as shown at the bottom of Fig. 9.

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Fig. 11
Fig. 11

Reconstructed topography for an array of three distance sensors with σ2/σ1 = 1.583 (blue curve at the top) in comparison to the high-resolution scan (orange) and the difference between both reconstructed topographies at the bottom. The vertical green lines mark the positions where the chirp specimen has a local wavelength corresponding to σ1/N (N = 1,2,3,…).

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Fig. 12
Fig. 12

Transfer functions for the two sensor distances σ1 and σ2 (top) and spectrum (bottom) of the difference between both reconstructed topographies as shown at the bottom of Fig. 11.

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Fig. 13
Fig. 13

Reconstructed topography for an array of three equally spaced distance sensors (blue curve at the top) in comparison to the high-resolution scan (orange) and the difference between both reconstructed topographies at the bottom. The sensor distances σ ˜ 1 = σ ˜ 2 = 1 / 36   σ 1 = 19   μm are here smaller than the reconstruction distance ds = 30.4 µm. The vertical green lines mark the positions where the chirp specimen has a local wavelength corresponding to σ1/N (N = 1,2,3,…).

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Fig. 14
Fig. 14

Transfer function for the two sensor distances σ ˜ 1 and σ ˜ 2 (top) and on the bottom the spectrum of the difference topography as shown at the bottom of Fig. 13.

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Equations (9)

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m i , j = − ∑ k = ⌊ x ˜ i , j d s ⌋ − o − 1 2 ⌈ x ˜ i , j d s ⌉ + o − 1 2 c k ( x ˜ i , j ) f ( x k ) + ε j + a i + b i s ( j )
c k ( x ˜ i , j ) = ∏ h = ⌊ x ˜ i , j d s ⌋ − o − 1 2 h ≠ k ⌈ x ˜ i , j d s ⌉ + o − 1 2 x ˜ i , j − x h x k − x h .
m i , 1 = − f ( x i ) + ε 1 + a i
m i , 2 = − f ( x i + σ 1 ) + ε 2 + a i + b i σ 1
m i , 3 = − f ( x i + σ 1 + σ 2 ) + ε 3 + a i + b i ( σ 1 + σ 2 ) .
m ˜ i , 1 = m i , 1 − m i , 2                     = f ( x i + σ 1 ) − f ( x i ) + ( ε 1 − ε 2 ) − b i σ 1                   = ( f ( x ) ∗ ( δ ( x + σ 1 ) − δ ( x ) ) ) | x = x i + ε ˜ 1 − b i σ 1
m ˜ i , 2 = m i , 2                                                     = − f ( x i + σ 1 ) + ε 2 + a i + b i σ 1                       = − f ( x i + σ 1 ) + ε ˜ 2 + a i + b i σ 1
m ˜ i , 3 = m i , 3 − m i , 2                     = f ( x i + σ 1 ) − f ( x i + σ 1 + σ 2 ) + b i σ 2 + ( ε 3 − ε 2 )                   = − ( f ( x ) ∗ ( δ ( x + σ 2 ) − δ ( x ) ) ) | x = x i + σ 1 + ε ˜ 3 + b i σ 2 .
T ( f t o p o ) = 𝔉 { δ ( x + σ ) − δ ( x ) } = e 2 i π σ f t o p o − 1

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