Abstract

We present a novel procedure for absolute, highly-accurate profile measurement with high dynamic range for large, moderately flat optical surfaces. The profile is reconstructed from many sub-profiles measured by a small interferometer which is scanned along the specimen under test. Additional angular and lateral distance measurements are used to account for the tilt of the interferometer and its precise lateral location during the measurements. Accurate positioning of the interferometer is not required. The algorithm proposed for the analysis of the data allows systematic errors of the interferometer and height offsets of the scanning stage to be eliminated and it does not reduce the resolution. By utilizing a realistic simulation scenario we show that accuracies in the nanometer range can be reached.

©2008 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing, (John Wiley & Sons Inc, 1992).
  2. R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. 161, 106–114 (1999).
    [Crossref]
  3. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006).
    [Crossref] [PubMed]
  4. M. Beyerlein, N. Lindlein, and J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics,“ Appl. Opt. 41, 2440–2447 (2002).
    [Crossref] [PubMed]
  5. S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolute calibration in wavefront testing interferometry,” Opt. Commun. 220, 23–32 (2003).
    [Crossref]
  6. S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology 45, 252–263 (2004).
  7. F. Simon, G. Khan, K. Mantel, N. Lindlein, and J. Schwider, “Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference,” Appl. Opt. 45, 8606–8612 (2006).
    [Crossref] [PubMed]
  8. T.W. Stuhlinger “Subaperture optical testing - Experimental verification,” Proc. SPIE. 656, pp. 118–127 (1986).
  9. M. Sjoedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
    [Crossref]
  10. J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. 5188, 296–307 (2003).
    [Crossref]
  11. K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
    [Crossref]
  12. C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. 30, 32–38 (2006).
    [Crossref]
  13. M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503-1–060503-3 (2006).
    [Crossref]
  14. A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
    [Crossref]
  15. 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).
  16. B. Jähne Digitale Bildverarbeitung, (Springer, 2005).
  17. B. Doerband and J. Hetzler , “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE. 5878, 587806 (2005).
    [Crossref]

2007 (1)

A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
[Crossref]

2006 (4)

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

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503-1–060503-3 (2006).
[Crossref]

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006).
[Crossref] [PubMed]

F. Simon, G. Khan, K. Mantel, N. Lindlein, and J. Schwider, “Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference,” Appl. Opt. 45, 8606–8612 (2006).
[Crossref] [PubMed]

2005 (2)

B. Jähne Digitale Bildverarbeitung, (Springer, 2005).

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

2004 (1)

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology 45, 252–263 (2004).

2003 (3)

S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolute calibration in wavefront testing interferometry,” Opt. Commun. 220, 23–32 (2003).
[Crossref]

J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. 5188, 296–307 (2003).
[Crossref]

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

2002 (2)

M. Sjoedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[Crossref]

M. Beyerlein, N. Lindlein, and J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics,“ Appl. Opt. 41, 2440–2447 (2002).
[Crossref] [PubMed]

1999 (1)

R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. 161, 106–114 (1999).
[Crossref]

1986 (1)

T.W. Stuhlinger “Subaperture optical testing - Experimental verification,” Proc. SPIE. 656, pp. 118–127 (1986).

Beyerlein, M.

Doerband, B.

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

Dörband, B.

R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. 161, 106–114 (1999).
[Crossref]

Dumas, P.

J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. 5188, 296–307 (2003).
[Crossref]

Elster, C.

A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
[Crossref]

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503-1–060503-3 (2006).
[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,” Prec. Eng. 30, 32–38 (2006).
[Crossref]

Endo, K.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Flannery, P. B.

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).

Fleig, J.

J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. 5188, 296–307 (2003).
[Crossref]

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,” Proc. SPIE. 5188, 296–307 (2003).
[Crossref]

Freimann, R.

R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. 161, 106–114 (1999).
[Crossref]

Geckeler, R.D.

A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
[Crossref]

Griesmann, U.

Hetzler, J.

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

Höller, F.

R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. 161, 106–114 (1999).
[Crossref]

Ishikawa, T.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Jähne, B.

B. Jähne Digitale Bildverarbeitung, (Springer, 2005).

Khan, G.

Lindlein, N.

Malacara, D.

D. Malacara, Optical Shop Testing, (John Wiley & Sons Inc, 1992).

Mantel, K.

Mimura, H.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Mori, Y.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

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,” Proc. SPIE. 5188, 296–307 (2003).
[Crossref]

Oreb, B. F.

M. Sjoedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[Crossref]

Press, W. H.

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).

Pruss, C.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology 45, 252–263 (2004).

Reichelt, S.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology 45, 252–263 (2004).

S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolute calibration in wavefront testing interferometry,” Opt. Commun. 220, 23–32 (2003).
[Crossref]

Saito, A.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Sano, Y.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Schulz, M.

A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
[Crossref]

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503-1–060503-3 (2006).
[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,” Prec. Eng. 30, 32–38 (2006).
[Crossref]

Schwider, J.

Simon, F.

Sjoedahl, M.

M. Sjoedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[Crossref]

Souvorov, A.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Stuhlinger, T.W.

T.W. Stuhlinger “Subaperture optical testing - Experimental verification,” Proc. SPIE. 656, pp. 118–127 (1986).

Tamasaku, K.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Teukolsky, S. A.

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).

Tiziani, H. J.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology 45, 252–263 (2004).

S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolute calibration in wavefront testing interferometry,” Opt. Commun. 220, 23–32 (2003).
[Crossref]

Ueno, K.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Vetterling, W. T.

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).

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,” Prec. Eng. 30, 32–38 (2006).
[Crossref]

Wiegmann, A.

A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
[Crossref]

Yabashi, M.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Yamamura, K.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Yamauchi, K.

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Appl. Opt. (3)

Opt. Commun. (2)

R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. 161, 106–114 (1999).
[Crossref]

S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolute calibration in wavefront testing interferometry,” Opt. Commun. 220, 23–32 (2003).
[Crossref]

Opt. Eng. (2)

M. Sjoedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[Crossref]

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503-1–060503-3 (2006).
[Crossref]

Optical Fabrication, Testing, and Metrology (1)

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology 45, 252–263 (2004).

Prec. Eng. (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,” Prec. Eng. 30, 32–38 (2006).
[Crossref]

Proc. SPIE. (4)

T.W. Stuhlinger “Subaperture optical testing - Experimental verification,” Proc. SPIE. 656, pp. 118–127 (1986).

J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. 5188, 296–307 (2003).
[Crossref]

A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. 6616, 661618 (2007).
[Crossref]

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

Rev. Sci. Instrum. (1)

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. 74, 2894–2898 (2003).
[Crossref]

Other (3)

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).

B. Jähne Digitale Bildverarbeitung, (Springer, 2005).

D. Malacara, Optical Shop Testing, (John Wiley & Sons Inc, 1992).

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Figures (12)

Fig. 1.
Fig. 1. Sketch of TMS. The thick red horizontal line represents the interferometer and the blue vertical lines represent the systematic sensor errors εj .

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Fig. 2.
Fig. 2. Photo of the new extended TMS measurement set-up (under construction) with an autocollimator (left red box), a compact interferometer (middle green box) and an additional displacement measurement interferometer (right blue box).

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Fig. 3.
Fig. 3. Transfer functions for the polynomial interpolation for different degrees of the polynomials; fnyq denotes the Nyquist frequency.

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Fig. 4.
Fig. 4. Sketch of simulation scenario. The simulated measurements p̃ i and bi account for systematic (psys , bsys ) and random measurement errors. N(μ,σ 2) represents a random value drawn from a Gaussian distribution with μ mean and variance σ 2.

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Fig. 5.
Fig. 5. Simulated surfaces used for the investigations of the extended TMS algorithm. On the left a steep surface with a PV of 15μm, on the right a sinusoidal surface with a PV of 200nm.

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Fig. 6.
Fig. 6. Comparison of TMS and extended TMS in dependence on the positioning error.

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Fig. 7.
Fig. 7. Reconstruction error for different positioning errors and spatial surface wavelength for TMS and extended TMS.

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Fig. 8.
Fig. 8. Influence of lateral measurement error (σ pos2) for high spatial frequencies for extended TMS. The maximum allowed degree of interpolation was o = 41.

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Fig. 9.
Fig. 9. Locations of interferometer measurements and choice of evaluation grid. Red vertical lines represent the measurement positions of the interferometer pixels, and the xk the chosen evaluation grid. The lower part of the figure shows the evaluation grid obtained from the pre-optimization procedure.

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Fig. 10.
Fig. 10. Influence of the choice of the evaluation grid for extended TMS for the surface on the left hand side of Fig. 5.

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Fig. 11.
Fig. 11. Influence of the relative pixel distance error on the reconstruction result of extended TMS.

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Fig. 12.
Fig. 12. Reconstruction errors for different surface extensions.

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Tables (1)

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Table 1. Summary of simulation parameters

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Equations (9)

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m i , j = − f ( p i + s ( j ) ) + ε j + a i + b i s ( j ) .
m i , j = − f ( x k ) + ε j + a i + b i s ( j ) ,
m i , j = − ∑ k = floor x̃ d s − o − 1 2 ceil ( x̃ d s ) + o − 1 2 c k ( x̃ ) f ( x k ) + ε j + a i + b i s ( j )
= c N T ⋅ f N + ε j + a i + b i s ( j )
c k ( x̃ ) = ∏ floor ( x̃ d s ) i ≠ k − o − 1 2 ceil ( x̃ d s ) + o − 1 2 x̃ − x i x k − x i , c N T = ( 0 , . . , 0 , c floor ( x̃ d s ) − o − 1 2 ( x̃ ) , . . , c ceil ( x̃ d s ) + o − 1 2 ( x̃ ) , 0 , . . , 0 )
f N T = ( f ( x 0 ) , . . , f ( x N − 1 ) ) .
m i , j = c N T ⋅ f N + ε j + a i + b i s ( j ) , A = ( L ∣ I N ∣ R ) .
= ( Ac N + o ) T ⋅ f N + ε j + a i + b i s ( j )
d s > 0.5 d pix 1 , d step ≤ d s .

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