## Direct phase extraction from interferograms with random phase shifts |

Optics Express, Vol. 18, Issue 20, pp. 20620-20627 (2010)

http://dx.doi.org/10.1364/OE.18.020620

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### Abstract

A new method that extracts phase directly from randomly phase-shifted interferograms is proposed. In this method, the background intensity and the modulation amplitude are determined by a temporal method and used as the system parameters and then the phase is extracted from one single interferogram by arc cosine function with range of (0, π). By partitioning the extracted phase distribution into watershed regions and determining the sign of phase in each region, the extracted phase is recovered to its principle phase with range of (–π, π). Numerical simulation and experiment are implemented to verify the effectiveness of this method. The method has applications in both static and dynamic phase measurement.

© 2010 OSA

## 1. Introduction

3. C. J. Stolz, “The National Ignition Facility: The world’s largest optical system,” Proc. SPIE **6834**, 683402 (2007). [CrossRef]

5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. **44**(32), 6861–6868 (2005). [CrossRef] [PubMed]

6. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express **16**(11), 7806–7817 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7806. [CrossRef] [PubMed]

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

10. P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. **34**(22), 3553–3555 (2009). [CrossRef] [PubMed]

*et al*[7

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

*et al*[8

8. 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]

*et al*[9

9. 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]

*et al*[10

10. P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. **34**(22), 3553–3555 (2009). [CrossRef] [PubMed]

*et al*[11

11. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. **34**(8), 1288–1290 (2009). [CrossRef] [PubMed]

12. J. Park and S.-W. Kim, “Vibration-desensitized interferometer by continuous phase shifting with high-speed fringe capturing,” Opt. Lett. **35**(1), 19–21 (2010). [CrossRef] [PubMed]

13. Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express **13**(8), 3110–3116 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3110. [CrossRef] [PubMed]

14. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express **18**(9), 9684–9689 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-9-9684. [CrossRef] [PubMed]

## 2. Principle

*N*frames of phase-shifted interferograms are collected, the intensity of an arbitrary pixel (

*x*,

*y*) in the

*n*th interferogram is expressed aswhere

*x*,

*y*), respectively.

*x*,

*y*) can be ergodic over

*N*frames if

*N*is large enough [11

11. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. **34**(8), 1288–1290 (2009). [CrossRef] [PubMed]

*N*frames of interferograms [11

11. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. **34**(8), 1288–1290 (2009). [CrossRef] [PubMed]

**34**(8), 1288–1290 (2009). [CrossRef] [PubMed]

### 2.1 Phase segmentation by watershed transformation

15. L. Vincent and P. Soille, “Watersheds in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Trans. Pattern Anal. Mach. Intell. **13**(6), 583–598 (1991). [CrossRef]

15. L. Vincent and P. Soille, “Watersheds in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Trans. Pattern Anal. Mach. Intell. **13**(6), 583–598 (1991). [CrossRef]

*x*approaches the 10

^{th},30

^{th}, 70th and 90th pixels. It is easy to identify the watershed regions since each watershed region has its corresponding number. As shown in Fig. 1(b), 1-value pixels belong to the first watershed region, 2-value pixels belong to the second watershed region, and so on.

### 2.2 Sign determination by phase difference

*n*th and (

*n-*1)th frames as

*n*th phase, the (

*n*-1)th phase and their phase difference. Since

*n*th and the (

*n*-1)th principle phases are in (0, π), and equals

*n*th and the (

*n*-1)th principle phase are in (-π, 0).Here

*m*th watershed region can be determined aswhere

*m*th watershed region. With the sign of the principle phase in one watershed region (e.g. the

*m*th watershed region) already known, the principle phase in all watershed regions can be easily recovered because the signs of the principle phase in the adjacent watershed regions (i.e., the

*m*th and the (

*m*+ 1)th watershed regions) have opposite signs. Figure 1(d) shows the recovered principle phase from Fig. 1(a) by the proposed method. After phase unwrapping, the phase

## 3. Numerical simulation and discussion

*φ*is obtained by phase unwrapping and shown in Fig. 2(e), and the residual phase error is calculated by comparing the obtained phase with the preset phase and shown in Fig. 2(f). Figure 2(f) shows that the residual phase error mainly locates at the place where

*N*is large enough and the intensity at (

*x*,

*y*) is ergodic over

*N*frames. If the number of fringe patterns used is not large enough, the intensity at (

*x*,

*y*) is not ergodic, and the obtained

*N*= 10, 20,40,60,80 and 100 and the result is shown in Table 1 . It shows that the residual phase error decreases with the number of fringe patterns used increasing. Since the preset phase shift

*x*,

*y*) is near ergodic over more than 20 frames. Thus, when

*N*is smaller than 20, the residual phase error is reduced effectively as

*N*increases; and when

*N*is larger than 20, the residual phase error is not reduced significantly as

*N*increases. Therefore, in order to reduce the residual phase error, the number of fringe patterns used should be larger than

*N*= 100. The reason is that our method fails to determine the sign of the principle phase where

*M*×

*M*) in interferogram is limited. We calculate the residual phase error for different numbers of pixels and values of ε and the result is shown in Table 2 . It shows that the residual phase error decreases with the number of pixels increasing and the value of ε decreasing. Thus in order to reduce the residual phase error of the proposed method, we should capture more interferograms, use high resolution CCD and choose a small value of ε.

## 4. Experiment

8. 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]

## 5. Conclusion

## References and links

1. | J. E. Greivenkamp, and J. H. Bruning, “Phase shifting interferometers, ” in |

2. | K. Creath, “Temporal phase measurement method,” in |

3. | C. J. Stolz, “The National Ignition Facility: The world’s largest optical system,” Proc. SPIE |

4. | R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. |

5. | M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. |

6. | G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express |

7. | K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. |

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

9. | 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. |

10. | P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. |

11. | Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. |

12. | J. Park and S.-W. Kim, “Vibration-desensitized interferometer by continuous phase shifting with high-speed fringe capturing,” Opt. Lett. |

13. | Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express |

14. | S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express |

15. | L. Vincent and P. Soille, “Watersheds in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Trans. Pattern Anal. Mach. Intell. |

**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: July 28, 2010

Revised Manuscript: August 26, 2010

Manuscript Accepted: August 27, 2010

Published: September 14, 2010

**Citation**

Jiancheng Xu, Qiao Xu, Liqun Chai, Yong Li, and Hui Wang, "Direct phase extraction from interferograms with random phase shifts," Opt. Express **18**, 20620-20627 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20620

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### References

- J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometers, ” in Optical Shop Testing, D. Malacara, ed., John Wiley and Sons, 1992, pp. 501–598.
- K. Creath, “Temporal phase measurement method,” in Interferogram Analysis, D.W. Robinson and G. T. Reid, eds. Institute of Physics, Bristol, UK, 1993, pp. 94–140.
- C. J. Stolz, “The National Ignition Facility: The world’s largest optical system,” Proc. SPIE 6834, 683402 (2007). [CrossRef]
- R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
- M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]
- G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16(11), 7806–7817 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7806 . [CrossRef] [PubMed]
- K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991). [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]
- 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]
- P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009). [CrossRef] [PubMed]
- Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009). [CrossRef] [PubMed]
- J. Park and S.-W. Kim, “Vibration-desensitized interferometer by continuous phase shifting with high-speed fringe capturing,” Opt. Lett. 35(1), 19–21 (2010). [CrossRef] [PubMed]
- Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3110 . [CrossRef] [PubMed]
- S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-9-9684 . [CrossRef] [PubMed]
- L. Vincent and P. Soille, “Watersheds in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 583–598 (1991). [CrossRef]

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