Abstract

A modified method for measuring the absolute figure of a large optical flat surface in synchrotron radiation by a small aperture interferometer is presented. The method consists of two procedures: the first step is oblique incidence measurement; the second is multiple rotating measurements. This simple method is described in terms of functions that are symmetric or antisymmetric with respect to reflections at the vertical axis. Absolute deviations of a large flat surface could be obtained when mirror antisymmetric errors are removed by N-position rotational averaging. Formulas are derived for measuring the absolute surface errors of a rectangle flat, and experiments on high-accuracy rectangle flats are performed to verify the method. Finally, uncertainty analysis is carried out in detail.

© 2014 Optical Society of America

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    [CrossRef]
  3. P. C. V. Mallik, C. Y. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
    [CrossRef]
  4. J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46, 8466–8474 (2007).
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  5. G. Ehret, M. Schulz, M. Stavridis, and C. Elster, “Deflectometric systems for absolute flatness measurements at PTB,” Meas. Sci. Technol. 23, 094007 (2012).
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  6. S. N. Qian, P. Takacs, G. Sostero, and D. Cocco, “Portable long trace profiler: concept and solution,” Rev. Sci. Instrum. 72, 3198–3204 (2001).
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  7. F. Siewert, J. Buchheim, and T. Zeschke, “Characterization and calibration of 2nd generation slope measuring profiler,” Nucl. Instrum. Methods Phys. Res. A 616, 119–127 (2010).
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    [CrossRef]
  12. S. Han, E. Novak, and M. Schurig, “Application of Ritchey-Common test in large flat measurements,” Proc. SPIE 4399, 131–136 (2001).
    [CrossRef]
  13. C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing’,” CIRP Ann 45, 617–634 (1996).
    [CrossRef]
  14. S. Han, E. Novak, and J. Sullivan, “Application of grazing incidence interferometer to rough surface measurements,” Proc. SPIE 5144, 391 (2003).
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  15. M. Vannoni, “Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data,” Opt. Express 22, 3538–3546 (2014).
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  24. W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
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  25. P. Hariharan, “Interferometric testing of optical surfaces: Absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
    [CrossRef]
  26. R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, and B. Loucks, “Measurements of the LIGO pathfinder optics,” Proc. SPIE 3134, 95–111 (1997).
    [CrossRef]
  27. R. E. Parks, L. Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
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    [CrossRef]
  30. M. F. Kuchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001).
    [CrossRef]
  31. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006).
    [CrossRef]
  32. U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
    [CrossRef]
  33. M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15, 6809–6816 (2007).
    [CrossRef]
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2014

2013

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

2012

G. Ehret, M. Schulz, M. Stavridis, and C. Elster, “Deflectometric systems for absolute flatness measurements at PTB,” Meas. Sci. Technol. 23, 094007 (2012).
[CrossRef]

2011

S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011).
[CrossRef]

2010

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, 21–31 (2010).
[CrossRef]

F. Siewert, J. Buchheim, and T. Zeschke, “Characterization and calibration of 2nd generation slope measuring profiler,” Nucl. Instrum. Methods Phys. Res. A 616, 119–127 (2010).
[CrossRef]

2008

2007

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15, 6809–6816 (2007).
[CrossRef]

R. D. Geckeler, “Optimal use of pentaprisms in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115–125 (2007).
[CrossRef]

P. C. V. Mallik, C. Y. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
[CrossRef]

J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46, 8466–8474 (2007).
[CrossRef]

2006

2003

S. Han, E. Novak, and J. Sullivan, “Application of grazing incidence interferometer to rough surface measurements,” Proc. SPIE 5144, 391 (2003).
[CrossRef]

2001

S. Han, E. Novak, and M. Schurig, “Application of Ritchey-Common test in large flat measurements,” Proc. SPIE 4399, 131–136 (2001).
[CrossRef]

S. N. Qian, P. Takacs, G. Sostero, and D. Cocco, “Portable long trace profiler: concept and solution,” Rev. Sci. Instrum. 72, 3198–3204 (2001).
[CrossRef]

L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001).
[CrossRef]

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637–1648 (2001).
[CrossRef]

M. F. Kuchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001).
[CrossRef]

1999

1998

1997

W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

P. Hariharan, “Interferometric testing of optical surfaces: Absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, and B. Loucks, “Measurements of the LIGO pathfinder optics,” Proc. SPIE 3134, 95–111 (1997).
[CrossRef]

1996

C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
[CrossRef]

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

1994

1993

1992

1984

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 234379 (1984).
[CrossRef]

1967

1966

G. D. Dew, “Measurement of optical flatness,” J. Sci. Instrum. 43, 409–415 (1966).
[CrossRef]

Ai, C. Y.

Artemiev, N. A.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Assoufid, L.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001).
[CrossRef]

Barber, S. K.

S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011).
[CrossRef]

Brown, N.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Buchheim, J.

S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011).
[CrossRef]

F. Siewert, J. Buchheim, and T. Zeschke, “Characterization and calibration of 2nd generation slope measuring profiler,” Nucl. Instrum. Methods Phys. Res. A 616, 119–127 (2010).
[CrossRef]

Burge, J. H.

Cocco, D.

S. N. Qian, P. Takacs, G. Sostero, and D. Cocco, “Portable long trace profiler: concept and solution,” Rev. Sci. Instrum. 72, 3198–3204 (2001).
[CrossRef]

Crews, D.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Dai, G. M.

Del Vecchio, C.

Dew, G. D.

G. D. Dew, “Measurement of optical flatness,” J. Sci. Instrum. 43, 409–415 (1966).
[CrossRef]

Ehret, G.

G. Ehret, M. Schulz, M. Stavridis, and C. Elster, “Deflectometric systems for absolute flatness measurements at PTB,” Meas. Sci. Technol. 23, 094007 (2012).
[CrossRef]

Elssner, K. E.

Elster, C.

G. Ehret, M. Schulz, M. Stavridis, and C. Elster, “Deflectometric systems for absolute flatness measurements at PTB,” Meas. Sci. Technol. 23, 094007 (2012).
[CrossRef]

Erdmann, M.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Estler, W. T.

W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

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

Evans, C. J.

R. E. Parks, L. Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
[CrossRef]

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, and B. Loucks, “Measurements of the LIGO pathfinder optics,” Proc. SPIE 3134, 95–111 (1997).
[CrossRef]

W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
[CrossRef]

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

Freischlad, K. R.

Fritz, B. S.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 234379 (1984).
[CrossRef]

Geckeler, R. D.

S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011).
[CrossRef]

R. D. Geckeler, “Optimal use of pentaprisms in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115–125 (2007).
[CrossRef]

Greco, V.

Griesmann, U.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

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

Grzanna, J.

Gubarev, M. V.

S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011).
[CrossRef]

Han, S.

S. Han, E. Novak, and J. Sullivan, “Application of grazing incidence interferometer to rough surface measurements,” Proc. SPIE 5144, 391 (2003).
[CrossRef]

S. Han, E. Novak, and M. Schurig, “Application of Ritchey-Common test in large flat measurements,” Proc. SPIE 4399, 131–136 (2001).
[CrossRef]

Hariharan, P.

P. Hariharan, “Interferometric testing of optical surfaces: Absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

Hignette, O.

L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001).
[CrossRef]

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, 617–634 (1996).
[CrossRef]

Howells, M.

L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001).
[CrossRef]

Irick, S.

L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001).
[CrossRef]

Jemian, P.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Kestner, R. N.

Kuchel, M. F.

M. F. Kuchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001).
[CrossRef]

Lammert, H.

L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001).
[CrossRef]

Loucks, B.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, and B. Loucks, “Measurements of the LIGO pathfinder optics,” Proc. SPIE 3134, 95–111 (1997).
[CrossRef]

Mahajan, V. N.

Mallik, P. C. V.

P. C. V. Mallik, C. Y. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
[CrossRef]

McKinney, W. R.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Merthe, D. J.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Molesini, G.

Novak, E.

S. Han, E. Novak, and J. Sullivan, “Application of grazing incidence interferometer to rough surface measurements,” Proc. SPIE 5144, 391 (2003).
[CrossRef]

S. Han, E. Novak, and M. Schurig, “Application of Ritchey-Common test in large flat measurements,” Proc. SPIE 4399, 131–136 (2001).
[CrossRef]

Okada, K.

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

Otsubo, M.

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

Parks, R. E.

Press, W. H.

W. H. Press, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 3, pp. 120–124.

Qian, J.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

Qian, S. N.

S. N. Qian, P. Takacs, G. Sostero, and D. Cocco, “Portable long trace profiler: concept and solution,” Rev. Sci. Instrum. 72, 3198–3204 (2001).
[CrossRef]

Schulz, G.

Schulz, M.

G. Ehret, M. Schulz, M. Stavridis, and C. Elster, “Deflectometric systems for absolute flatness measurements at PTB,” Meas. Sci. Technol. 23, 094007 (2012).
[CrossRef]

Schurig, M.

S. Han, E. Novak, and M. Schurig, “Application of Ritchey-Common test in large flat measurements,” Proc. SPIE 4399, 131–136 (2001).
[CrossRef]

Schwider, J.

Shao, L. Z.

R. E. Parks, L. Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
[CrossRef]

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, and B. Loucks, “Measurements of the LIGO pathfinder optics,” Proc. SPIE 3134, 95–111 (1997).
[CrossRef]

W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997).
[CrossRef]

Siewert, F.

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011).
[CrossRef]

F. Siewert, J. Buchheim, and T. Zeschke, “Characterization and calibration of 2nd generation slope measuring profiler,” Nucl. Instrum. Methods Phys. Res. A 616, 119–127 (2010).
[CrossRef]

Soons, J.

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007).
[CrossRef]

Sostero, G.

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Nucl. Instrum. Methods Phys. Res. A

F. Siewert, J. Buchheim, and T. Zeschke, “Characterization and calibration of 2nd generation slope measuring profiler,” Nucl. Instrum. Methods Phys. Res. A 616, 119–127 (2010).
[CrossRef]

L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013).
[CrossRef]

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[CrossRef]

P. C. V. Mallik, C. Y. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
[CrossRef]

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[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of the absolute flatness testing for long optics. (a) Configuration of oblique incidence measurement in top view. The large surface under test is arranged by its external normal at an angle α to the optical axis of the interferometer. (b) N-position rotating measurements between transmission flat and reference flat in side view. The reference flat is rotated by an angle ΔΦ=2π/N in N equal steps about the optical axis. The coordinate system of the interferometer is indicated with bold arrows.

Fig. 2.
Fig. 2.

Simulation of the absolute oblique incidence test at an 82° angle with 12-position averaging. Top row: simulated long flat wavefront; middle row: reconstructed long flat wavefront; bottom row: difference between the simulated wavefront and the reconstructed wavefront by enlarging 500×.

Fig. 3.
Fig. 3.

Decrease of rotating average error rms by N rotational position averaging with different flatness of the reference flat B which is rotated.

Fig. 4.
Fig. 4.

Relative interpolation error due to oblique incidence angle for different flatness of the long surface under test.

Fig. 5.
Fig. 5.

Measurement results of the flatness of the rectangular plane at oblique incidence angles of 38.9°, 61.0°, 72.8°, and 81.7°. To the left are the relative test results, and to the right are the corresponding reconstructed absolute results with 24-position averaging.

Fig. 6.
Fig. 6.

Comparison of reconstructed oblique incidence test results from oblique incidence angle of 81.7° with the normal incidence absolute result. Top row: reconstructed relative flat figure; middle row: reconstructed absolute flat figure with 24-position averaging; bottom row: absolute flat figure obtained from the normal incidence measurements using modified Griesmann’s approach.

Fig. 7.
Fig. 7.

Differences of measured figure results of rectangular plane by oblique incidence and absolute normal incidence methods. To the left are the differences of the relative oblique incidence result and normal incidence result, and to the right are the differences of the absolute oblique incidence result and normal incidence result.

Tables (1)

Tables Icon

Table 1. PV and RMS Values of Flatness Errors of the Rectangular Plane by Absolute Oblique Incidence Test with Different Oblique Incidence Angles and Rotating Numbers

Equations (26)

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We(x,y)=12[W(x,y)+W(x,y)],Wo(x,y)=12[W(x,y)W(x,y)].
WR(x,y)=limN(1Nk=0N1[W(x,y)]k·ΔΦ).
ΩR=(WWR)R=WRWR=0.
[We]R=WR,[Wo]R=0.
W1(x,y)=WA(x,y)+WB(x,y)+2cosα·WC(xcosα,y),W2(x,y)=WA(x,y)+WB(x,y),W3(x,y)=WA(x,y)+[WB(x,y)]R.
W1e(x,y)+W1o(x,y)=WAe(x,y)WAo(x,y)+WBe(x,y)WBo(x,y)+2cosα·[WCe(xcosα,y)+WCo(xcosα,y)],W2e(x,y)+W2o(x,y)=WAe(x,y)WAo(x,y)+WBe(x,y)+WBo(x,y),W3e(x,y)+W3o(x,y)=WAe(x,y)WAo(x,y)+[WB(x,y)]R+0.
[W1o(x,y)W2o(x,y)W3o(x,y)]=[112cosα110100][WAo(x,y)WBo(x,y)WCo(x/cosα,y)].
[WAo(x,y)WBo(x,y)WCo(x/cosα,y)]=[0010111/2cosα1/2cosα1/cosα][W1o(x,y)W2o(x,y)W3o(x,y)].
WCe(xcosα,y)=12cosα(W1e(x,y)W2e(x,y)).
WC(xcosα,y)=WCe(xcosα,y)+WCo(xcosα,y)=W1e(x,y)+W1o(x,y)W2e(x,y)+W2o(x,y)2W3o(x,y)2cosα=W1(x,y)W2e(x,y)+W2o(x,y)2W3o(x,y)2cosα.
Wf=W1W2e+W2o2W3o.
dWC=d(Wf2cosα)=dWf2cosα+sinα·dα2cos2α·Wf,
cosα=LCL.
sinα·dα=L·dLCLC·dLL2.
dWC=dWf2cosα(1cosα)·dLCLC·WC.
ΔWCΔWf2cosα(1cosα)·ΔLCLC·WC,
Ω(r,θ)=n=1(an·cosnθ+bn·sinnθ),
W3o=WAo+ΔΩR.
ΔWf=2·ΔΩR.
ΔWCΔΩRcosα(1cosα)·ΔLCLC·WC.
ΔWC=ΔΩRcosα(1cosα)·ΔLCLC·WC+ΔS,
δ2(W2e)=δ2(W2o)=12δ02,δ2(W3o)=12δ2(W3)=12Nδ02.
δr=12cosα·1+1N·δ0.
δr=12cos81.7°·1+112·δ0=2.6nm.
δs=(δΔΩRcos81.7°)2+((1cos81.7°)×0.37150·δWc)2+(0.06%·δWc)2=0.036nm.
δ=δs2+δr2=2.6nm.

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