## Two-step self-tuning phase-shifting interferometry |

Optics Express, Vol. 19, Issue 2, pp. 638-648 (2011)

http://dx.doi.org/10.1364/OE.19.000638

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### Abstract

A two-step self-tuning phase-shifting method is presented. The phase-step between the two interferograms is not known when the experiment is performed. Our demodulating method finds, in a robust way, this unknown phase-step. Once the phase-step is estimated we proceed to phase demodulate the interferograms. Moreover our method only requires the fringe patterns to have a constant unknown phase-shift between them. As a consequence, this technique can be used to demodulate open and closed-fringed patterns without phase-sign ambiguity. The method may be regarded as a self-tuning quadrature filter, which determines the phase-shift between the two fringe patterns and finally estimates the demodulated phase map. The proposed technique has been tested with simulated and real interferograms obtaining satisfactory results.

© 2011 OSA

## 1. Introduction

*N*interferograms is obtained having a known temporal carrier. Usually a minimum of three phase-shifted interferograms are need to retrieve the phase. Other phase-shifting methods have been reported as the four-step, five-step or the least-squares algorithm, for example. A good general overview of the phase-shifting algorithms can be found in [2

2. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express **17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

4. S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. **42**(12), 3524–3531 (2003). [CrossRef]

5. Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) **119**(9), 424–428 (2008). [CrossRef]

4. S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. **42**(12), 3524–3531 (2003). [CrossRef]

5. Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) **119**(9), 424–428 (2008). [CrossRef]

6. X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. **273**(1), 54–59 (2007). [CrossRef]

7. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. **33**(8), 776–778 (2008). [CrossRef] [PubMed]

## 2. Proposed method

*t*denotes the

*t*

^{th}interferogram obtained at time

*t*. In our case

*t*= [0, 1]. For simplicity in the following we will drop the spatial dependence. Expression (1) can be rewritten as follow,The Fourier transform of expression (2) is,where

*δ*is the Dirac delta function. In order to obtain the desired phase map, we can define a complex quadrature filter with frequency response

2. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express **17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*t*= 0 obtaining the angle and the norm of

8. M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express **17**(19), 16423–16428 (2009). [CrossRef] [PubMed]

9. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. **224**(4-6), 221–227 (2003). [CrossRef]

9. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. **224**(4-6), 221–227 (2003). [CrossRef]

2. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express **17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*Φ*is determined using expressions (4) to (6).

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*Φ*is obtained.

## 3. Simulations

*π*] and the processing time is of 5 s using a 1.6 GHz laptop and MATLAB. The detected minimum corresponds to

*Φ*. In Fig. 4 , it is shown the obtained wrapped phase. Finally, in Fig. 5 are shown the reconstructed phase and the computed error between actual and obtained phases. The

*rms*(root-mean-square) error of the difference between the actual and measured phases is 0.08 rad.

*rms*error between the actual and computed phase maps and the processing times. The computed phase map is retrieved by interferograms without noise in order to show the detuning error caused by the mismatch between the actual and obtained temporal frequencies. In Table 1 , the first row corresponds to the different signal to noise ratios that quantify how much the phase has been corrupted by noise. As can be seen from Table 1, the proposed two-step self-tuning algorithm gives appropriate results even in the case of interferograms with high noise.

## 4. Experimental results

10. 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]

*rms*(root-mean-square) error of the difference between both reconstructed phases is 0.3 rad. On the other hand, the temporal frequency between interferograms obtained by the proposed and the least-squares methods is

## 5. Conclusions

## References and links

1. | D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998) |

2. | M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express |

3. | F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. |

4. | S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. |

5. | Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) |

6. | X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. |

7. | X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. |

8. | M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express |

9. | J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. |

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

**OCIS Codes**

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 8, 2010

Revised Manuscript: October 14, 2010

Manuscript Accepted: October 15, 2010

Published: January 5, 2011

**Citation**

J. Vargas, J. Antonio Quiroga, T. Belenguer, M. Servín, and J. C. Estrada, "Two-step self-tuning phase-shifting interferometry," Opt. Express **19**, 638-648 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-638

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### References

- D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)
- M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]
- F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).
- S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003). [CrossRef]
- Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008). [CrossRef]
- X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007). [CrossRef]
- X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]
- M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009). [CrossRef] [PubMed]
- J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]
- 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]

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