Abstract

Positioning errors and miscalibrations of the phase-stepping device in a phase-stepping interferometer lead to systematic errors proportional to twice the measured phase distribution. We discuss the historical development of various error-compensating phase-shift algorithms from a unified mathematical point of view. Furthermore, we demonstrate experimentally that systematic errors can also be removed a posteriori. A Twyman–Green-type microlens test interferometer was used for the experiments.

© 1999 Optical Society of America

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References

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  1. H. J. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. J. Schwider, B. Burow, K. E. Elsner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef]
  3. K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef]
  4. D. Malacara, Optical Shop Testing, Wiley Series in Pure and Applied Physics (Wiley, New York, 1990).
  5. 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]
  6. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
    [CrossRef]
  7. J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
    [CrossRef]
  8. M. F. Kuechel, “Some progress in phase measurement techniques,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 27–44.
  9. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [CrossRef]
  10. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  11. J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  12. K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, 2084–2093 (1997).
    [CrossRef] [PubMed]
  13. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1997).
    [CrossRef]
  14. K. Freischlad, Ch. L. Koliopoulos, “Fourier description of digital measuring interferometry,” J. Opt. Soc. Am. 7, 542–551 (1990).
    [CrossRef]
  15. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  16. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. 9, 481–489 (1992).
    [CrossRef]
  17. D. Malacara, D. Malacara-Doblado, “Error analysis of phase detection algorithms,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 45–51.
  18. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
    [CrossRef] [PubMed]
  19. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1992).
    [CrossRef]
  20. D. W. Robinson, Interferogram Analysis (Institute of Physics, Bristol, UK, 1990).
  21. J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975 (1995).
    [CrossRef]
  22. S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
    [CrossRef]
  23. M. Eisner, J. Schwider, “Transferring resist microlenses into silicon by reactive ion etching,” Opt. Eng. 35, 2979–2982 (1996).
    [CrossRef]

1997 (3)

1996 (2)

M. Eisner, J. Schwider, “Transferring resist microlenses into silicon by reactive ion etching,” Opt. Eng. 35, 2979–2982 (1996).
[CrossRef]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

1995 (2)

1993 (2)

S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
[CrossRef]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

1992 (2)

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. 9, 481–489 (1992).
[CrossRef]

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1992).
[CrossRef]

1990 (2)

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

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

1989 (1)

1988 (2)

1987 (1)

1983 (1)

1974 (1)

Brangaccio, D. J.

Bruning, H. J.

Burow, B.

Creath, K.

Eiju, T.

Eisner, M.

M. Eisner, J. Schwider, “Transferring resist microlenses into silicon by reactive ion etching,” Opt. Eng. 35, 2979–2982 (1996).
[CrossRef]

Elsner, K. E.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

Farrant, D. I.

Freischlad, K.

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

Gallagher, J. E.

Grzanna, J.

Hariharan, P.

Haselbeck, S.

S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
[CrossRef]

Herriot, D. R.

Hibino, K.

Kinnstätter, K.

Koliopoulos, Ch. L.

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

Kuechel, M. F.

M. F. Kuechel, “Some progress in phase measurement techniques,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 27–44.

Larkin, K. G.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. 9, 481–489 (1992).
[CrossRef]

Lohmann, A. W.

Macy, W. W.

Malacara, D.

D. Malacara, D. Malacara-Doblado, “Error analysis of phase detection algorithms,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 45–51.

D. Malacara, Optical Shop Testing, Wiley Series in Pure and Applied Physics (Wiley, New York, 1990).

Malacara-Doblado, D.

D. Malacara, D. Malacara-Doblado, “Error analysis of phase detection algorithms,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 45–51.

Merkel, K.

Oreb, B. F.

Rathjen, C.

Robinson, D. W.

D. W. Robinson, Interferogram Analysis (Institute of Physics, Bristol, UK, 1990).

Rosenfeld, D. P.

Schmit, J.

Schreiber, H.

S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
[CrossRef]

Schwider, J.

M. Eisner, J. Schwider, “Transferring resist microlenses into silicon by reactive ion etching,” Opt. Eng. 35, 2979–2982 (1996).
[CrossRef]

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

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

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

J. Schwider, B. Burow, K. E. Elsner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef]

Spolaczyk, R.

Streibl, N.

S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
[CrossRef]

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

Surrel, Y.

White, A. D.

Appl. Opt. (10)

H. J. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

J. Schwider, B. Burow, K. E. Elsner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef]

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1992).
[CrossRef]

K. Kinnstätter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

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

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, 2084–2093 (1997).
[CrossRef] [PubMed]

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

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]

J. Opt. Soc. Am. (2)

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

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. 9, 481–489 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (3)

J. Schwider, O. Falkenstörfer, “Twyman–Green interferometer for testing microspheres,” Opt. Eng. 34, 2972–2975 (1995).
[CrossRef]

S. Haselbeck, H. Schreiber, J. Schwider, N. Streibl, “Microlenses fabricated by melting photoresist,” Opt. Eng. 6, 1322–1324 (1993).
[CrossRef]

M. Eisner, J. Schwider, “Transferring resist microlenses into silicon by reactive ion etching,” Opt. Eng. 35, 2979–2982 (1996).
[CrossRef]

Prog. Opt. (2)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

Other (4)

M. F. Kuechel, “Some progress in phase measurement techniques,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 27–44.

D. Malacara, D. Malacara-Doblado, “Error analysis of phase detection algorithms,” in Fringe 97, Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, Bremen, Akademie-Verlag Series in Optical Metrology, W. Jueptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 45–51.

D. W. Robinson, Interferogram Analysis (Institute of Physics, Bristol, UK, 1990).

D. Malacara, Optical Shop Testing, Wiley Series in Pure and Applied Physics (Wiley, New York, 1990).

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Figures (3)

Fig. 1
Fig. 1

Form deviation of a SiO2 microsphere measured with a Twyman–Green interferometer. Using a well-calibrated phase shifter, the simplest four-phase algorithm has been used for fringe evaluation.

Fig. 2
Fig. 2

Same microsphere measured with a miscalibrated phase shifter. The phase steps have been miscalibrated to 0.6π/2 on purpose.

Fig. 3
Fig. 3

A posteriori removal of the reference phase error applied to the data of Fig. 2.

Equations (23)

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Ir=I01+V cosΦ-φr,
N :=2Nr Ir sin φr,
D :=2Nr Ir cos φr,
Φ=arctanND.
Ir=I01+V cosΦ-φr+r,
ΔN=E cos Φ-C cos Φ-S sin Φ,
ΔD=-E sin Φ-C sin Φ+S cos Φ,
E=1Rr r,
C=1Rr r cos2φr,
S=1Rr r sin2φr.
ΔΦ=DΔN-NΔDN2+D2=DΔN-NΔD,
ΔΦ=E-C cos2Φ-S sin2Φ.
Φ=arctanN1+N2D1+D2.
r=a+br.
RC1=a r=0R-1cos2φr=0+b r=0R-1 r cos2φr=b r=0R-1 r cos2φr
RC2=a r=1Rcos2φr=0+b r=1R r cos2φr=b r=0R-1r+1cos2φr+π=-b r=0R-1 r cos2φr=-RC1.
r=a+br+cr2
RC1=a r=0R-1cos2φr=0+b r=0R-1 r cos2φr+c r=0R-1 r2 cos2φr
RC2=a r=1Rcos2φr=0+b r=1R r cos2φr+c r=1R r2 cos2φr=b r=0R-1r+1cos2φr+π+c r=0R-1r+12 cos2φr+π=-RC1-2c r=0R-1 r cos2φr
RC3=RC1+4c r=0R-1 r cos2φr.
Δ2f=fi+1-2fi+fi-1,
Φ=arctan3I2-4I4+I6I1-4I3+3I5
ΔΦ=a+b cos2Φ+c sin2Φ

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