Abstract

White-light interferometry measuring an optically rough surface commonly does not resolve the lateral structure of the surface. This means that there are height differences within one resolution cell that exceed one-fourth of the wavelength of the light used. Thus the following questions arise: Which height is measured by white-light interferometry? How does the surface roughness affect the measurement uncertainty? The goal of the presented paper is to answer these questions by means of numerical simulations. Before the aforementioned questions can be answered, the distribution of the intensity of individual speckles, the influence of surface roughness, and the spectral width of the light source used are discussed.

© 2008 Optical Society of America

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  1. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775-3783 (1990).
    [CrossRef] [PubMed]
  2. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784-3788 (1990).
    [CrossRef] [PubMed]
  3. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832-843 (1996).
    [CrossRef]
  4. G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.
  5. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
    [CrossRef] [PubMed]
  6. G. Häusler, “Speckle and Coherence,” Encyclopedia of Modern Optics, B. D. Guenther, ed. (Academic, 2005), pp. 114 - 123.
    [CrossRef]
  7. P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
    [CrossRef]
  8. J. W. Goodman, “Statistical properties of laser speckle patterns,” Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 9-75.
  9. J. W. Goodman, “Speckle with finite number of steps,” Appl. Opt. 47, A111-A118 (2008).
    [CrossRef] [PubMed]
  10. S. G. Rabinovich, Measurement Errors and Uncertainties (Springer-Verlag, 2000).
  11. T. Dresel, “Grundlagen und Grenzen der 3D-Datengewinnung mit dem Kohärenzradar,” Master's thesis (University Erlangen-Nuremberg, 1991).
  12. P. Ettl, “Über die Signalentstehung bei Weißlichtinterferometrie,” Ph.D. dissertation (University Erlangen-Nuremberg, 2001).
  13. C. Richter, B. Wiesner, R. Groß, and G. Häusler, “White-light interferometry with higher accuracy and more speed,” in Proceedings of Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer-Verlag, 2005), pp. 605-612.
  14. P. Pavliček and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42, 1809-1813 (2003).
    [CrossRef] [PubMed]
  15. N. George and A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202-1212 (1973).
    [CrossRef] [PubMed]
  16. A. Harasaki and J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101-2106 (2000).
    [CrossRef]
  17. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 77-121.
  18. I. Yamaguchi, A. Yamamoto, and S. Kuwamura, “Speckle decorrelation in surface profilometry by wavelength scanning interferometry,” Appl. Opt. 37, 6721-6728 (1998).
    [CrossRef]
  19. R. Windecker and H. J. Tiziani, “Optical roughness measurements using extended white-light interferometry,” Opt. Eng. 38, 1081-1087 (1999).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2003).
  21. P. Lehmann, “Aspect ratio of elongated polychromatic far-field speckles of continuous and discrete spectral distribution with respect to surface roughness characterization,” Appl. Opt. 41, 2008-2014 (2002).
    [CrossRef] [PubMed]
  22. W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992).
  23. R. Onodera, H. Watanebe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29-36 (2005).
    [CrossRef]
  24. Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
    [CrossRef]
  25. B. Wiesner and G. Häusler, “A new method to reduce the measuring uncertainty and the number of outliers in white-light interferometry,” in DGaO Proceedings (2005), http://www.dgao-proceedings.de.
  26. The distribution has been estimated based on 800 000 simulations. The number of scattering regions N=200.
  27. The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
    [CrossRef]
  28. P. Horváth, M. Hrabovský, and Z. Bača, “Statistical properties of a speckle pattern,” Proc. SPIE 4888, 99-108 (2002).
    [CrossRef]
  29. J. N. Kapur and H. C. Saxena, Mathematical Statistics (S. Chand & Company Ltd., 2006).
  30. The distribution has been estimated based on 40 000 simulations. The number of scattering regions N=200.
  31. R.-J. Recknagel and G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122-128(1998).
    [CrossRef]
  32. P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
    [CrossRef] [PubMed]
  33. R. G. Dorsch, G. Häusler, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306-1314 (1994).
    [CrossRef] [PubMed]

2008 (1)

2005 (1)

R. Onodera, H. Watanebe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29-36 (2005).
[CrossRef]

2004 (2)

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
[CrossRef] [PubMed]

2003 (2)

The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
[CrossRef]

P. Pavliček and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42, 1809-1813 (2003).
[CrossRef] [PubMed]

2002 (2)

2000 (1)

1999 (1)

R. Windecker and H. J. Tiziani, “Optical roughness measurements using extended white-light interferometry,” Opt. Eng. 38, 1081-1087 (1999).
[CrossRef]

1998 (3)

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

R.-J. Recknagel and G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122-128(1998).
[CrossRef]

I. Yamaguchi, A. Yamamoto, and S. Kuwamura, “Speckle decorrelation in surface profilometry by wavelength scanning interferometry,” Appl. Opt. 37, 6721-6728 (1998).
[CrossRef]

1996 (1)

1994 (1)

1992 (1)

1990 (2)

1973 (1)

Baca, Z.

P. Horváth, M. Hrabovský, and Z. Bača, “Statistical properties of a speckle pattern,” Proc. SPIE 4888, 99-108 (2002).
[CrossRef]

Bohn, G.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

Chim, S. S. C.

Dorsch, R. G.

Dresel, T.

T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
[CrossRef] [PubMed]

T. Dresel, “Grundlagen und Grenzen der 3D-Datengewinnung mit dem Kohärenzradar,” Master's thesis (University Erlangen-Nuremberg, 1991).

Ettl, P.

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.

P. Ettl, “Über die Signalentstehung bei Weißlichtinterferometrie,” Ph.D. dissertation (University Erlangen-Nuremberg, 2001).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992).

George, N.

Goodman, J. W.

J. W. Goodman, “Speckle with finite number of steps,” Appl. Opt. 47, A111-A118 (2008).
[CrossRef] [PubMed]

J. W. Goodman, “Statistical properties of laser speckle patterns,” Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 9-75.

Groß, R.

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

C. Richter, B. Wiesner, R. Groß, and G. Häusler, “White-light interferometry with higher accuracy and more speed,” in Proceedings of Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer-Verlag, 2005), pp. 605-612.

Haderka, O.

The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
[CrossRef]

Harasaki, A.

Häusler, G.

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

R. G. Dorsch, G. Häusler, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306-1314 (1994).
[CrossRef] [PubMed]

T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
[CrossRef] [PubMed]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.

G. Häusler, “Speckle and Coherence,” Encyclopedia of Modern Optics, B. D. Guenther, ed. (Academic, 2005), pp. 114 - 123.
[CrossRef]

C. Richter, B. Wiesner, R. Groß, and G. Häusler, “White-light interferometry with higher accuracy and more speed,” in Proceedings of Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer-Verlag, 2005), pp. 605-612.

B. Wiesner and G. Häusler, “A new method to reduce the measuring uncertainty and the number of outliers in white-light interferometry,” in DGaO Proceedings (2005), http://www.dgao-proceedings.de.

Hendrych, M.

The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
[CrossRef]

Herrmann, J. M.

Horváth, P.

P. Horváth, M. Hrabovský, and Z. Bača, “Statistical properties of a speckle pattern,” Proc. SPIE 4888, 99-108 (2002).
[CrossRef]

Hrabovský, M.

P. Horváth, M. Hrabovský, and Z. Bača, “Statistical properties of a speckle pattern,” Proc. SPIE 4888, 99-108 (2002).
[CrossRef]

Ishii, Y.

R. Onodera, H. Watanebe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29-36 (2005).
[CrossRef]

Jain, A.

Kapur, J. N.

J. N. Kapur and H. C. Saxena, Mathematical Statistics (S. Chand & Company Ltd., 2006).

Kino, G. S.

Kuwamura, S.

Larkin, K. G.

Laszlo, I.

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.

Lee, B. S.

Lehmann, P.

Notni, G.

R.-J. Recknagel and G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122-128(1998).
[CrossRef]

Onodera, R.

R. Onodera, H. Watanebe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29-36 (2005).
[CrossRef]

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 77-121.

Pavlicek, P.

P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
[CrossRef] [PubMed]

The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
[CrossRef]

P. Pavliček and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42, 1809-1813 (2003).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992).

Rabinovich, S. G.

S. G. Rabinovich, Measurement Errors and Uncertainties (Springer-Verlag, 2000).

Recknagel, R.-J.

R.-J. Recknagel and G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122-128(1998).
[CrossRef]

Richter, C.

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

C. Richter, B. Wiesner, R. Groß, and G. Häusler, “White-light interferometry with higher accuracy and more speed,” in Proceedings of Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer-Verlag, 2005), pp. 605-612.

Saraç, Z.

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

Saxena, H. C.

J. N. Kapur and H. C. Saxena, Mathematical Statistics (S. Chand & Company Ltd., 2006).

Schenk, M.

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.

Schmidt, B.

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

Soubusta, J.

P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
[CrossRef] [PubMed]

The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
[CrossRef]

P. Pavliček and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42, 1809-1813 (2003).
[CrossRef] [PubMed]

Strand, T. C.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992).

Tiziani, H. J.

R. Windecker and H. J. Tiziani, “Optical roughness measurements using extended white-light interferometry,” Opt. Eng. 38, 1081-1087 (1999).
[CrossRef]

Venzke, H.

Vettering, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992).

Watanebe, H.

R. Onodera, H. Watanebe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29-36 (2005).
[CrossRef]

Wiesner, B.

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

B. Wiesner and G. Häusler, “A new method to reduce the measuring uncertainty and the number of outliers in white-light interferometry,” in DGaO Proceedings (2005), http://www.dgao-proceedings.de.

C. Richter, B. Wiesner, R. Groß, and G. Häusler, “White-light interferometry with higher accuracy and more speed,” in Proceedings of Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer-Verlag, 2005), pp. 605-612.

Windecker, R.

R. Windecker and H. J. Tiziani, “Optical roughness measurements using extended white-light interferometry,” Opt. Eng. 38, 1081-1087 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

Wyant, J. C.

Yamaguchi, I.

Yamamoto, A.

Appl. Opt. (11)

G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775-3783 (1990).
[CrossRef] [PubMed]

B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784-3788 (1990).
[CrossRef] [PubMed]

T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
[CrossRef] [PubMed]

J. W. Goodman, “Speckle with finite number of steps,” Appl. Opt. 47, A111-A118 (2008).
[CrossRef] [PubMed]

P. Pavliček and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42, 1809-1813 (2003).
[CrossRef] [PubMed]

N. George and A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202-1212 (1973).
[CrossRef] [PubMed]

A. Harasaki and J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101-2106 (2000).
[CrossRef]

I. Yamaguchi, A. Yamamoto, and S. Kuwamura, “Speckle decorrelation in surface profilometry by wavelength scanning interferometry,” Appl. Opt. 37, 6721-6728 (1998).
[CrossRef]

P. Lehmann, “Aspect ratio of elongated polychromatic far-field speckles of continuous and discrete spectral distribution with respect to surface roughness characterization,” Appl. Opt. 41, 2008-2014 (2002).
[CrossRef] [PubMed]

P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
[CrossRef] [PubMed]

R. G. Dorsch, G. Häusler, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306-1314 (1994).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

R.-J. Recknagel and G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122-128(1998).
[CrossRef]

Opt. Eng. (1)

R. Windecker and H. J. Tiziani, “Optical roughness measurements using extended white-light interferometry,” Opt. Eng. 38, 1081-1087 (1999).
[CrossRef]

Opt. Rev. (1)

R. Onodera, H. Watanebe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29-36 (2005).
[CrossRef]

Optik (Jena) (1)

Z. Saraç, R. Groß, C. Richter, B. Wiesner, and G. Häusler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik (Jena) 115, 351-357 (2004).
[CrossRef]

Proc. SPIE (3)

The random numbers used for the simulations have been generated by quantum random number generator developed in the Joint Laboratory of Optics, Olomouc: J. Soubusta, O. Haderka, M. Hendrych, and P. Pavliček, “Experimental realization of quantum random generator,” Proc. SPIE 5259, 7-13(2003).
[CrossRef]

P. Horváth, M. Hrabovský, and Z. Bača, “Statistical properties of a speckle pattern,” Proc. SPIE 4888, 99-108 (2002).
[CrossRef]

P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, and G. Häusler, “Roughness parameters and surface deformation measured by 'Coherence Radar',” Proc. SPIE 3407, 133-140 (1998).
[CrossRef]

Other (14)

J. W. Goodman, “Statistical properties of laser speckle patterns,” Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 9-75.

G. Häusler, “Speckle and Coherence,” Encyclopedia of Modern Optics, B. D. Guenther, ed. (Academic, 2005), pp. 114 - 123.
[CrossRef]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, and I. Laszlo, “Limits of optical range sensors and how to exploit them,” in International Trends in Optics and Photonics ICO IV, T. Asakura, ed., Springer Series in Optical Sciences (Springer-Verlag, 1999), Vol. 74, pp. 328-342.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 77-121.

S. G. Rabinovich, Measurement Errors and Uncertainties (Springer-Verlag, 2000).

T. Dresel, “Grundlagen und Grenzen der 3D-Datengewinnung mit dem Kohärenzradar,” Master's thesis (University Erlangen-Nuremberg, 1991).

P. Ettl, “Über die Signalentstehung bei Weißlichtinterferometrie,” Ph.D. dissertation (University Erlangen-Nuremberg, 2001).

C. Richter, B. Wiesner, R. Groß, and G. Häusler, “White-light interferometry with higher accuracy and more speed,” in Proceedings of Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer-Verlag, 2005), pp. 605-612.

J. N. Kapur and H. C. Saxena, Mathematical Statistics (S. Chand & Company Ltd., 2006).

The distribution has been estimated based on 40 000 simulations. The number of scattering regions N=200.

B. Wiesner and G. Häusler, “A new method to reduce the measuring uncertainty and the number of outliers in white-light interferometry,” in DGaO Proceedings (2005), http://www.dgao-proceedings.de.

The distribution has been estimated based on 800 000 simulations. The number of scattering regions N=200.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992).

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Figures (9)

Fig. 1
Fig. 1

Schematic of white-light interferometry on rough surface.

Fig. 2
Fig. 2

Intensity distribution for λ = 820 nm , FWHM ( λ ) = 38 nm , and σ h = 1.2 μm numerically calculated (solid curve) and analytical solution according to Parry (dashed curve).

Fig. 3
Fig. 3

Numerically calculated distribution of the measured error caused by surface roughness for λ = 820 nm , FWHM ( λ ) = 38 nm , σ h = 1.2 μm , and I outobj = I outobj .

Fig. 4
Fig. 4

Measured interferograms for λ = 820 nm , FWHM ( λ ) = 44 nm , R a = 1.6 μm . (a)  I outobj / I outobj = 1.2 . (b)  I outobj / I outobj = 4.5 .

Fig. 5
Fig. 5

Numerically calculated distribution of difference Δ form for FWHM ( λ ) = 38 nm , σ h = 1.2 μm , and I outobj = I outobj .

Fig. 6
Fig. 6

Limit spectral width FWHM lim as a function of roughness σ h for three values of the ratio I outobj / I outobj .

Fig. 7
Fig. 7

Numerically calculated measurement uncertainties δ X z and δ G z as functions of spectral width F W H M ( λ ) for λ = 820 nm , σ h = 1.2 μm , and I outobj = I outobj .

Fig. 8
Fig. 8

Numerically calculated measurement uncertainty δ G z as a function of spectral width FWHM ( λ ) for λ = 820 nm , σ h = 1.2 μm , and three values of the ratio I outobj / I outobj .

Fig. 9
Fig. 9

Numerically calculated measurement uncertainty δ G z as a function of spectral width FWHM ( λ ) for λ = 820 nm , I outobj = I outobj , and three values of roughness σ h .

Equations (19)

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δ z = 1 2 I I σ h .
S ( k ) = 1 2 π Δ k exp [ - ( k - k 0 2 Δ k ) 2 ] ,
Δ k = π ln 2 1 FWHM ( λ ) { 1 + [ FWHM ( λ ) λ ] 2 - 1 } π 2 ln 2 FWHM ( λ ) λ 2 .
l c = 1 2 Δ k ln 2 π λ 2 FWHM ( λ ) .
A ^ ( k ) = j = 1 N a j N exp [ i 2 k ( z O + h j ) ] ,
B ^ ( k ) = B exp ( i 2 k z R ) ,
I ( k ) = | A ^ ( k ) + B ^ ( k ) | 2 .
I ( k , z O z R ) = ( j = 1 N a j N cos φ j ) 2 + ( j = 1 N a j N sin φ j ) 2 + B 2 + 2 B j = 1 N a j N cos φ j ,
φ j = 2 k ( z O z R + h j ) .
I out ( z O z R ) = k 0 - 3 2 Δ k k 0 + 3 2 Δ k S ( k ) I ( k , z O z R ) d k .
Δ z = z OM z R .
I obj ( k ) = ( j = 1 N a j N cos φ j ) 2 + ( j = 1 N a j N sin φ j ) 2 ,
I outobj = k 0 - 3 2 Δ k k 0 + 3 2 Δ k S ( k ) I obj ( k ) d k .
p ( I ) = M M I M - 1 I M Γ ( M ) exp ( - M I I ) ,
M = 1 + 32 ( σ h Δ k ) 2 = 1 + 8 ( σ h l c ) 2
I = S ( k ) I ( k ) d k .
δ z = 1 2 I outobj I outobj σ h .
Δ form = z OMmax z OMgra ,
l clim = 4 I outobj I outobj σ h .

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