Abstract

In this paper, we present a practical approach for phase analysis of sinusoidally phase shifted interference signals, which are generally used to detect optical path length changes in one arm of the interferometer based on an algorithm introduced by de Groot. We describe the original algorithm from our point of view and try to emphasize the limitations and some details that need to be known for practical implementation. We introduce methods for how to overcome these limitations, and in addition, we provide an extension of the algorithm to a temporal high-resolution mode, which provides a possibility to calculate a phase value for each sampled point of an interference signal and opens new applications for the existing measurement devices without any hardware changes. Simulated and experimental results verify our extensions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).
    [Crossref]
  2. P. J. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7, 1–65 (2015).
    [Crossref]
  3. K. Creath, “Phase-measurement interferometry techniques,” in Progress in OpticsXXVI, E. Wolf, ed. (Elsevier, 1988).
    [Crossref]
  4. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref] [PubMed]
  5. P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [Crossref]
  6. O. Sasaki and H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25, 3137–3140 (1986).
    [Crossref] [PubMed]
  7. O. Sasaki and H. Okazaki, “Analysis of measurement accuracy in sinusoidal phase modulating interferometry,” Appl. Opt. 25, 3152–3158 (1986).
    [Crossref] [PubMed]
  8. U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
    [Crossref]
  9. A. Dubois, “Phase-map measurements by interferometry with sinusoidal phase modulation and four integrating buckets,” J. Opt. Soc. Am. A 18, 1972–1979 (2001).
    [Crossref]
  10. K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
    [Crossref]
  11. M. Schulz and P. Lehmann, “Measurement of distance changes using a fibre-coupled common-path interferometer with mechanical path length modulation,” Meas. Sci. Technol. 24, 065202 (2013).
    [Crossref]
  12. H. Knell, S. Laubach, G. Ehret, and P. Lehmann, “Continuous measurement of optical surfaces using a line-scan interferometer with sinusoidal path length modulation,” Opt. Express 22, 29787–29797 (2014).
    [Crossref]
  13. S. Tereschenko, P. Lehmann, L. Zellmer, and A. Brueckner-Foit, “Passive vibration compensation in scanning white-light interferometry,” Appl. Opt. 55, 6172–6182 (2016).
    [Crossref] [PubMed]
  14. P. J. de Groot, “Design of error-compensating algorithms for sinusoidal phase shifting interferometry,” Appl. Opt. 48, 6788–6796 (2009).
    [Crossref] [PubMed]
  15. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).
  16. P. J. de Groot, “Sinusoidal phase shifting interferometry,” Patent No. US7,933,025B2, Apr.26, 2011.
  17. S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
    [Crossref]
  18. K. Wang, Z. Ding, Y. Zeng, J. Meng, and M. Chen, “Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact,” Opt. Express 17, 16820–16833 (2009).
    [Crossref] [PubMed]
  19. S. Bochkanov, “ALGLIB,” http://www.alglib.net .
  20. J. H. Galeti, P. L. Berton, C. Kitano, R. T. Higuti, R. C. Carbonari, and E. C. N. Silva, “Wide dynamic range homodyne interferometry method and its application for piezoactuator displacement measurements,” Appl. Opt. 52, 6919–6930 (2013).
    [Crossref] [PubMed]
  21. B. J. Pernick, “Self-consistent and direct reading laser homodyne measurement technique,” Appl. Opt. 12, 607–610 (1973).
    [Crossref] [PubMed]
  22. K. Deb, “Multi-objective optimization,” in Search Methodologies, E. K. Burke and G. Kendall, eds. (Springer, 2014).
    [Crossref]
  23. A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: a tutorial,” Reliab. Eng. Syst. Saf. 91, 992–1007 (2006).
    [Crossref]
  24. R. Fletcher, Practical Methodes of Optimization(John Wiley & Sonsa, Ltd, 1980).
  25. P. J. de Groot, “Error compensation in phase shifting interferometry,” Patent No. US7,948,637B2, May. 24, 2011.
  26. E. Jacobsen and R. Lyons, “The sliding dft,” IEEE Signal Process. Mag. 20, 74–80 (2003).
    [Crossref]
  27. P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).
  28. K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
    [Crossref]
  29. M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
    [Crossref]
  30. S. S. C. Chim and G. S. Kino, “Phase measurements using the mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
    [Crossref] [PubMed]
  31. A. Olszak and J. Schmit, “High-stability white-light interferometry with reference signal for real-time correction of scanning errors,” Opt. Eng. 42, 54–59 (2003).
    [Crossref]
  32. M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “A new single-phase PLL structure based on second order generalized integrator,” in 37th IEEE Power Electronics Specialists Conference, (IEEE, 2006), pp. 1–6.

2017 (1)

S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
[Crossref]

2016 (1)

2015 (1)

P. J. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7, 1–65 (2015).
[Crossref]

2014 (1)

2013 (2)

J. H. Galeti, P. L. Berton, C. Kitano, R. T. Higuti, R. C. Carbonari, and E. C. N. Silva, “Wide dynamic range homodyne interferometry method and its application for piezoactuator displacement measurements,” Appl. Opt. 52, 6919–6930 (2013).
[Crossref] [PubMed]

M. Schulz and P. Lehmann, “Measurement of distance changes using a fibre-coupled common-path interferometer with mechanical path length modulation,” Meas. Sci. Technol. 24, 065202 (2013).
[Crossref]

2009 (3)

2008 (1)

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

2006 (1)

A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: a tutorial,” Reliab. Eng. Syst. Saf. 91, 992–1007 (2006).
[Crossref]

2003 (2)

E. Jacobsen and R. Lyons, “The sliding dft,” IEEE Signal Process. Mag. 20, 74–80 (2003).
[Crossref]

A. Olszak and J. Schmit, “High-stability white-light interferometry with reference signal for real-time correction of scanning errors,” Opt. Eng. 42, 54–59 (2003).
[Crossref]

2001 (1)

1995 (1)

1992 (1)

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

1991 (2)

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

S. S. C. Chim and G. S. Kino, “Phase measurements using the mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
[Crossref] [PubMed]

1987 (2)

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[Crossref] [PubMed]

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
[Crossref]

1986 (2)

1973 (1)

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

Berton, P. L.

Blaabjerg, F.

M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “A new single-phase PLL structure based on second order generalized integrator,” in 37th IEEE Power Electronics Specialists Conference, (IEEE, 2006), pp. 1–6.

Brueckner-Foit, A.

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).
[Crossref]

Carbonari, R. C.

Chen, M.

Chim, S. S. C.

Ciobotaru, M.

M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “A new single-phase PLL structure based on second order generalized integrator,” in 37th IEEE Power Electronics Specialists Conference, (IEEE, 2006), pp. 1–6.

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
[Crossref]

Coit, D. W.

A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: a tutorial,” Reliab. Eng. Syst. Saf. 91, 992–1007 (2006).
[Crossref]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in OpticsXXVI, E. Wolf, ed. (Elsevier, 1988).
[Crossref]

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
[Crossref]

de Groot, P. J.

P. J. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7, 1–65 (2015).
[Crossref]

P. J. de Groot, “Design of error-compensating algorithms for sinusoidal phase shifting interferometry,” Appl. Opt. 48, 6788–6796 (2009).
[Crossref] [PubMed]

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[Crossref]

P. J. de Groot, “Error compensation in phase shifting interferometry,” Patent No. US7,948,637B2, May. 24, 2011.

P. J. de Groot, “Sinusoidal phase shifting interferometry,” Patent No. US7,933,025B2, Apr.26, 2011.

Deb, K.

K. Deb, “Multi-objective optimization,” in Search Methodologies, E. K. Burke and G. Kendall, eds. (Springer, 2014).
[Crossref]

Deck, L. L.

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

Ding, Z.

Docchio, F.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

Dubois, A.

Ehret, G.

Eiju, T.

Falaggis, K.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[Crossref]

Fleming, J.

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

Fletcher, R.

R. Fletcher, Practical Methodes of Optimization(John Wiley & Sonsa, Ltd, 1980).

Galeti, J. H.

Gelmini, E.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

Gollor, P.

S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
[Crossref]

Ha, S.

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

Hariharan, P.

Higuti, R. T.

Jacobsen, E.

E. Jacobsen and R. Lyons, “The sliding dft,” IEEE Signal Process. Mag. 20, 74–80 (2003).
[Crossref]

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
[Crossref]

Kino, G. S.

Kitano, C.

Knell, H.

Konak, A.

A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: a tutorial,” Reliab. Eng. Syst. Saf. 91, 992–1007 (2006).
[Crossref]

Kuehnhold, P.

S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
[Crossref]

Laubach, S.

Lehmann, P.

S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
[Crossref]

S. Tereschenko, P. Lehmann, L. Zellmer, and A. Brueckner-Foit, “Passive vibration compensation in scanning white-light interferometry,” Appl. Opt. 55, 6172–6182 (2016).
[Crossref] [PubMed]

H. Knell, S. Laubach, G. Ehret, and P. Lehmann, “Continuous measurement of optical surfaces using a line-scan interferometer with sinusoidal path length modulation,” Opt. Express 22, 29787–29797 (2014).
[Crossref]

M. Schulz and P. Lehmann, “Measurement of distance changes using a fibre-coupled common-path interferometer with mechanical path length modulation,” Meas. Sci. Technol. 24, 065202 (2013).
[Crossref]

Lyons, R.

E. Jacobsen and R. Lyons, “The sliding dft,” IEEE Signal Process. Mag. 20, 74–80 (2003).
[Crossref]

Marioli, D.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
[Crossref]

Meng, J.

Minoni, U.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

Okazaki, H.

Olszak, A.

A. Olszak and J. Schmit, “High-stability white-light interferometry with reference signal for real-time correction of scanning errors,” Opt. Eng. 42, 54–59 (2003).
[Crossref]

Oreb, B. F.

Pernick, B. J.

Sardini, E.

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

Sasaki, O.

Schmit, J.

A. Olszak and J. Schmit, “High-stability white-light interferometry with reference signal for real-time correction of scanning errors,” Opt. Eng. 42, 54–59 (2003).
[Crossref]

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).
[Crossref]

Schulz, M.

M. Schulz and P. Lehmann, “Measurement of distance changes using a fibre-coupled common-path interferometer with mechanical path length modulation,” Meas. Sci. Technol. 24, 065202 (2013).
[Crossref]

Shinpaugh, K.

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

Silva, E. C. N.

Simpson, R.

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

Smith, A. E.

A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: a tutorial,” Reliab. Eng. Syst. Saf. 91, 992–1007 (2006).
[Crossref]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

Teodorescu, R.

M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “A new single-phase PLL structure based on second order generalized integrator,” in 37th IEEE Power Electronics Specialists Conference, (IEEE, 2006), pp. 1–6.

Tereschenko, S.

S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
[Crossref]

S. Tereschenko, P. Lehmann, L. Zellmer, and A. Brueckner-Foit, “Passive vibration compensation in scanning white-light interferometry,” Appl. Opt. 55, 6172–6182 (2016).
[Crossref] [PubMed]

Towers, C. E.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[Crossref]

Towers, D. P.

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[Crossref]

Wang, K.

Wicks, A.

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

Zellmer, L.

Zeng, Y.

Adv. Opt. Photonics (1)

P. J. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7, 1–65 (2015).
[Crossref]

Appl. Opt. (9)

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[Crossref] [PubMed]

P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[Crossref]

O. Sasaki and H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25, 3137–3140 (1986).
[Crossref] [PubMed]

O. Sasaki and H. Okazaki, “Analysis of measurement accuracy in sinusoidal phase modulating interferometry,” Appl. Opt. 25, 3152–3158 (1986).
[Crossref] [PubMed]

S. Tereschenko, P. Lehmann, L. Zellmer, and A. Brueckner-Foit, “Passive vibration compensation in scanning white-light interferometry,” Appl. Opt. 55, 6172–6182 (2016).
[Crossref] [PubMed]

P. J. de Groot, “Design of error-compensating algorithms for sinusoidal phase shifting interferometry,” Appl. Opt. 48, 6788–6796 (2009).
[Crossref] [PubMed]

J. H. Galeti, P. L. Berton, C. Kitano, R. T. Higuti, R. C. Carbonari, and E. C. N. Silva, “Wide dynamic range homodyne interferometry method and its application for piezoactuator displacement measurements,” Appl. Opt. 52, 6919–6930 (2013).
[Crossref] [PubMed]

B. J. Pernick, “Self-consistent and direct reading laser homodyne measurement technique,” Appl. Opt. 12, 607–610 (1973).
[Crossref] [PubMed]

S. S. C. Chim and G. S. Kino, “Phase measurements using the mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
[Crossref] [PubMed]

Exp. Fluids (1)

K. Shinpaugh, R. Simpson, A. Wicks, S. Ha, and J. Fleming, “Signal-processing techniques for low signal-to-noise ratio laser doppler velocimetry signals,” Exp. Fluids 12, 319–328 (1992).
[Crossref]

IEEE Signal Process. Mag. (1)

E. Jacobsen and R. Lyons, “The sliding dft,” IEEE Signal Process. Mag. 20, 74–80 (2003).
[Crossref]

J. Opt. A Pure Appl. Opt. (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Phase measurement through sinusoidal excitation with application to multi-wavelength interferometry,” J. Opt. A Pure Appl. Opt. 11, 054008 (2009).
[Crossref]

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

Meas. Sci. Technol. (1)

M. Schulz and P. Lehmann, “Measurement of distance changes using a fibre-coupled common-path interferometer with mechanical path length modulation,” Meas. Sci. Technol. 24, 065202 (2013).
[Crossref]

Opt. Eng. (1)

A. Olszak and J. Schmit, “High-stability white-light interferometry with reference signal for real-time correction of scanning errors,” Opt. Eng. 42, 54–59 (2003).
[Crossref]

Opt. Express (2)

Proc. SPIE (3)

S. Tereschenko, P. Lehmann, P. Gollor, and P. Kuehnhold, “Vibration compensated high-resolution scanning white-light Linnik-interferometer,” Proc. SPIE 10329, 1032940 (2017).
[Crossref]

P. J. de Groot and L. L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 706301 (2008).

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 0775, 233–247 (1987).
[Crossref]

Reliab. Eng. Syst. Saf. (1)

A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: a tutorial,” Reliab. Eng. Syst. Saf. 91, 992–1007 (2006).
[Crossref]

Rev. Sci. Instrum. (1)

U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation,” Rev. Sci. Instrum. 62, 2579–2583 (1991).
[Crossref]

Other (9)

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).
[Crossref]

K. Creath, “Phase-measurement interferometry techniques,” in Progress in OpticsXXVI, E. Wolf, ed. (Elsevier, 1988).
[Crossref]

S. Bochkanov, “ALGLIB,” http://www.alglib.net .

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

P. J. de Groot, “Sinusoidal phase shifting interferometry,” Patent No. US7,933,025B2, Apr.26, 2011.

R. Fletcher, Practical Methodes of Optimization(John Wiley & Sonsa, Ltd, 1980).

P. J. de Groot, “Error compensation in phase shifting interferometry,” Patent No. US7,948,637B2, May. 24, 2011.

K. Deb, “Multi-objective optimization,” in Search Methodologies, E. K. Burke and G. Kendall, eds. (Springer, 2014).
[Crossref]

M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “A new single-phase PLL structure based on second order generalized integrator,” in 37th IEEE Power Electronics Specialists Conference, (IEEE, 2006), pp. 1–6.

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Figures (8)

Fig. 1
Fig. 1 Sinusoidally phase modulated interference signal (blue curve) sampled at the temporally equidistant points with the sampling interval Δα during one period of the phase shift function ϕ(t) = π cos(2πt).
Fig. 2
Fig. 2 Algorithm for the simultaneous estimation of the phase shift amplitude a, the phase offset φ and the height dependent phase Θ. A discrete search parameter range (up to 36 values for each parameter) limits the total number of iterations, but still covers all possible variations in the parameters that can occur in a given setup.
Fig. 3
Fig. 3 Variation of the normalized filter functions Foddodd and Feveneven as a function of the phase shift amplitude a for 10 harmonics and the design amplitude a0 = 5.175. The function values in the ranges a1 to a3 are used for the optimization.
Fig. 4
Fig. 4 Simple implementation of the sliding evaluation algorithm with a set of P different sampling vectors, where each sampling vector is shifted by Δα to the previous one.
Fig. 5
Fig. 5 (a) Theoretical error in height determination depending on the phase offset error φerr, given by Eq. (162) in [16]. (b) Height deviation from linear ramp with 2 nm slope per evaluation period for different phase offsets φerr. Interference signal generated with a = 5, φ = 0, P = 50, nmax = 7 and λ = 850 nm. Maximal deviations: Δhφ=0 = 0.019 nm; Δhφ=π/8 = 28.8 nm; Δhφ=π/4 = 3.85 nm; Δhφ=π/2 = 104.5 nm.
Fig. 6
Fig. 6 Comparison of parameter estimation algorithms depending on the signal quality. (a) and (b) standard deviation and maximal error in phase estimation φ. (c) and (d) standard deviation and maximal error in estimation of phase shift amplitude a. "QA": quasi-analytical, "BF 1./(2.) run": brute force 1./(2.) run, "BF 2. run + fit": brute force 2. run with additional LSQ fit.
Fig. 7
Fig. 7 Distance measurement during a CSI depth-scan without vibrations and in presence of high aperiodic vibration caused by a hammer stroke. Several phase jumps (shown in lower inset) appear at the time 425 ms in the result obtained by the conventional SinPSI algorithm. Upper inset shows the height data density of the sliding SinPSI result. The SNR of the interference signal is 27 dB. For better comparability, an offset of 1 μm was added to the blue curve.
Fig. 8
Fig. 8 Height errors in sliding SinPSI evaluation in dependance on height change Δh per sampling point and initial phase offset φ. a). Δh = 1 nm; b). Δh = 0.1 nm. Curves with dot markers represent evaluation with conventional SinPSI algorithm, solid curves show sliding SinPSI evaluation results. For better comparability additional height offsets (100, 200 and 300 nm respectively 10, 20 and 30 nm) were added to the profiles with φ ≠ 0.

Equations (35)

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I ( t ) = I mean [ 1 + V cos  ( 4 π λ { z ^ cos  ( 2 π f 0 t + φ ) + z ( t ) } ) ] ,
Δ I ( t ) = I 0 [ cos  ( Θ ( t ) ) cos  ( ϕ ( t ) ) sin  ( Θ ( t ) ) sin  ( ϕ ( t ) ) ] ,
cos  ( a cos  ( b ) ) = J 0 ( a ) + 2 n = 0 ( 1 ) n J 2 n ( a ) cos  ( 2 n b ) ,
sin  ( a cos  ( b ) ) = 2 n = 1 ( 1 ) n J 2 n + 1 ( a ) cos  ( ( 2 n + 1 ) b ) ,
Δ I ( t ) = I 0 J 0 ( a ) cos  ( Θ ) + 2 I 0 { cos  ( Θ ) n = 2 , 4 , ... ( 1 ) n / 2 J n ( a ) cos  ( n [ α ( t ) + φ ] ) + sin ( Θ ) n = 1 , 3 , ... ( 1 ) ( n + 1 ) / 2 J n ( a ) cos ( n [ α ( t ) + φ ] ) } ,
I ¯ ( t ) = t β / 2 t + β / 2 I ( t ) d t
Δ I ¯ ( t ) = I 0 J 0 ( a ) cos ( Θ ) + 2 I 0 { cos ( Θ ) n = 2 , 4 , ... ( 1 ) n / 2 J n ( a ) B ( n ) cos ( n [ α ( t ) + φ ] ) + sin ( Θ ) n = 1 , 3 , ... ( 1 ) ( n + 1 ) / 2 J n ( a ) B ( n ) cos ( n [ α ( t ) + φ ] ) } .
Θ = arctan  ( Γ even Γ odd sin  ( Θ ) n = 1 , 3 , ... n max γ n Q odd ( n ) cos  ( Θ ) n = 2 , 4 , ... n max γ n Q even ( n ) ) ,
Q odd ( n ) = 2 I 0 ( 1 ) ( n + 1 ) / 2 B ( n ) J n ( a 0 ) and Q even ( n ) = 2 I 0 ( 1 ) n / 2 B ( n ) J n ( a 0 ) .
H ( Θ , n ) = j h n , j I ¯ j ( Θ ) = j cos  ( n α j ) I ¯ j ( Θ ) j cos  ( n α j ) 2
h odd , j = n = odd γ n h n , j and h even , j = n = even γ n h n , j .
H odd ( Θ ) = sin  ( Θ ) n = 1 , 3 , ... n max γ n Q odd ( n ) = j h odd , j I ¯ j ( Θ )
H even ( Θ ) = cos  ( Θ ) n = 2 , 4 , ... n max γ n Q even ( n ) = j h even , j I ¯ j ( Θ ) .
Γ odd = 2 n = 1 , 3 , ... n max γ n ( 1 ) ( n + 1 ) / 2 B ( n ) J n ( a 0 ) and Γ even = 2 n = 2 , 4 , ... n max γ n ( 1 ) n / 2 B ( n ) J n ( a 0 ) .
F odd ( a ) = 2 n = 1 , 3 , ... n max ( 1 ) ( n + 1 ) / 2 B ( n ) J n ( a ) j h odd , j cos ( n α j ) ,
F even ( a ) = 2 n = 2 , 4 , ... n max ( 1 ) n / 2 B ( n ) J n ( a ) j h even , j cos ( n α j ) .
h ˜ ( n , j ) = cos  ( n α j ) cos  ( n φ ) sin  ( n α j ) sin  ( n φ ) .
φ = arctan ( { H _ n ( Θ ) } { H _ n ( Θ ) } ) / n .
J n ( a 0 ) J n + 2 ( a 0 ) = B ( n + 2 ) B ( n ) H n ( Θ ) H n + 2 ( Θ ) .
f min ( γ n , w , a ) = w 1 ( ( F o d d ( a 1 ) Γ o d d ) 2 + ( F e v e n ( a 1 ) Γ e v e n ) 2 ) + w 2 ( F o d d ( a 2 ) Γ o d d F e v e n ( a 2 ) Γ e v e n ) 2 + w 3 ( ( F o d d ( a 3 ) Γ o d d ) 2 + ( F e v e n ( a 3 ) Γ e v e n ) 2 ) ,
F { h ( t t 0 ) } = e i 2 π f t 0 H ( f ) .
D F T n { x ( j k ) } = e i 2 π n k / N X _ n ,
X _ n = j = 0 P 1 x ( j ) e i 2 π j n / P .
X _ n ( m ) = [ X _ n ( m 1 ) + x m x m P ] e i 2 π n / P .
x ( j ) = X 0 cos  ( n [ 2 π j / P + φ ] ) j [ 0 , ... , P 1 ] ,
X n ( m = 0 ) = X 0 = j = 0 P 1 x ( j ) e i 2 π j n / P e i n φ .
X n ( m = p ) = X 0 = j = 0 P 1 x ( j + p ) e i 2 π j n / P e i n φ e i 2 π n p / P .
X n ( m = p ) = [ X n ( m = p 1 ) e i n φ e i 2 π n ( p 1 ) / P + x m x m P ] e i 2 π n / P e i n φ e i 2 π n p / P .
X n ( m = p ) = X n ( m = p 1 ) + [ x m x m P ] e i n ( φ + 2 π ( p 1 ) / P ) .
X n ( m = p ) = X n ( m = p 1 ) + [ x m x m P ] cos  ( n [ φ + 2 π ( p 1 ) / P ] ) .
Θ ( m = 0 ) = arctan  ( Γ even j = 0 P 1 n = odd n max γ n h ˜ ( n , j ) I ¯ ( j , Θ ) Γ odd j = 0 P 1 n = even n max γ n h ˜ ( n , j ) I ¯ ( j , Θ ) ) .
Θ ( m > 0 ) = arctan  ( X odd ( m 1 ) + ( I ¯ ( m , Θ ) I ¯ ( m P , Θ ) ) S odd ( p ) X even ( m 1 ) + ( I ¯ ( m , Θ ) I ¯ ( m P , Θ ) ) S even ( p ) ) ,
S odd ( p ) = Γ even n = 1 , 3.. n max γ n cos  ( n [ φ + 2 π ( p 1 ) / P ] ) ,
S even ( p ) = Γ odd n = 2 , 4.. n max γ n cos  ( n [ φ + 2 π ( p 1 ) / P ] ) .
SNR = 10 log 10 ( mean square of signal variance of noise ( bandwidth equal to Nyquist frequency ) ) .

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