Abstract

A correlation algorithm to recover the phase in phase-shifting interferometry is presented. We make numerical simulations to test the proposed algorithm and apply it to real interferograms with satisfactory results.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, ISBN 0-8247-9940-2 Marcel Dekker Inc., New York, (1998).
  2. J. R. P. Angel and P. L. Wizinowich, �??A method of Phase Shifting in the Presence of Vibration,�?? European Southern Observatory Conf. Proc. 30, 561 (1988).
  3. P. L. Wizinowich, �??System for Phase Shifting Interferometry in the Presence of Vibration,�?? SPIE 1164, 25-32 (1989).
    [CrossRef]
  4. P. L. Wizinowich, �??System for Phase Shifting Interferometry in the Presence of Vibration: a New Algorithm and System,�?? Appl. Opt. 29, 3271 - 3315 (1990).
    [CrossRef] [PubMed]
  5. P. Hariharan, �??Digital Phase-Stepping Interferometry: Effects of Multiple Reflected Beams,�?? Appl. Opt. 26, 2506-2508 (1987).
    [CrossRef] [PubMed]
  6. J. E. Gallagher and D. R. Herriott, �??Wave front Measurement,�?? U.S. Patent 3,694,088 (1972/1972).
  7. K. Creath, �??Phase Measurement Interferometry Techniques,�?? in Progress in Optics, Vol. XXVI, E. Wolf, Ed. Elsevier Science Publishers, Amsterdan, pp. 349-393(1988).
    [CrossRef]
  8. Y. Morimoto and M. Fujisawa, �??Fringe-Pattern Analysis by phase-shifting Method using extraction of characteristic,�?? Exp. Tech. 20(4), 25-29 (1996).
    [CrossRef]
  9. J. H. Brunning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, �??Digital Wave-front Measuring Inteferometer for Testing Optical Surfaces and Lenses,�?? Appl. Opt. 13, 2693-2703(1974).
    [CrossRef]
  10. J. E. Greivenkamp, �??Generalized Data Reduction for Heterodyne Interferometry,�?? Opt. Eng. 23, 350-352 (1984).
  11. Z. Wang and B. Han, �??Advance iterative algorithm for phase extraction of randomly phase-shifted interferograms,�?? Opt. Lett. 29, 1671-1673 (2004).
    [CrossRef] [PubMed]
  12. Y. Surrel, �??Design of Algorithms for phase measurements by the use of phase stepping,�?? Appl. Opt. 35, 51-60 (1996).
    [CrossRef] [PubMed]
  13. H. van Brug, �??Phase-step calibration for phase-stepped interferometry,�?? Appl. Opt. 38, 3549-3555 (1999).
    [CrossRef]
  14. Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, and T. Yatagai, �??Phase-resolved correlation and its application to analysis of low-coherence interferograms,�?? Opt. Lett. 26, 90-92 (2001).
    [CrossRef]
  15. X. Chen, M. Gramaglia, and J. A. Yeazell, �??Phase-shifting interferometry with uncalibrated phase shifts,�?? Appl. Opt. 39, 585-591 (2000).
    [CrossRef]
  16. Creath, K. and Morales, A. �??Contact and noncontac profiles,�?? in Optical Shop Testing 2nd. ed. D. Malacara (Ed. Wiley, New York), Chap. 17, 687 (1992).
  17. L. Salas, E. Luna, J. Salinas, V. García, and M. Servín, �??Profilometry by fringe projections,�?? Opt. Eng. 42, 3307-3315 (2003).
    [CrossRef]

Appl. Opt. (6)

European Southern Observatory Conf. Proc (1)

J. R. P. Angel and P. L. Wizinowich, �??A method of Phase Shifting in the Presence of Vibration,�?? European Southern Observatory Conf. Proc. 30, 561 (1988).

Exp. Tech. (1)

Y. Morimoto and M. Fujisawa, �??Fringe-Pattern Analysis by phase-shifting Method using extraction of characteristic,�?? Exp. Tech. 20(4), 25-29 (1996).
[CrossRef]

Opt. Eng. (2)

J. E. Greivenkamp, �??Generalized Data Reduction for Heterodyne Interferometry,�?? Opt. Eng. 23, 350-352 (1984).

L. Salas, E. Luna, J. Salinas, V. García, and M. Servín, �??Profilometry by fringe projections,�?? Opt. Eng. 42, 3307-3315 (2003).
[CrossRef]

Opt. Lett. (2)

Optical Shop Testing (1)

Creath, K. and Morales, A. �??Contact and noncontac profiles,�?? in Optical Shop Testing 2nd. ed. D. Malacara (Ed. Wiley, New York), Chap. 17, 687 (1992).

Progress in Optics (1)

K. Creath, �??Phase Measurement Interferometry Techniques,�?? in Progress in Optics, Vol. XXVI, E. Wolf, Ed. Elsevier Science Publishers, Amsterdan, pp. 349-393(1988).
[CrossRef]

SPIE (1)

P. L. Wizinowich, �??System for Phase Shifting Interferometry in the Presence of Vibration,�?? SPIE 1164, 25-32 (1989).
[CrossRef]

Other (2)

J. E. Gallagher and D. R. Herriott, �??Wave front Measurement,�?? U.S. Patent 3,694,088 (1972/1972).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, ISBN 0-8247-9940-2 Marcel Dekker Inc., New York, (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 (7)

Fig. 1.
Fig. 1.

Results for N=15 interferograms. (a) Asterisk indicates the Intensity as a function of phase shifting. (b) Shows the correlation results as a function of k and indicates the value kmax where the maximum is located. The recovered phase is 4.77 rad and the cosine function corresponding to it is shown with a solid line in (a)

Fig. 2.
Fig. 2.

Phase error as a function of the number of interferograms. The position uncertainty ε=5, 10, 20, 50 and 100 nm is indicated on each curve.

Fig. 3.
Fig. 3.

Interferograms of sample surface acquired with a Linnik interferometer. (a) 0 steps, (b) 2 steps, (c) 4 steps.

Fig. 4.
Fig. 4.

Intensity variations for one arbitrary pixel from the nine interferograms. Continuum curve represent the fitting to obtained Intensity values.

Fig. 5.
Fig. 5.

Resultant phase image that results from the correlation algorithm for N=9. Phase wrapped image.

Fig. 6.
Fig. 6.

(a) Unwrapped phase image. The box indicates a region to be analyzed to measure the (b) noise and the (c) micro-roughness.

Fig. 7.
Fig. 7.

Phase error as a function of the number of interferograms for numerical simulations (connect dots) and experimental results(stars).

Tables (2)

Tables Icon

Table 2. Comparison of phase error from real interferograms results with numerical experiment results.

Equations (11)

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

I l ( x , y ) = a ( x , y ) Cos [ ϕ ( x , y ) α l ] + b ( x , y ) ,
corr ( f , g ) = + f ( z ) g ( z + ξ ) dz
f ( z ) = I ( z ) I ( z ) ¯
g ( z + ξ ) = a Cos ( z + ξ ) ,
corr ( f , g ) = l = 1 N a Cos [ ϕ α l ] Cos [ 2 π m k α l ] Δ ,
ϕ α l = 2 π m k max α l ,
ϕ = 2 π m k max .
δ ϕ = 2 π m δ k = π m ,
error = η ε 2 π λ ,
I ( z ) = a Cos [ ϕ α l + error ] + b ,
Φ o = [ [ Φ o ] 2 π [ Φ r ] 2 π ] 2 π + Φ r ,

Metrics