Abstract

A blind self-calibrating algorithm for phase-shifting interferometry is presented, with which the nonlinear interaction introduced by phase shift errors, between the reconstructed phases and the reconstructed amplitudes of the reference wave, is measured with cross-bispectrum. Minimizing an objective function based on this cross-bispectrum allows accurately estimating the true phase shifts from only three interferograms in the absence of any supplementary assumptions and knowledge about these interferograms.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
    [CrossRef] [PubMed]
  2. J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  3. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [CrossRef] [PubMed]
  4. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
    [CrossRef]
  5. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
    [CrossRef] [PubMed]
  6. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
    [CrossRef]
  7. C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
    [CrossRef]
  8. I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
    [CrossRef]
  9. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000).
    [CrossRef]
  10. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
    [CrossRef]
  11. H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
    [CrossRef]
  12. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
    [CrossRef] [PubMed]
  13. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
    [CrossRef] [PubMed]
  14. L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1-3), 21–26 (2004).
    [CrossRef]
  15. X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
    [CrossRef] [PubMed]
  16. P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
    [CrossRef] [PubMed]
  17. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
    [CrossRef] [PubMed]
  18. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
    [CrossRef]
  19. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13(10), 3743–3753 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-10-3743 .
    [CrossRef] [PubMed]
  20. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
    [CrossRef]
  21. K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981).
    [CrossRef]

2009

2007

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007).
[CrossRef]

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[CrossRef]

2006

2005

2004

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1-3), 21–26 (2004).
[CrossRef]

2003

2001

2000

1995

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

1993

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
[CrossRef] [PubMed]

1992

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[CrossRef]

1991

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

1987

1984

J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
[CrossRef] [PubMed]

1982

1981

K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981).
[CrossRef]

Bokor, J.

Cai, L. Z.

Chen, M.

Chen, X.

Dong, G. Y.

Eiju, T.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[CrossRef]

Gao, P.

Geist, E.

Goldberg, K. A.

Gramaglia, M.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Guo, H.

Han, B.

Harder, I.

Hariharan, P.

Helland, K. N.

K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981).
[CrossRef]

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

Kwon, O. Y.

Lii, K. S.

K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981).
[CrossRef]

Lindlein, N.

Liu, Q.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1-3), 21–26 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[CrossRef] [PubMed]

Mantel, K.

Meng, X. F.

Morgan, C. J.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

Oreb, B. F.

Player, M. A.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[CrossRef]

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Shen, X. X.

Soloviev, O.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Surrel, Y.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

Vdovin, G.

Wang, Z.

Wei, C.

Xu, X. F.

Yang, X. L.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1-3), 21–26 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[CrossRef] [PubMed]

Yao, B.

Yeazell, J. A.

Yu, Y.

Zhao, Z.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[CrossRef]

Zöller, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

ACM Trans. Math. Softw.

K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[CrossRef]

Opt. Commun.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1-3), 21–26 (2004).
[CrossRef]

Opt. Eng.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[CrossRef]

J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007).
[CrossRef]

Opt. Lett.

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 (4)

Fig. 1
Fig. 1

Numerical simulation results. (a) Phase map. (b) One of three interferograms. (c) The errors in the reconstructed amplitudes of the reference wave and (d) the errors in the reconstructed phase map by using nominal phase shifts.

Fig. 2
Fig. 2

Numerical simulation results. (a) Selected cross-sections in the fringe patterns. (b) Magnitude of cross-bispectrum when (δ 1, δ 2) = (1.7, 2.9). (c) In the search regions, β as a function of (δ 1, δ 2). (d) The contracted search regions.

Fig. 3
Fig. 3

Measurement results of a silicon plate. (a) One of three interferograms. (b) Search regions. (c) The amplitudes of the reference wave and (d) the phase map, reconstructed by using the nominal phase shifts. (e) and (f) are similar to (c) and (d) but the used phase shifts are estimated with the proposed algorithm.

Fig. 4
Fig. 4

Measurement results of a grating profile. (a) One of three interferograms. (b) Search regions. (c) The amplitudes of the reference wave and (d) the phase map, reconstructed by using the nominal phase shifts. (e) and (f) are similar to (c) and (d) but the used phase shifts are estimated with the proposed algorithm.

Tables (1)

Tables Icon

Table 1 Estimated phase shifts in the presence of noise

Equations (12)

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

I k = | r | 2 + | o | 2 + 2 | r | | o | cos ( ϕ + δ k ) , k = 0 , 1 , , K 1
I k = c 0 + c 1 cos δ k + c 2 sin δ k , k = 0 ,   1 ,   2 .
c 0 = I 0 sin ( δ 2 δ 1 ) I 1 sin δ 2 + I 2 sin δ 1 sin ( δ 2 δ 1 ) sin δ 2 + sin δ 1 ,
c 1 = I 0 ( sin δ 1 sin δ 2 ) + I 1 sin δ 2 I 2 sin δ 1 sin ( δ 2 δ 1 ) sin δ 2 + sin δ 1 ,
c 2 = I 0 ( cos δ 2 cos δ 1 ) + I 1 ( 1 cos δ 2 ) I 2 ( 1 cos δ 1 ) sin ( δ 2 δ 1 ) sin δ 2 + sin δ 1 .
ϕ = arctan ( c 2 c 1 ) ,
| r | = c 0 + c 0 2 c 1 2 c 2 2 2 ,
| o | = c 0 c 0 2 c 1 2 c 2 2 2 .
ψ = exp ( i ϕ ) = cos ϕ + i sin ϕ ,
B ( ω 1 , ω 2 ) = E { Ψ ( ω 1 ) Ψ ( ω 2 ) R ( ω 1 + ω 2 ) } ,
B ^ ( δ 1 , δ 2 ) ( ω 1 , ω 2 ) = n = 1 N Ψ n ( ω 1 ) Ψ ( ω 2 ) n R n ( ω 1 + ω 2 ) ,
β ( δ 1 , δ 2 ) = u = π π v = π π | B ^ ( δ 1 , δ 2 ) ( ω 1 , ω 2 ) | .

Metrics