## Correlation Algorithm to recover the phase of a test surface using Phase-Shifting Interferometry

Optics Express, Vol. 12, Issue 22, pp. 5296-5306 (2004)

http://dx.doi.org/10.1364/OPEX.12.005296

Acrobat PDF (325 KB)

### 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

## 1. Introduction

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]

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]

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 steping,” Appl. Opt. **35**, 51–60 (1996). [CrossRef] [PubMed]

13. H. van Brug, “Phase-step calibration for phase-stepped interferometry,” 1999. 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]

## 2. Correlation algorithm

*a*(

*x*,

*y*) is the fringe contrast,

*b*(

*x*,

*y*) is the average intensity and

*α*is the phase-shifting given as

_{l}*α*we get an interferogram

_{l}*I*(

_{l}*x*,

*y*), and we can get up to N such interferograms. Hence, for each pixel (x,y) we have N intensity values that should describe a cosine function. We can use a correlation function to measure how close this cosine law is followed by the data and obtain the phase from this measurement.

*x*,

*y*) of the image and we will omit the notation (

*x*,

*y*) for brevity. The correlations will be analyzed in the

*z*direction that corresponds to the interferogram number or phase step.

*f*(

*z*) and

*g*(

*z*) as one of them is shifted by an amount ξ with respect to the other in the z direction:

*f*(

*z*) be determined by the observed data points while

*g*(

*z*) is a simple cosine function, that is:

*I*(

*z*) represent one of the N interferograms,

*z*= −

*α*and

_{l}*k*between 1 and

*m*,

*m*is the number of points where the two functions are to be compared.

*f*(

*z*) should in turn be a function of the unknown phase of the object, according to Eq. (1)

*f*(

*z*)=

*a*

*Cos*(

*ϕ*+

*z*), and thus the correlation of

*f*(

*z*) and

*g*(

*z*) may be written,

*k*=

*k*such that,

_{max}*k*where the maximum correlation is attained, we can measure the phase

_{max}*ϕ*.

## 3. Numerical simulations

*N*will be modified to evaluate the precision of the method in the presence of varying amounts of noise. Even in the case of zero noise the uncertainty in the phase value

*δϕ*is given (see Eq.7) by,

*δϕ*=0.06

*rad*), we will then require

*m*=50 for the numerical simulations.

*η*is a random number, taken from a Gaussian distribution with zero mean and unity standard deviation,

*ε*is the uncertainty in position in nm,

*error*is added to the phase, so Eq. (1) is re-written as,

*a*(

*x*,

*y*)=100,

*Ī*=1000,

*λ*=0.5

*µm*an optical path difference of

*OPD*=0.38

*µm*corresponding to

*ϕ*=4.77 rad. We considered several values for the uncertainty in position:

*ε*=5, 10, 20, 50, y 100

*nm*. For each case of

*ε*we simulated N interferograms, with N=4, 6, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90 and 100, according to Eq. (10). In each case we found the maximum

*k*from the correlation of this data with the expected cosine given by Eq. (4). We further repeated each experiment 100 times corresponding to different realizations of the random number

_{max}*η*. Therefore we obtained 100 different phase values

*ϕ*, which allowed us to calculate the mean value, and more important, the RMS dispersion σ.

*N*=15 interferograms. The graphs on (a) show the intensity values (vertical axis) as a function of the phase-shifting amount indicated in the horizontal axis. Asterisks indicate the simulated data points, and the continuum line is the cosine function of Eq. (4) corresponding to the value of

*k*found with the algorithm; (b) shows the results of the correlation (vertical axis) as a function of the displacement position

_{max}*k*(horizontal axis). We can localize the position

*k*where the correlation between the functions is maximum. We can notice that although the intensity dispersion increases greatly when the position uncertainty ε increases, the correlation algorithm works well giving correct results for the phase.

_{max}*N*and position uncertainty

*ε*the table indicates σ measured in percentage. This measurement will be referred to as phase error, and is a measurement of the precision of the algorithm.

*N*>4 value when

*ε*=5, 10, 20, 50 nm, and for

*N*>60 when

*ε*=100 nm.

*N*increases for any value of ε in a predictable way. This will allow us to choose the number

*N*of interferograms needed to recover the phase of a test surface as a function of the required precision and of the accuracy of the mechanism to shift the phase in the particular interferometer that is going to be used.

## 4. Experimental results

*N*here. The solid line in Fig. 4 shows a cosine function fitting from which we identify

_{c}*Nc*=4.2, and notice that

*N*can be an fractional number and that

_{c}*N*can be larger than

*N*resulting in a better precision in the phase value. We applied the correlation algorithm with

_{c}*N*taken for the process was varying between 4 and 9. The resulting phase image for the case

*N*=9 (in 2

*π*module) is shown in Fig. 5. The image is later unwrapped using an algorithm given by Salas [17

17. L. Salas, E. Luna, J. Salinas, V. García, and M. Servín, “Profilometry by fringe projections,” Opt. Eng. **42**, 3307–3315 (2003). [CrossRef]

*π*indicates the wrapping operation, Φ

*is the object’s phase, Φ*

_{o}*is the reference phase. After applying the algorithm we obtained the unwrapped image of the object phase shown in Fig. 6.*

_{r}## 4.1. Image analysis

*µm*(5×5 pixels) we notice sharp structures with high frequency pixel to pixel variations. Furthermore, these structures change when the number of images

*N*into the correlation algorithm is changed from 4 to 9, indicating that they are not real structures but phase fluctuations that measure the precision of the correlation algorithm with the present experimental setup. We then measure the variance of such structures σ to determine the phase error. Table 2 present these results and makes a comparison with the numerical simulation results for

*ε*=5 nm and

*ε*=10 nm values. We notice that the behavior of real interferograms is similar to numerical results. The noise that is produced by the uncertainty of the shift position of the step-motor of between 5 and 10 nm, gives a similar phase error as the real interferograms. This suggests that the uncertainty of the steper motors used in the real experiments is on the order of one half step, which is expected. We also conclude from this table that the phase can be measured with precision better than 1% in al cases of

*N*=4, ..9.

*µm*. In Fig. 6 we mark a small region with a box, indicating a typical 4.4

*µm*region (20×20 pixels) to be analyzed for micro-roughness. We took 15 such similar regions, and we found a micro-roughness value of 11.2±2.8 nm. The result is in good agreement with the one obtained with a mechanical profile, which indicates an average value of 15 nm roughness given by reference [18

18. E. Luna, J. Nuñez, J. Salinas, L. Salas, E. Ruiz, I. Cruz-Gonzalez, A. Nava-Vega, E. Sohn, and B. Martínez, “Caracterización de un interferómetro de Linnik construido para medir microrugosidades en superficies ópticas generadas por la HyDra,” Technical Report RT-2004-18, Instituto de Astronomía, Universidad Autonoma Nacional de México, Apartado Postal 877, C.P. 22800, Ensenada, B.C., México (2004).

## 5. Conclusions

## Acknowledgments

## References and links

1. | D. Malacara, M. Servin, and Z. Malacara, |

2. | J.R.P. Angel and P.L. Wizinowich, “A method of Phase Shifting in the Presence of Vibration,” European Southern Observatory Conf. Proc. |

3. | P.L. Wizinowich, “System for Phase Shifting Interferometry in the Presence of Vibration,” SPIE |

4. | P.L. Wizinowich, “System for Phase Shifting Interferometry in the Presence of Vibration: a New Algorithm and System,” Appl. Opt. |

5. | P. Hariharan, “Digital Phase-Stepping Interferometry: Effects of Multiple Reflected Beams,” Appl. Opt. |

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. |

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. |

10. | J.E. Greivenkamp, “Generalized Data Reduction for Heterodyne Interferometry,” Opt. Eng. |

11. | Z. Wang and B. Han, “Advance iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. |

12. | Y. Surrel, “Design of Algorithms for phase measurements by the use of phase steping,” Appl. Opt. |

13. | H. van Brug, “Phase-step calibration for phase-stepped interferometry,” 1999. Appl. Opt. |

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. |

15. | X. Chen, M. Gramaglia, and J.A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. |

16. | K. Creath and A. Morales “Contact and noncontac profiles,” in Optical Shop Testing2nd. 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. |

18. | E. Luna, J. Nuñez, J. Salinas, L. Salas, E. Ruiz, I. Cruz-Gonzalez, A. Nava-Vega, E. Sohn, and B. Martínez, “Caracterización de un interferómetro de Linnik construido para medir microrugosidades en superficies ópticas generadas por la HyDra,” Technical Report RT-2004-18, Instituto de Astronomía, Universidad Autonoma Nacional de México, Apartado Postal 877, C.P. 22800, Ensenada, B.C., México (2004). |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 3, 2004

Revised Manuscript: October 13, 2004

Published: November 1, 2004

**Citation**

Adriana Nava-Vega, L. Salas, E. Luna, and A. Cornejo-Rodríguez, "Correlation algorithm to recover the phase of a test surface using phase-shifting interferometry," Opt. Express **12**, 5296-5306 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-22-5296

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### References

- D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, ISBN 0-8247-9940-2 Marcel Dekker Inc., New York, (1998).
- 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).
- P. L. Wizinowich, �??System for Phase Shifting Interferometry in the Presence of Vibration,�?? SPIE 1164, 25-32 (1989). [CrossRef]
- 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]
- P. Hariharan, �??Digital Phase-Stepping Interferometry: Effects of Multiple Reflected Beams,�?? Appl. Opt. 26, 2506-2508 (1987). [CrossRef] [PubMed]
- J. E. Gallagher and D. R. Herriott, �??Wave front Measurement,�?? U.S. Patent 3,694,088 (1972/1972).
- K. Creath, �??Phase Measurement Interferometry Techniques,�?? in Progress in Optics, Vol. XXVI, E. Wolf, Ed. Elsevier Science Publishers, Amsterdan, pp. 349-393(1988). [CrossRef]
- Y. Morimoto and M. Fujisawa, �??Fringe-Pattern Analysis by phase-shifting Method using extraction of characteristic,�?? Exp. Tech. 20(4), 25-29 (1996). [CrossRef]
- 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]
- J. E. Greivenkamp, �??Generalized Data Reduction for Heterodyne Interferometry,�?? Opt. Eng. 23, 350-352 (1984).
- Z. Wang and B. Han, �??Advance iterative algorithm for phase extraction of randomly phase-shifted interferograms,�?? Opt. Lett. 29, 1671-1673 (2004). [CrossRef] [PubMed]
- Y. Surrel, �??Design of Algorithms for phase measurements by the use of phase stepping,�?? Appl. Opt. 35, 51-60 (1996). [CrossRef] [PubMed]
- H. van Brug, �??Phase-step calibration for phase-stepped interferometry,�?? Appl. Opt. 38, 3549-3555 (1999). [CrossRef]
- 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]
- X. Chen, M. Gramaglia, and J. A. Yeazell, �??Phase-shifting interferometry with uncalibrated phase shifts,�?? Appl. Opt. 39, 585-591 (2000). [CrossRef]
- 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).
- L. Salas, E. Luna, J. Salinas, V. García, and M. Servín, �??Profilometry by fringe projections,�?? Opt. Eng. 42, 3307-3315 (2003). [CrossRef]

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