Abstract

Statistical properties of phase-shift algorithms are investigated for the case of additive Gaussian intensity noise. Based on a bivariate normal distribution, a generally valid probability-density function for the random phase error is derived. This new description of the random phase error shows properties that cannot be obtained through Gaussian error propagation. The assumption of a normally distributed phase error is compared with the derived probability-density function. For small signal-to-noise ratios the assumption of a normally distributed phase error is not valid. Additionally, it is shown that some advanced systematic-error-compensating algorithms have a disadvantageous effect on the random phase error.

© 1995 Optical Society of America

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References

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  1. M. Küchel, “Verfahren zur Messung eines phasenmodulierten Signals, Offenlegungsschrift,” German patentDE 4014019 A1. G 01 J 9/00 (1991).
  2. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  3. P. L. Ransom, J. V. Kokal, “Interferogram analysis by a modified sinusoid fitting technique,” Appl. Opt. 25, 4199–4204 (1986).
    [CrossRef] [PubMed]
  4. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  5. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. XXVI, pp. 351–393.
  6. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  7. Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  8. B. Dörband, “Die 3-Interferogramm-Methode zur automatischen Streifenauswertung in rechnergesteuerten digitalen Zweistrahlinterferometern,” Optik (Stuttgart) 60, 161–174 (1982).
  9. C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).
  10. J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  11. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]
  12. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  13. C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef] [PubMed]
  14. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  15. G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 230–231 (1990).
    [CrossRef]
  16. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  17. C. Liu, J. Chen, Z. Li, “A method of eliminating the measurement error for phase shifting interferometry,” in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 706–707 (1993).
  18. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  19. B. Dörband, “Analyse optischer Systeme,” Ph.D. dissertation (Berichte aus dem Institut für Technische Optik der Universität Stuttgart, Stuttgart, Germany, 1986).
  20. X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
    [CrossRef]
  21. N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
    [CrossRef]
  22. G. K. Larkin, B. F. Oerb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 219–227 (1992).
    [CrossRef]
  23. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
    [CrossRef] [PubMed]
  24. G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttgart) 82, 161–164 (1989).
  25. R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
    [CrossRef]
  26. W. R. Bennet, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
    [CrossRef]
  27. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 409–437.
  28. N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
    [CrossRef]
  29. C. P. Brophy, “Effect of intensity error correlation on computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  30. I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1977). See also standard statistical textbooks.
  31. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
    [CrossRef]
  32. This test is not the only possible one. The advantage of the test is the simple idea on which it is based.
  33. Averaging the single intensity pattern Ii(x, y) first will not yield this effect. This is the important difference between averaging before and averaging after phase calculation. The difference is caused by the nonlinearity of the arctangent function. Therefore the difference vanishes with an increasing SNR.
  34. In Ref. 5 an asymmetric distribution of the reference phase is published. Such an asymmetry may be caused by the calibration algorithm.
  35. J. Minkoff, Signals, Noise & Active Sensors (Wiley, New York, 1992).
  36. Regarding k1as a quadratic equation of α, one can show that there are no real roots for ρ2< 1. Therefore k1is always positive. For ρ2= 1, which corresponds to completely correlated noise terms of Dand N, no practical importance can be seen. However, the phase error Δφcan easily be derived from Fig. 1. Then we can obtain the probability-density function of Δφby transforming the probability-density function of the noise term in a manner similar to that in Eq. (A5).

1993 (1)

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1992 (2)

X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

1991 (1)

1990 (2)

1989 (2)

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttgart) 82, 161–164 (1989).

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[CrossRef] [PubMed]

1988 (2)

1987 (2)

1986 (2)

P. L. Ransom, J. V. Kokal, “Interferogram analysis by a modified sinusoid fitting technique,” Appl. Opt. 25, 4199–4204 (1986).
[CrossRef] [PubMed]

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

1985 (2)

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
[CrossRef]

Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1983 (2)

1982 (1)

B. Dörband, “Die 3-Interferogramm-Methode zur automatischen Streifenauswertung in rechnergesteuerten digitalen Zweistrahlinterferometern,” Optik (Stuttgart) 60, 161–174 (1982).

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
[CrossRef]

1956 (1)

W. R. Bennet, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
[CrossRef]

Ai, C.

Bennet, W. R.

W. R. Bennet, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
[CrossRef]

Böhme, H.

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttgart) 82, 161–164 (1989).

Bönsch, G.

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttgart) 82, 161–164 (1989).

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1977). See also standard statistical textbooks.

Brophy, C. P.

Bruning, J. H.

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 409–437.

Burow, R.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
[CrossRef]

Chen, J.

C. Liu, J. Chen, Z. Li, “A method of eliminating the measurement error for phase shifting interferometry,” in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 706–707 (1993).

Cheng, Y.-Y.

Cornejo-Rodriguez, A.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. XXVI, pp. 351–393.

Dändliker, R.

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
[CrossRef]

Dörband, B.

B. Dörband, “Die 3-Interferogramm-Methode zur automatischen Streifenauswertung in rechnergesteuerten digitalen Zweistrahlinterferometern,” Optik (Stuttgart) 60, 161–174 (1982).

B. Dörband, “Analyse optischer Systeme,” Ph.D. dissertation (Berichte aus dem Institut für Technische Optik der Universität Stuttgart, Stuttgart, Germany, 1986).

Eiju, T.

Elssner, K.-E.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Frankena, H. J.

Freischlad, K.

Grzanna, J.

Hariharan, P.

Honda, T.

N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
[CrossRef]

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

Kinnstaetter, K.

Kinoshita, S.

Kokal, J. V.

Koliopoulos, C. L.

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

Küchel, M.

M. Küchel, “Verfahren zur Messung eines phasenmodulierten Signals, Offenlegungsschrift,” German patentDE 4014019 A1. G 01 J 9/00 (1991).

Lai, G.

G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 230–231 (1990).
[CrossRef]

Larkin, G. K.

G. K. Larkin, B. F. Oerb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 219–227 (1992).
[CrossRef]

Larkin, K. G.

Li, Z.

C. Liu, J. Chen, Z. Li, “A method of eliminating the measurement error for phase shifting interferometry,” in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 706–707 (1993).

Liu, C.

C. Liu, J. Chen, Z. Li, “A method of eliminating the measurement error for phase shifting interferometry,” in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 706–707 (1993).

Lohmann, A. W.

Macy, W. W.

Merkel, K.

Minkoff, J.

J. Minkoff, Signals, Noise & Active Sensors (Wiley, New York, 1992).

Oerb, B. F.

G. K. Larkin, B. F. Oerb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 219–227 (1992).
[CrossRef]

Ohyama, N.

N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
[CrossRef]

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

Oreb, B. F.

Ransom, P. L.

Schreiber, H.

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schreiber, O.

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1977). See also standard statistical textbooks.

Shimano, T.

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

Smorenburg, C.

Spolaczyk, R.

Strebl, N.

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Streibl, N.

Su, X.-Y.

X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Thalmann, R.

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
[CrossRef]

Tsujiuchi, J.

N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
[CrossRef]

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

van Wingerden, J.

von Bally, G.

X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Vukicevic, D.

X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Wyant, J. C.

Yatagai, T.

G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 230–231 (1990).
[CrossRef]

Zhou, W.-S.

X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Zöller, A.

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt. (9)

J. Opt. Soc. Am. A (4)

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
[CrossRef]

Opt. Commun. (2)

X.-Y. Su, W.-S. Zhou, G. von Bally, D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

N. Ohyama, T. Shimano, J. Tsujiuchi, T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[CrossRef]

Opt. Eng. (3)

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

J. Schwider, O. Falkenstörfer, O. Schreiber, H. Schreiber, A. Zöller, N. Strebl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Optik (Stuttgart) (2)

B. Dörband, “Die 3-Interferogramm-Methode zur automatischen Streifenauswertung in rechnergesteuerten digitalen Zweistrahlinterferometern,” Optik (Stuttgart) 60, 161–174 (1982).

G. Bönsch, H. Böhme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttgart) 82, 161–164 (1989).

Proc. IRE (1)

W. R. Bennet, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
[CrossRef]

Other (14)

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 409–437.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1977). See also standard statistical textbooks.

This test is not the only possible one. The advantage of the test is the simple idea on which it is based.

Averaging the single intensity pattern Ii(x, y) first will not yield this effect. This is the important difference between averaging before and averaging after phase calculation. The difference is caused by the nonlinearity of the arctangent function. Therefore the difference vanishes with an increasing SNR.

In Ref. 5 an asymmetric distribution of the reference phase is published. Such an asymmetry may be caused by the calibration algorithm.

J. Minkoff, Signals, Noise & Active Sensors (Wiley, New York, 1992).

Regarding k1as a quadratic equation of α, one can show that there are no real roots for ρ2< 1. Therefore k1is always positive. For ρ2= 1, which corresponds to completely correlated noise terms of Dand N, no practical importance can be seen. However, the phase error Δφcan easily be derived from Fig. 1. Then we can obtain the probability-density function of Δφby transforming the probability-density function of the noise term in a manner similar to that in Eq. (A5).

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. XXVI, pp. 351–393.

M. Küchel, “Verfahren zur Messung eines phasenmodulierten Signals, Offenlegungsschrift,” German patentDE 4014019 A1. G 01 J 9/00 (1991).

G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 230–231 (1990).
[CrossRef]

C. Liu, J. Chen, Z. Li, “A method of eliminating the measurement error for phase shifting interferometry,” in 16th Congress of the International Commission for Optics: Optics as a Key to High Technology, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 706–707 (1993).

B. Dörband, “Analyse optischer Systeme,” Ph.D. dissertation (Berichte aus dem Institut für Technische Optik der Universität Stuttgart, Stuttgart, Germany, 1986).

G. K. Larkin, B. F. Oerb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 219–227 (1992).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Vector representation of the phase error Δφ.

Fig. 2
Fig. 2

Bivariate normal distribution centered on IS defining the probability-density function of IR, where [·] indicates an operation that gives the dimension of d and n.

Fig. 3
Fig. 3

Probability-density functions of the phase error for several values of SNR. Case: α = 1 and ρ = 0.

Fig. 4
Fig. 4

Comparison of the probability-density function of the phase error fφ) with a normally distributed phase error fGφ) for small values of SNR.

Fig. 5
Fig. 5

Maximal deviation of the cumulative distribution of the phase error from a normally distributed phase error.

Fig. 6
Fig. 6

Comparison of the confidence interval limits of the phase error with a Gaussian limit for three confidence coefficients Pi.

Fig. 7
Fig. 7

Comparison of the variance of the phase error with the variance of a corresponding normal distribution.

Fig. 8
Fig. 8

Probability-density functions of the phase error for several phase values. Case: α ≠ 1 and ρ = 0.

Fig. 9
Fig. 9

Probability-density functions of the phase error for several values of SNR. Case: α ≠ 1 and ρ = 0.

Fig. 10
Fig. 10

Average of the phase error for several values of SNR. Case: α ≠ 1 and ρ = 0.

Fig. 11
Fig. 11

Deviation of the confidence interval limits of the phase error from Gaussian confidence interval limits for a small value of SNR. Case: α ≠ 1 and ρ = 0.

Fig. 12
Fig. 12

Confidence limits of the phase error approaching a limit for increasing values of SNR. Case: α ≠ 1 and ρ = 0.

Fig. 13
Fig. 13

Deviation of the confidence limits of the phase error from a phase-dependent Gaussian distribution. The deviation increases with α deviating from 1. Case: α ≠ 1 and ρ = 0.

Fig. 14
Fig. 14

Probability-density functions of the phase error for several phase values. Case: α = 1 and ρ ≠ 0.

Fig. 15
Fig. 15

Probability-density functions of the phase error for several values of SNR. Case: α = 1 and ρ ≠ 0.

Fig. 16
Fig. 16

Average of the phase error for several values of SNR. Case: α = 1 and ρ ≠ 0.

Fig. 17
Fig. 17

Deviation of the confidence interval limits of the phase error from Gaussian confidence interval limits for a small value of SNR. Case: α = 1 and ρ ≠ 0.

Fig. 18
Fig. 18

Confidence limits of the phase error approaching a limit for increasing values of SNR. Case: α = 1 and ρ ≠ 0.

Fig. 19
Fig. 19

Deviation of the confidence limits of the phase error from a phase-dependent Gaussian distribution. The deviation increases with an increasing ρ. Case: α = 1 and ρ ≠ 0.

Tables (1)

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Table 1 List of Phase-Shift Algorithms with α ≠ 1 or ρ ≠ 0

Equations (47)

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I ( x , y ) = I 0 ( x , y ) { 1 + m ( x , y ) cos [ φ ( x , y ) ] } ,
I i = I 0 [ 1 + m cos ( φ + ψ i ) ] ,             i = 1 , , K .
tan φ = I 4 - I 2 I 1 - I 3 = 2 I 0 m sin φ 2 I 0 m cos φ .
tan φ = N D = I S sin φ I S cos φ ,
tan φ = N + Δ N D + Δ D ,
σ D 2 = i = 1 K j = 1 K D I i D I j c i j ,             σ N 2 = i = 1 K j = 1 K N I i N I j c i j ,
c i i = σ I i 2 ,             c i j = COV ( I i , I j ) = ρ i j σ I i σ I j .
σ D 2 = i = 1 K ( D I i ) 2 σ I i 2 ,             σ N 2 = i = 1 K ( N I i ) 2 σ I i 2 .
f ( d , n ) = 1 2 π σ D σ N 1 - ρ 2 exp [ - 1 2 h ( d , n ) ] ,
h ( d , n ) = 1 1 - ρ 2 [ ( d - μ D σ D ) 2 + ( n - μ N σ N ) 2 - 2 ρ ( d - μ D σ D ) ( n - μ N σ N ) ] .
ρ = COV ( D , N ) σ D σ N = σ D N σ D σ N .
f ( Δ φ ) = 0 f ( r , Δ φ ) d r = 1 8 π k 1 3 / 2 σ D 2 α 1 - ρ 2 { 4 k 1 exp ( - 1 2 k 3 ) + 2 π k 2 exp ( - 4 k 1 k 3 - k 2 2 8 k 1 ) × [ erf ( 2 k 2 4 k 1 ) - 1 ] } ,
k 1 = 1 σ D 2 ,             k 2 = - 2 2 SNR cos ( Δ φ ) σ D ,             k 3 = 2 SNR ,
f ( Δ φ ) = 1 2 π exp ( - SNR ) ( 1 + π SNR cos ( Δ φ ) × exp [ SNR cos 2 ( Δ φ ) ] { 1 + erf [ SNR cos ( Δ φ ) ] } ) .
sin ( Δ φ ) Δ φ , cos ( Δ φ ) 1 , sin ( φ + Δ φ ) sin φ + Δ φ cos φ , cos ( φ + Δ φ ) cos φ - Δ φ sin φ .
h ( r , Δ φ ) = 1 σ D 2 [ r 2 ( 1 + Δ φ 2 ) - 2 r I S + I S 2 ] = 1 σ D 2 [ ( r - I S ) 2 + r 2 Δ φ 2 ] .
f ( r , Δ φ ) = exp [ - 1 2 ( r - I S ) 2 σ D 2 ] 2 π σ D r exp ( - 1 2 r 2 Δ φ 2 σ D 2 ) 2 π σ D .
f ( r , Δ φ ) = f ( r ) f G ( Δ φ ) = exp [ - 1 2 ( r - I S ) 2 σ D 2 ] 2 π σ D × SNR exp ( - SNR Δ φ 2 ) π .
σ G 2 = 1 2 SNR .
CONV { - σ G Δ φ σ G } : P 1 = P ( - σ G Δ φ σ G ) 0.683 = 68.3 % , CONV { - 2 σ G Δ φ 2 σ G } : P 2 = P ( - 2 σ G Δ φ 2 σ G ) 0.955 = 95.5 % , CONV { - 3 σ G Δ φ 3 σ G } : P 3 = P ( - 3 σ G Δ φ 3 σ G ) 0.997 = 99.7 % .
R i = Limit [ CONV ( P i ) ] i σ G ,             i = 1 , 2 , 3.
CONV { - π Δ φ - 3 σ V } : P - 3 = P ( - π Δ φ - 3 σ V ) 0.00135 = 0.135 % , CONV { - π Δ φ - 2 σ V } : P - 2 = P ( - π Δ φ - 2 σ V ) 0.02275 = 2.275 % , CONV { - π Δ φ - 1 σ V } : P - 1 = P ( - π Δ φ - 1 σ V ) 0.15866 = 15.866 % , CONV { - π Δ φ + 1 σ V } : P + 1 = P ( - π Δ φ + 1 σ V ) 0.84134 = 84.134 % , CONV { - π Δ φ + 2 σ V } : P + 2 = P ( - π Δ φ + 2 σ V ) 0.97725 = 97.725 % , CONV { - π Δ φ + 3 σ V } : P + 3 = P ( - π Δ φ + 3 σ V ) 0.99865 = 99.865 % .
R i = Right _ Limit [ CONV ( P i ) ] i σ V , i = - 3 , - 2 , - 1 , 1 , 2 , 3.
σ V α = { 1 2 SNR [ 1 + ( α 2 - 1 ) cos 2 ( φ ) ] } 1 / 2 = R Limit σ V .
σ V ρ = { 1 2 SNR [ 1 - ρ sin ( 2 φ ) ] } 1 / 2 ,
R max = 1 + ρ .
d = r cos φ = r cos ( φ + Δ φ ) , μ D = D = I S cos φ , n = r sin φ = r sin ( φ + Δ φ ) , μ N = N = I S sin φ ,
h ( r , Δ φ ) = k 1 r 2 + k 2 r + k 3 ,
k 1 = 1 2 σ D 2 α 2 ( 1 - ρ 2 ) { α 2 [ 1 + cos ( 2 φ + 2 Δ φ ) ] - 2 α ρ sin ( 2 φ + 2 Δ φ ) - cos ( 2 φ + 2 Δ φ ) + 1 } , k 2 = - 2 SNR σ D α 2 ( 1 - ρ 2 ) { α 2 [ cos ( 2 φ + Δ φ ) + cos ( Δ φ ) ] - 2 α ρ sin ( 2 φ + Δ φ ) - cos ( 2 φ + Δ φ ) + cos ( Δ φ ) } , k 3 = SNR α 2 ( 1 - ρ 2 ) [ α 2 - 2 α ρ sin ( 2 φ ) + ( α 2 - 1 ) cos ( 2 φ ) + 1 ] .
α = σ N σ D ,             SNR = I S 2 2 σ D 2
f ( d , n ) d d d n = f [ d ( r , Δ φ ) , n ( r , Δ φ ) ] r d φ d r = f ( r , Δ φ ) d φ d r
f ( r , Δ φ ) = r 2 π α σ D 2 1 - ρ 2 exp [ - 1 2 ( k 1 r 2 + k 2 r + k 3 ) ] .
h ( r , Δ φ ) = 1 α 2 σ D 2 ( 1 - ρ 2 ) ( t 1 + t 2 + t 3 + t 4 ) ,
t 1 = r 2 [ α 2 cos 2 ( φ + Δ φ ) + sin 2 ( φ + Δ φ ) ] , t 2 = - 2 r I S [ α 2 cos φ cos ( φ + Δ φ ) + sin φ sin ( φ + Δ φ ) ] , t 3 = I S 2 [ α 2 cos 2 ( φ ) + sin 2 ( φ ) ] , t 4 = - ρ α [ r 2 sin ( 2 φ + 2 Δ φ ) - 2 r I S sin ( 2 φ + Δ φ ) + I S 2 sin ( 2 φ ) ] .
t 1 ~ cos ( 2 φ + 2 Δ φ ) , t 2 ~ cos ( Δ φ ) + α cos ( 2 φ ) cos ( Δ φ ) + b sin ( 2 φ ) sin ( Δ φ ) , t 3 ~ cos ( 2 φ ) , t 4 ~ sin ( 2 φ + 2 Δ φ ) + c sin ( 2 φ + Δ φ ) + d sin ( 2 φ ) .
f ( Δ φ , φ ) = f ( Δ φ , π + φ ) .
f 123 ( Δ φ , φ ) = f 123 ( - Δ φ , - φ ) , f 123 ( Δ φ , π / 2 + φ ) = f 123 ( - Δ φ , π / 2 - φ ) .
f 4 ( Δ φ , π / 4 + φ ) = f 4 ( - Δ φ , π / 4 - φ ) , f 4 ( Δ φ , - π / 4 + φ ) = f 4 ( - Δ φ , - π / 4 - φ ) .
t 1 = r 2 ,             t 2 = - 2 r I S cos ( Δ φ ) ,             t 3 = I S 2 ,             t 4 = 0.
t 1 = r 2 , t 2 = - 2 r I S cos ( Δ φ ) , t 3 = I S 2 , t 4 = - ρ [ r 2 sin ( 2 φ + 2 Δ φ ) - 2 r I S sin ( 2 φ + Δ φ ) + I S 2 sin ( 2 φ ) ] .
tan φ = I 1 - 2 I 2 + I 3 I 1 - I 3 .
σ D 2 = σ I 1 2 + σ I 3 2 = 2 σ i 2 , σ N 2 = σ I 1 2 + 4 σ I 2 2 + σ I 3 2 = 6 σ i 2 .
α = 3 ,             S N R = I S 2 2 σ D 2 = ( 2 I 0 m ) 2 4 σ i 2 = ( I 0 m σ i ) 2 .
COV ( D , N ) = σ D N = D N - D N = I 1 2 - 2 I 1 I 2 + 2 I 2 I 3 - I 3 2 - I 1 2 + 2 I 1 I 2 - 2 I 2 I 3 + I 3 2 = σ I 1 2 - 2 σ I 1 I 2 + 2 σ I 2 I 3 - σ I 3 2 = σ i 2 - σ i 2 = 0.
Δ φ ( φ D ) Δ D + ( φ N ) Δ N = 1 1 + tan 2 ( φ ) ( - N D 2 Δ D + 1 D Δ N ) = 1 D 2 + N 2 ( - N Δ D + D Δ N ) .
σ φ 2 = σ Δ φ 2 = Δ φ 2 ( 1 D 2 + N 2 ) 2 ( N 2 Δ D 2 - 2 D N Δ D Δ N + D 2 Δ N 2 ) = ( 1 D 2 + N 2 ) 2 ( N 2 σ D 2 - 2 D N σ D N + D 2 σ N 2 ) .
σ φ 2 = 1 2 SNR [ 1 + ( α 2 - 1 ) cos 2 ( φ ) - α ρ sin ( 2 φ ) ] .

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