Abstract

White-light interferometry on rough surfaces is an optical method for the measurement of the geometrical form of objects. The longitudinal coordinate of the measured surface is obtained from the measured interferogram by means of an evaluation method. However, the longitudinal coordinate cannot be determined completely accurately because the interferogram is affected by noise. We calculate the lower limit of the longitudinal measurement uncertainty caused by noise by use of the Cramer–Rao inequality. Additionally, we calculate the lower limit of the longitudinal measurement uncertainty caused by shot noise only.

© 2012 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]
  2. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
    [CrossRef]
  3. T. Dresel, G. Haüsler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef]
  4. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
    [CrossRef]
  5. L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
    [CrossRef]
  6. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
    [CrossRef]
  7. 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.
  8. G. Häusler, “Speckle and Coherence,” in Encyclopedia of Modern Optics, B. D. Guenther, ed. (Elsevier, Academic, 2005), pp. 114–123.
  9. 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]
  10. Z. Saraç, R. Gross, C. Richter, B. Wiesner, and G. Haüsler, “Optimization of white light interferometry on rough surfaces based on error analysis,” Optik 115, 351–357 (2004).
    [CrossRef]
  11. G. Haüsler, “Three-dimensional sensors—potential and limitations,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 485–506.
  12. M. Hering, K. Körner, and B. Jähne, “Correlated speckle noise in white-light interferometry: theoretical analysis of measurement uncertainty,” Appl. Opt. 48, 525–538 (2009).
    [CrossRef]
  13. M. Fleischer, R. Windecker, and H. J. Tiziani, “Theoretical limits of scanning white-light interferometry signal evaluation algorithms,” Appl. Opt. 40, 2815–2820 (2001).
    [CrossRef]
  14. M. Fox, Quantum Optics (Oxford University, 2006).
  15. B. Jähne, “Continuous and digital signals,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 9–34.
  16. T. Seiffert, “Verfahren zur schnellen Signalaufnahme in der Weisslichtinterferometrie,” Ph.D. dissertation (University Erlangen-Nuremberg, 2007).
  17. H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, 2003).
  18. M. Born and E. Wolf, Principles of Optics (Cambridge University, 2003).
  19. K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2004).
  20. L. Kubáček, Foundations of Estimation Theory (Elsevier, 1988).
  21. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984), pp. 9–75.
  22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).
  23. M. Fleischer, R. Windecker, and H. J. Tiziani, “Fast algorithms for data reduction in modern optical three-dimensional profile measurement systems with MMX technology,” Appl. Opt. 39, 1290–1297 (2000).
    [CrossRef]
  24. A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Signal evaluation for high-speed confocal measurements,” Appl. Opt. 41, 7410–7415 (2002).
    [CrossRef]
  25. T. Wilson, Confocal Microscopy (Academic, 1990).
  26. W. Gong, K. Si, and C. J. R. Sheppard, “Optimization of axial resolution in a confocal microscope with D-shaped apertures,” Appl. Opt. 48, 3998–4002 (2009).
    [CrossRef]
  27. S. K. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
    [CrossRef]
  28. Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.
  29. R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric phase-measurement using a one-dimensional discrete Hilbert transform,” Opt. Rev. 12, 29–36 (2005).
    [CrossRef]
  30. M. Kendall, A. Stuart, and J. K. Ord, The Advanced Theory of Statistics (Charles Griffin, 1983).
  31. J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer Academic, 1991).
  32. P. Pavliček and O. Hýbl, “White-light interferometry on rough surfaces—measurement uncertainty caused by surface roughness,” Appl. Opt. 47, 2941–2949 (2008).
    [CrossRef]
  33. T. Dresel, “Grundlagen und Grenzen der 3D-Datengewinnung mit dem Kohärenzradar,” Master’s thesis (University Erlangen-Nuremberg, 1991).
  34. R. Gross, O. Hýbl, B. Knapp, and G. Häusler, “Ultrafast 4-Megapixel white-light interferometry,” in DGaO Proceedings (German Society for Applied Optics, 2009), http://www.dgao-proceedings.de/download/110/110_p17.pdf.

2009 (2)

2008 (1)

2005 (1)

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

2004 (1)

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

2002 (1)

2001 (1)

2000 (1)

1998 (1)

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]

1996 (1)

1994 (2)

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
[CrossRef]

S. K. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

1993 (1)

1992 (1)

1990 (2)

Albrecht, H.-E.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, 2003).

An, Y.

Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.

Bence, S. J.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2004).

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, 2003).

Borys, M.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, 2003).

Caber, P. J.

Chim, S. S. C.

Chung, H.-S.

Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.

Damaschke, N.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, 2003).

de Groot, P.

Deck, L.

Dresel, T.

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

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.

Fleischer, M.

Fox, M.

M. Fox, Quantum Optics (Oxford University, 2006).

Gong, W.

Goodman, J. W.

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

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Gross, R.

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

R. Gross, O. Hýbl, B. Knapp, and G. Häusler, “Ultrafast 4-Megapixel white-light interferometry,” in DGaO Proceedings (German Society for Applied Optics, 2009), http://www.dgao-proceedings.de/download/110/110_p17.pdf.

Haüsler, G.

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

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

G. Haüsler, “Three-dimensional sensors—potential and limitations,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 485–506.

Häusler, G.

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.

G. Häusler, “Speckle and Coherence,” in Encyclopedia of Modern Optics, B. D. Guenther, ed. (Elsevier, Academic, 2005), pp. 114–123.

R. Gross, O. Hýbl, B. Knapp, and G. Häusler, “Ultrafast 4-Megapixel white-light interferometry,” in DGaO Proceedings (German Society for Applied Optics, 2009), http://www.dgao-proceedings.de/download/110/110_p17.pdf.

Hering, M.

Hobson, M. P.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2004).

Hýbl, O.

P. Pavliček and O. Hýbl, “White-light interferometry on rough surfaces—measurement uncertainty caused by surface roughness,” Appl. Opt. 47, 2941–2949 (2008).
[CrossRef]

R. Gross, O. Hýbl, B. Knapp, and G. Häusler, “Ultrafast 4-Megapixel white-light interferometry,” in DGaO Proceedings (German Society for Applied Optics, 2009), http://www.dgao-proceedings.de/download/110/110_p17.pdf.

Ishii, Y.

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

Jähne, B.

M. Hering, K. Körner, and B. Jähne, “Correlated speckle noise in white-light interferometry: theoretical analysis of measurement uncertainty,” Appl. Opt. 48, 525–538 (2009).
[CrossRef]

B. Jähne, “Continuous and digital signals,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 9–34.

Kang, G.

Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.

Kendall, M.

M. Kendall, A. Stuart, and J. K. Ord, The Advanced Theory of Statistics (Charles Griffin, 1983).

Kim, I.-J.

Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.

Kino, G. S.

Knapp, B.

R. Gross, O. Hýbl, B. Knapp, and G. Häusler, “Ultrafast 4-Megapixel white-light interferometry,” in DGaO Proceedings (German Society for Applied Optics, 2009), http://www.dgao-proceedings.de/download/110/110_p17.pdf.

Körner, K.

Kubácek, L.

L. Kubáček, Foundations of Estimation Theory (Elsevier, 1988).

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.

Nakagawa, Y.

S. K. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

Nayar, S. K.

S. K. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

Onodera, R.

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

Ord, J. K.

M. Kendall, A. Stuart, and J. K. Ord, The Advanced Theory of Statistics (Charles Griffin, 1983).

Park, J.

Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.

Pavlicek, P.

Perina, J.

J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer Academic, 1991).

Richter, C.

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

Riley, K. F.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2004).

Ruprecht, A. K.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Saraç, Z.

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

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]

Seiffert, T.

T. Seiffert, “Verfahren zur schnellen Signalaufnahme in der Weisslichtinterferometrie,” Ph.D. dissertation (University Erlangen-Nuremberg, 2007).

Sheppard, C. J. R.

Si, K.

Strand, T. C.

Stuart, A.

M. Kendall, A. Stuart, and J. K. Ord, The Advanced Theory of Statistics (Charles Griffin, 1983).

Tiziani, H. J.

Tropea, C.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, 2003).

Venzke, H.

Watanabe, H.

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

Wiesendanger, T. F.

Wiesner, B.

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

Wilson, T.

T. Wilson, Confocal Microscopy (Academic, 1990).

Windecker, R.

Wolf, E.

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

Appl. Opt. (11)

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

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

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[CrossRef]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
[CrossRef]

M. Fleischer, R. Windecker, and H. J. Tiziani, “Fast algorithms for data reduction in modern optical three-dimensional profile measurement systems with MMX technology,” Appl. Opt. 39, 1290–1297 (2000).
[CrossRef]

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

M. Fleischer, R. Windecker, and H. J. Tiziani, “Theoretical limits of scanning white-light interferometry signal evaluation algorithms,” Appl. Opt. 40, 2815–2820 (2001).
[CrossRef]

A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Signal evaluation for high-speed confocal measurements,” Appl. Opt. 41, 7410–7415 (2002).
[CrossRef]

P. Pavliček and O. Hýbl, “White-light interferometry on rough surfaces—measurement uncertainty caused by surface roughness,” Appl. Opt. 47, 2941–2949 (2008).
[CrossRef]

M. Hering, K. Körner, and B. Jähne, “Correlated speckle noise in white-light interferometry: theoretical analysis of measurement uncertainty,” Appl. Opt. 48, 525–538 (2009).
[CrossRef]

W. Gong, K. Si, and C. J. R. Sheppard, “Optimization of axial resolution in a confocal microscope with D-shaped apertures,” Appl. Opt. 48, 3998–4002 (2009).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. K. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

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

Opt. Rev. (1)

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

Optik (1)

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

Proc. SPIE (1)

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 (18)

G. Haüsler, “Three-dimensional sensors—potential and limitations,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 485–506.

M. Fox, Quantum Optics (Oxford University, 2006).

B. Jähne, “Continuous and digital signals,” in Handbook of Computer Vision and Applications, B. Jähne, H. Haussecker, and P. Geissler, eds. (Academic, 1999), pp. 9–34.

T. Seiffert, “Verfahren zur schnellen Signalaufnahme in der Weisslichtinterferometrie,” Ph.D. dissertation (University Erlangen-Nuremberg, 2007).

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, 2003).

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

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2004).

L. Kubáček, Foundations of Estimation Theory (Elsevier, 1988).

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

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

T. Wilson, Confocal Microscopy (Academic, 1990).

M. Kendall, A. Stuart, and J. K. Ord, The Advanced Theory of Statistics (Charles Griffin, 1983).

J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer Academic, 1991).

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

R. Gross, O. Hýbl, B. Knapp, and G. Häusler, “Ultrafast 4-Megapixel white-light interferometry,” in DGaO Proceedings (German Society for Applied Optics, 2009), http://www.dgao-proceedings.de/download/110/110_p17.pdf.

Y. An, G. Kang, I.-J. Kim, H.-S. Chung, and J. Park, “Shape from focus through Laplacian using 3D window,” in Proceedings of IEEE Second International Conference on Future Generation Communication and Networking (IEEE, 2008), pp. 46–50.

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,” in Encyclopedia of Modern Optics, B. D. Guenther, ed. (Elsevier, Academic, 2005), pp. 114–123.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of white-light interferometry.

Fig. 2.
Fig. 2.

White-light interferogram.

Fig. 3.
Fig. 3.

Signal without modulation.

Fig. 4.
Fig. 4.

Demodulation of a simulated interferogram by means of Hilbert transform (Δz=λ0/10). (a) Signal before and after the demodulation. (b) Noise extracted from the signal. (c) Correlation function of the noise.

Fig. 5.
Fig. 5.

Demodulation of a simulated interferogram by means of five-step phase-shifting algorithm (Δz=λ0/10). (a) Signal before and after the demodulation. (b) Noise extracted from the signal. (c) Correlation function of the noise.

Fig. 6.
Fig. 6.

Example of a model interferogram: λ0=850nm, FWHM=40nm, IM=255DN, I0=100DN, and IA=80DN.

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

I(z=zk)=Ik=mk+nk,k=0,1,,(N1).
m(z)=I0+IAexp[(zzMlc)2]cos[4πλ0(zzM)+φ],
m(z)=IR+IA24IR+IAexp[(zzMlc)2]cos[4πλ0(zzM)+φ].
p(Ik,ai)=(12πσ2)N2exp[12σ2k=0N1(Ikmk)2],
σai2(J1)ii,
Jij=E[2lnpaiaj],
Jij=1σ2k=0N1(mkaimkaj).
m(z)=I0+IAexp[(zzMlc)2]cos(4πλ0z+φ1),
J11=1σ2k=0N1{exp[2(zkzMlc)2]cos2(4πλ0zk+φ1)},
J22=4IA2σ2lc4k=0N1{exp[2(zkzMlc)2](zkzM)2cos2(4πλ0zk+φ1)},
J12=J21=2IAσ2lc2k=0N1{exp[2(zkzMlc)2](zkzM)cos2(4πλ0zk+φ1)}=0.
(J1)ii=(Jii)1.
σzM2J221.
J22=4IA2σ2lc4k=sns+n{exp[2(zkzMlc)2](zkzM)2cos2(4πλ0zk+φ1)}.
k=sns+n{exp[2(zkzMlc)2](zkzM)2cos2(4πλ0zk+φ1)}1ΔzLLexp[2(zlc)2]z2cos2(4πzλ0+φ)dz.
J2212π2IA2σ21lcΔz[erf(2u)22πuexp(2u2)],
δz=σzMC22π4σIAΔzlc,
C=[erf(2u)22πuexp(2u2)]1/2.
δz22π4σIAΔzlc.
m(z)=I0+IAexp[(zzMlc)2].
δz=σzMC2π4σIAΔzlc.
δz2π4σIAΔzlc.
R(k)=1(Nk)σ2t=0Nk1ntnt+k.
4M2sin4ψ=(I2I4)2(I1I3)(I3I5),
p(Ik,ai)=(12πσ2)N2exp{12σ2k=0N1l=0N1[(Ikmk)rkl(Ilml)]},
Jij=1σ2k=0N1l=0N1(mkairklmlaj).
J22=4IA2σ2lc4k=0N1l=0N1{exp[(zkzMlc)2](zkzM)rklexp[(zlzMlc)2](zlzM)}.
(Δnp)2=np.
I=IMnpnFW,
σ=ΔI=IMnFWI,
σ¯2=1Nk=0N1σk2=1NIMnFWk=0N1Ik.
σ¯21NΔzIMnFW0NΔz{I0+IAexp[(zzMlc)2]cos[4πλ0(zzM)+φ]}dz.
σ¯IMI0nFW.
δz=σzMC22π4IMnFWI0IAΔzlc.
δz22π4IMnFWI0IAΔzlc.
lcln2πλ02FWHM=4.8μm.
δzrough=12IobjIobjRq,

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