Abstract

Temporal phase shifting in automatic interferogram analysis offers very high accuracy of phase retrieval providing that several experimental conditions are met. The paper is focused on the calibration error of unequal phase changes across the interferogram field, i.e., tilt-shift error. For its detection the lattice-site representation of phase shift angles is proposed. The error can be readily discerned using (N+1) algorithms with the last frame overlapping the first one. Four and five frame algorithms are considered. The influence of experimental parameters on the error detection sensitivity is discussed. Numerical studies are complemented by experimental results.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf ed., 28, 271-359 (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990), Chap. 4.
  2. J. E. Greivenkamp, and J. H. Brunning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, Singapore, 1992) Chap. 14, pp. 501-598.
  3. K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement, D.W. Robinson and G. Reid, eds., (Institute of Physics Publishing, Bristol, Philadelphia, 1993), Chap 4, pp 94-140.
  4. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).
  5. K. Kinnstaetter, A. W. Lohmann, J. Schwider, and J. Streibl, "Accuracy of phase shifting interferometry," Appl. Opt. 27, 5082-5089 (1988).
    [CrossRef] [PubMed]
  6. J. van Wingerden, H. J. Frankena, and C. Smorenburg, "Linear approximation for measurement errors in phase shifting interferometry," Appl. Opt. 30, 2718-2729 (1991).
    [CrossRef] [PubMed]
  7. K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt Eng. 44, 065601; published online June 7, 2005 (2005).
    [CrossRef]
  8. K. Patorski, and A. Styk, "Theory and practice of two-beam interferogram modulation determination," Proc. SPIE 5958, 119-129 (2005).
  9. P. de Groot, "Phase-shift calibration errors in interferometers with spherical Fizeau cavities," Appl. Opt. 34, 2856-2863 (1995).
    [CrossRef]
  10. M. Chen, H. Guo, and C. Wei, "Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
    [CrossRef]
  11. A. Dobroiu, A. Apostol, V. Nascov, and V. Damian, "Tilt-compensating algorithm for phase-shift interferometry," Appl. Opt. 41, 2435-2439 (2002).
    [CrossRef] [PubMed]
  12. X. Ding, G. L. Cloud, and B. B. Raju, "Noise tolerance of the improved max-min scanning method for phase determination," Opt. Eng. 44, 035605; published online March 18, 2005, (2005).
    [CrossRef]
  13. B. Gutmann, and H. Weber, "Phase-shifter calibration and error detection in phase-shifting applications: a new method," Appl. Opt. 37, 7624-7631 (1998).
    [CrossRef]
  14. J. Schwider, R. Burrow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef] [PubMed]
  15. P. Hariharan, B. Oreb, and T. Eiju, "Digital phase shifting interferometry: a simple error compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987).
    [CrossRef] [PubMed]
  16. A. Styk, and K. Patorski, "Identification of nonlinear recording error in phase shifting interferometry," Opt. Lasers. Eng., in press (2006).
  17. Y. Y. Cheng, and J. C. Wyant, "Phase shifter calibration in phase-shifting interferometry," Appl. Opt. 24, 3049-3052 (1985).
    [CrossRef] [PubMed]
  18. Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32, 3598-3599 (1993).
    [CrossRef] [PubMed]
  19. Q. Yu, X. Liu, and K. Andresen, "New spin filters for interferometric fringe patterns and grating patterns," Appl. Opt. 33, 3705-3711 (1994).
    [CrossRef] [PubMed]
  20. T. Kreis, Holographic Interferometry: Principles and Methods, (Akademie Verlag GmbH, Berlin, 1996).
  21. J. Schmit, and K. Creath, "Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry," Appl. Opt. 34, 3610-3619 (1995).
    [CrossRef] [PubMed]
  22. R. Schödel, A. Nicolaus, and G. Bönsch, "Phase-stepping interferometry: methods for reducing errors caused by camera nonlinearities," Appl. Opt. 41, 55-63 (2002).
    [CrossRef] [PubMed]

2006 (1)

A. Styk, and K. Patorski, "Identification of nonlinear recording error in phase shifting interferometry," Opt. Lasers. Eng., in press (2006).

2005 (3)

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt Eng. 44, 065601; published online June 7, 2005 (2005).
[CrossRef]

K. Patorski, and A. Styk, "Theory and practice of two-beam interferogram modulation determination," Proc. SPIE 5958, 119-129 (2005).

X. Ding, G. L. Cloud, and B. B. Raju, "Noise tolerance of the improved max-min scanning method for phase determination," Opt. Eng. 44, 035605; published online March 18, 2005, (2005).
[CrossRef]

2002 (2)

2000 (1)

1998 (1)

1995 (2)

1994 (1)

1993 (1)

1991 (1)

1988 (1)

1987 (1)

1985 (1)

1983 (1)

Andresen, K.

Apostol, A.

Bönsch, G.

Burrow, R.

Chen, M.

Cheng, Y. Y.

Cloud, G. L.

X. Ding, G. L. Cloud, and B. B. Raju, "Noise tolerance of the improved max-min scanning method for phase determination," Opt. Eng. 44, 035605; published online March 18, 2005, (2005).
[CrossRef]

Creath, K.

Damian, V.

de Groot, P.

Ding, X.

X. Ding, G. L. Cloud, and B. B. Raju, "Noise tolerance of the improved max-min scanning method for phase determination," Opt. Eng. 44, 035605; published online March 18, 2005, (2005).
[CrossRef]

Dobroiu, A.

Eiju, T.

Elssner, K. E.

Frankena, H. J.

Grzanna, J.

Guo, H.

Gutmann, B.

Hariharan, P.

Kinnstaetter, K.

Liu, X.

Lohmann, A. W.

Merkel, K.

Nascov, V.

Nicolaus, A.

Oreb, B.

Patorski, K.

A. Styk, and K. Patorski, "Identification of nonlinear recording error in phase shifting interferometry," Opt. Lasers. Eng., in press (2006).

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt Eng. 44, 065601; published online June 7, 2005 (2005).
[CrossRef]

K. Patorski, and A. Styk, "Theory and practice of two-beam interferogram modulation determination," Proc. SPIE 5958, 119-129 (2005).

Raju, B. B.

X. Ding, G. L. Cloud, and B. B. Raju, "Noise tolerance of the improved max-min scanning method for phase determination," Opt. Eng. 44, 035605; published online March 18, 2005, (2005).
[CrossRef]

Schmit, J.

Schödel, R.

Schwider, J.

Sienicki, Z.

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt Eng. 44, 065601; published online June 7, 2005 (2005).
[CrossRef]

Smorenburg, C.

Spolaczyk, R.

Streibl, J.

Styk, A.

A. Styk, and K. Patorski, "Identification of nonlinear recording error in phase shifting interferometry," Opt. Lasers. Eng., in press (2006).

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt Eng. 44, 065601; published online June 7, 2005 (2005).
[CrossRef]

K. Patorski, and A. Styk, "Theory and practice of two-beam interferogram modulation determination," Proc. SPIE 5958, 119-129 (2005).

Surrel, Y.

van Wingerden, J.

Weber, H.

Wei, C.

Wyant, J. C.

Yu, Q.

Appl. Opt. (13)

J. Schwider, R. Burrow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
[CrossRef] [PubMed]

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

K. Kinnstaetter, A. W. Lohmann, J. Schwider, and J. Streibl, "Accuracy of phase shifting interferometry," Appl. Opt. 27, 5082-5089 (1988).
[CrossRef] [PubMed]

J. van Wingerden, H. J. Frankena, and C. Smorenburg, "Linear approximation for measurement errors in phase shifting interferometry," Appl. Opt. 30, 2718-2729 (1991).
[CrossRef] [PubMed]

Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32, 3598-3599 (1993).
[CrossRef] [PubMed]

Q. Yu, X. Liu, and K. Andresen, "New spin filters for interferometric fringe patterns and grating patterns," Appl. Opt. 33, 3705-3711 (1994).
[CrossRef] [PubMed]

B. Gutmann, and H. Weber, "Phase-shifter calibration and error detection in phase-shifting applications: a new method," Appl. Opt. 37, 7624-7631 (1998).
[CrossRef]

P. de Groot, "Phase-shift calibration errors in interferometers with spherical Fizeau cavities," Appl. Opt. 34, 2856-2863 (1995).
[CrossRef]

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

M. Chen, H. Guo, and C. Wei, "Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
[CrossRef]

R. Schödel, A. Nicolaus, and G. Bönsch, "Phase-stepping interferometry: methods for reducing errors caused by camera nonlinearities," Appl. Opt. 41, 55-63 (2002).
[CrossRef] [PubMed]

A. Dobroiu, A. Apostol, V. Nascov, and V. Damian, "Tilt-compensating algorithm for phase-shift interferometry," Appl. Opt. 41, 2435-2439 (2002).
[CrossRef] [PubMed]

P. Hariharan, B. Oreb, and T. Eiju, "Digital phase shifting interferometry: a simple error compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987).
[CrossRef] [PubMed]

Opt Eng. (1)

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt Eng. 44, 065601; published online June 7, 2005 (2005).
[CrossRef]

Opt. Eng. (1)

X. Ding, G. L. Cloud, and B. B. Raju, "Noise tolerance of the improved max-min scanning method for phase determination," Opt. Eng. 44, 035605; published online March 18, 2005, (2005).
[CrossRef]

Opt. Lasers. Eng. (1)

A. Styk, and K. Patorski, "Identification of nonlinear recording error in phase shifting interferometry," Opt. Lasers. Eng., in press (2006).

Proc. SPIE (1)

K. Patorski, and A. Styk, "Theory and practice of two-beam interferogram modulation determination," Proc. SPIE 5958, 119-129 (2005).

Other (5)

T. Kreis, Holographic Interferometry: Principles and Methods, (Akademie Verlag GmbH, Berlin, 1996).

J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf ed., 28, 271-359 (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990), Chap. 4.

J. E. Greivenkamp, and J. H. Brunning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, Singapore, 1992) Chap. 14, pp. 501-598.

K. Creath, "Temporal phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement, D.W. Robinson and G. Reid, eds., (Institute of Physics Publishing, Bristol, Philadelphia, 1993), Chap 4, pp 94-140.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1.
Fig. 1.

Phase shift angle histograms and their lattice-site representations calculated using Eqs. (1) and (2) for the case of tilt-shift error of 1.0 µrad. Tilt axis parallel to fringe direction. Left and right column results are for four and five frame algorithms, respectively. Upper row-conventional histograms, bottom row-lattice-site representations. Very similar results are obtained for the same tilt-shift error value for the perpendicular orientation of the tilt axis.

Fig. 2.
Fig. 2.

Phase shift angle histograms and lattice-site representations calculated using Eqs. (1) and (2) for tilt-shift error of 10 µrad. Tilt axis parallel to fringe direction.

Fig. 3.
Fig. 3.

Phase shift angle histograms and lattice-site representations calculated using Eqs. (1) and (2) for tilt-shift error of 10 µrad. Tilt axis perpendicular to fringe direction.

Fig. 4.
Fig. 4.

Phase shift angle histograms and lattice-site representations calculated using Eq. (2) for simultaneous tilt-shift error of — 5 µrad and 10 µrad about x and y axes, respectively (left column) and -10 µrad and 10 µrad about x and y axes, respectively (right column). Linear detection, γ=1. Compare with Fig. 3.

Fig. 5.
Fig. 5.

Phase shift angle histograms and lattice-site representations for the case of the tilt-shift error of 1.0 µrad. Tilt axis parallel to fringe direction. Nonlinear detection, γ≠1. Very similar results are obtained for the tilt axis perpendicular to fringe direction. Compare with Fig. 1.

Fig. 6.
Fig. 6.

Phase shift angle histograms and lattice-site representations calculated using Eqs. (1) and (2) for tilt-shift error of 10 µrad. Tilt axis parallel to fringe direction. Nonlinear detection, γ≠1. Compare with Fig. 2.

Fig. 7.
Fig. 7.

Phase shift angle histograms and lattice-site representations calculated using Eqs. (1) and (2) for tilt-shift error of 10 µrad. Tilt axis perpendicular to fringe direction, γ≠1. Compare with Fig. 3.

Fig. 8.
Fig. 8.

Phase shift histograms and lattice-sites for two cases of constant phase steps of 70 deg (left) and 110 deg (right). Five frame data; tilt-shift error of 1 µrad, γ≠1. The results are almost the same for the beam tilt axis being parallel and perpendicular to fringe direction. Compare with Figs. 1 and 5.

Fig. 9.
Fig. 9.

Phase shift histograms and lattice-sites for two cases of constant phase step values of 70 deg (left) and 110 deg (right). Five frame data; tilt-shift error of 10 µrad, γ≠1. Tilt axis parallel to fringe direction. Compare with Figs. 2 and 6.

Fig. 10.
Fig. 10.

Phase shift histograms and lattice-sites for constant phase steps of 70 deg (left) and 110deg (right). Five frame data; tilt-shift error of 10 µrad, γ≠1. Tilt axis perpendicular to fringe direction. Compare with Figs. 3 and 7.

Fig. 11.
Fig. 11.

Phase shift angle histograms and lattice-sites calculated using Eqs. (1) and (2) for the tilt-shift error of 1.0 µrad. Unequal phase steps between the frames (see text). Tilt axis parallel to fringe direction, γ=1. Very similar results are obtained for the same tilt-shift error value and perpendicular orientation of the tilt axis.

Fig. 12.
Fig. 12.

Phase shift angle histograms and lattice-sites calculated using Eqs. (1) and (2) for the tilt-shift error of 10 µrad. Unequal phase steps between the frames (see text), γ=1. Tilt axis parallel to fringe direction.

Fig. 13.
Fig. 13.

Phase shift angle histograms and lattice-sites calculated using Eqs. (1) and (2) for the tilt-shift error of 10 µrad. Unequal phase steps between the frames (see text), γ=1. Tilt axis perpendicular to fringe direction.

Fig. 14.
Fig. 14.

Phase shift angle histograms, upper row, and lattice-site representations, bottom row, calculated using Eq. (1) from two sets of four frames and one set of five component interferograms, Eq. (2). Null-fringe interferometer adjustment, nominally equal steps realized by three PZTs. See text for more detailed explanation.

Fig. 15.
Fig. 15.

Phase shift angle histograms and lattice-sites calculated in case of deliberately introduced step error to one of the PZTs. See text for details and compare with Fig. 14.

Fig. 16.
Fig. 16.

Calculation results for a similar tilt-shift error case as in Fig. 15, but with eight reference fringes introduced into the interferogram.

Fig. 17.
Fig. 17.

Calculated phase shift angle histograms and lattice-sites for an experiment with an increased tilt-shift error and changed sign, realized by another PZT, as compared with the case of Fig. 16. Five frame lattice-site representation provides very good display of experimental errors (tilt-shift error, unequal phase steps and detection nonlinearity).

Fig. 18.
Fig. 18.

Exemplary data and calculations for a Twyman-Green interferometer experiment: a) selected interferogram; b) interferogram intensity cross-section; c) and d) grey-level interferogram modulation maps calculated using frames 1 to 4 and 2 to 5, respectively.

Fig. 19.
Fig. 19.

Calculated phase shift angle histograms and lattice-sites from the experiment with the average phase step of 120 deg. Observe considerable noise influence.

Fig. 20.
Fig. 20.

Same as in Fig. 19 but with spin filtered component TPS interferograms. Observe dramatic improvement with respect to experimental errors detection and identification.

Equations (4)

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

α ( x , y ) = arc cos { 1 2 [ I 1 I 2 + I 3 I 4 I 2 I 3 ] } ,
α ( x , y ) = arc cos { 1 2 [ I 5 I 1 I 4 I 2 ] } ,
α ( x , y ) = 2 arc tan { 3 ( I 2 I 3 ) ( I 1 I 4 ) ( I 2 I 3 ) + ( I 1 I 4 ) } .
I ' = I + a I 2 + b I 3 + c I 4 +

Metrics