## Phase-shift extraction for phase-shifting interferometry by histogram of phase difference |

Optics Express, Vol. 18, Issue 23, pp. 24368-24378 (2010)

http://dx.doi.org/10.1364/OE.18.024368

Acrobat PDF (1356 KB)

### Abstract

We propose a non-iterative approach to extract the unknown phase shift in phase shifting interferometry without the assumption of equal distribution of measured phase in [0,2π]. According to the histogram of the phase difference between two adjacent frames, the phase shift can be accurately extracted by finding the bin of histogram with the highest frequency. The main factors that influence the accuracy of the proposed method are analyzed and discussed, such as the random noise, the quantization bit of CCD, the number of fringe patterns used and the bin width of histogram. Numerical simulations and optical experiments are also implemented to verify the effectiveness of this method.

© 2010 OSA

## 1. Introduction

1. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. **48**(20), 3948–3960 (2009). [CrossRef] [PubMed]

2. A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. **20**(6), 635–637 (1995). [CrossRef] [PubMed]

5. 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/abstract.cfm?URI=oe-16-11-7806. [CrossRef] [PubMed]

2. A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. **20**(6), 635–637 (1995). [CrossRef] [PubMed]

4. 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]

5. 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/abstract.cfm?URI=oe-16-11-7806. [CrossRef] [PubMed]

6. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. **24**(18), 3049–3052 (1985). [CrossRef] [PubMed]

13. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A **14**(4), 918–930 (1997). [CrossRef]

*K*, where

*K*is a positive integer), which imposes a strict requirement on the environmental stability. To deal with the problem of the random phase shifts caused by the environmental vibration, iterative algorithms based on the least-square method have been proposed to determine the phase-shift amounts and the phase distribution simultaneously [14

14. 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]

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]

*et al*proposed the statistical self-calibrating algorithms to extract the phase distribution on the assumption of a constant fringe contrast [18

18. A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. **8**(7), 738–745 (1997). [CrossRef]

19. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. **9**(5), 744–750 (1998). [CrossRef]

20. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. **9**(9), 1451–1455 (1998). [CrossRef]

*et al*[21

21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. **29**(2), 183–185 (2004). [CrossRef] [PubMed]

*et al*[22

22. 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]

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. **27**(2), 024215 (2010). [CrossRef]

*et al*[24

24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. **34**(8), 1210–1212 (2009). [CrossRef] [PubMed]

*et al*[25

25. 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*[26

26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. **39**(4), 585–591 (2000). [CrossRef]

27. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A **17**(11), 2061–2066 (2000). [CrossRef]

26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. **39**(4), 585–591 (2000). [CrossRef]

27. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A **17**(11), 2061–2066 (2000). [CrossRef]

## 2. Principle

*N*frames of random phase-shifting interferograms are collected and the intensity of an arbitrary pixel (x, y) in the

*n*th interferogram is expressed aswhere

*x*,

*y*), respectively.

*n*th frame.

*n*th frame, denoted as

*x*,

*y*) are omitted for brevity)

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

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. **27**(2), 024215 (2010). [CrossRef]

*n*th and the

*(n-*1)th frames as

*n*th and the (

*n-*1)th frames and the phase difference between these two frames. Since

*φ*is larger than

*π*, which is usually met in practical experiment. Thus there is always more than half of the area over the interferogram where

*m*is positive integer and means the

*m*th bin of histogram. Because the phase shown in Fig. 1 (a) has a very small noise,

*ε*is chosen to be a small value (

*ε*= 0.001 rad) when calculating the histogram. From the Fig. 1(b), we can obtain that the 500th bin has the highest frequency and its range is (0.4995, 0.5005).However, if the extracted phase

*ε*should be chosen larger, such as

*ε*= 0.01 rad. According to the property of histogram of

*m*th bin.

## 3. Numerical simulation and discussion

*N*frames are generated according to Eq. (1) by setting the parameters as follows.

*,*

*.*

*N*frames, where

*N*is the number of fringe patterns used in the simulation. Then the main factors that influence the accuracy of the proposed method are analyzed and discussed as follows.

### 3.1 Influence of random noise

*SNR*) as the ratio of the average of the modulation amplitude to the root mean square of the random noise. We assume that ε equals to 0.01 rad, the number of fringe pattern used (

*N*) is 50 and the bit of the CCD is 10. Then we perform numerical simulations at different signal-to-noise ratios to obtain the phase-shift extraction error. The relation between the average phase-shift extraction error and the SNR of interferogram is shown in Fig. 2(a) .It shows that the average phase-shift extraction error decreases with the SNR increasing. When

*SNR*= 60dB, the phase-shift extraction error of the

*N*frames is shown in Fig. 2(b).It shows that the phase-shift extraction error is less than 0.02rad.

### 3.2 Influence of quantization error

*INT*indicates the nearest integer representation. We assume that the maximum of

*-1, where*

^{t}*t*means the quantization bit of CCD. Figure 3(a) shows the average phase-shift extraction errors for different bits of CCD when

*N*= 30,

*SNR*= 100dB and ε = 0.01rad. It shows that the average phase-shift extraction error decreases as the bit of CCD increases. To reduce the influence of quantization error, the bit of CCD should be chosen no less than 8 and the intensity of laser source should be chosen to make the maximum of

*-1. When the bit of CCD is 8, the phase-shift extraction errors of these 30 frames are shown in Fig. 3(b).It shows that the phase-shift extraction error is less than 0.01rad.*

^{t}### 3.3 Influence of the number of frame used

*N*is large enough and the intensity at (

*x*,

*y*) is ergodic over

*N*frames. If

*N*is not large enough, the intensity at (

*x*,

*y*) is not ergodic, and the obtained

*SNR*= 100dB, ε = 0.01rad and the bit of CCD is 8. It shows that the average phase-shift extraction error decreases as the number of fringe patterns used increases. Since the preset phase shift

*x*,

*y*) is near ergodic over more than 16 frames. Thus, when

*N*is smaller than 16, the average phase-shift extraction error is reduced effectively as

*N*increases; and when

*N*is larger than 16, the average phase-shift extraction error is reduced as

*N*increases but not significantly. Therefore, in order to reduce the phase-shift extraction error,

*N*should be larger than

*N*should be about 2~5 times as large as

*N*= 50, the phase-shift extraction errors of 50 frames are shown in Fig. 4(b). It shows that the phase-shift extraction error is less than 0.015 rad.

### 3.4 Influence of bin width

*m*th bin, thus the bin width certainly has effect on the phase-shift extraction error. According to the practical phase measurement condition, we assume that the number of fringe pattern used is 50, the SNR of each fringe pattern is 60dB and the bit of CCD is 8. Then we calculate the average phase-shift extraction errors at different bin widths and the result is shown in Fig. 5 . It shows that the average phase-shift extraction error increases with the bin width increasing for ε>0.02 rad but increases with the bin width decreasing for ε<0.01 rad. Thus, a small value of ε may lead to large phase-shift extraction error because our method is sensitive to noise when ε is very small. For the case that

*N*= 50,

*SNR*= 60dB and the bit of CCD is 8, the average phase-shift extraction error is less than 0.01 rad when 0.001rad <ε<0.03 rad.

### 3.5 Comparison with other method

*et al*[21

21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. **29**(2), 183–185 (2004). [CrossRef] [PubMed]

*et al*[22

22. 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]

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. **27**(2), 024215 (2010). [CrossRef]

*et al*[24

24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. **34**(8), 1210–1212 (2009). [CrossRef] [PubMed]

**27**(2), 024215 (2010). [CrossRef]

*SNR*of the interferogram is 60dB and the bit of CCD is 8. The real phase shift

21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. **29**(2), 183–185 (2004). [CrossRef] [PubMed]

24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. **34**(8), 1210–1212 (2009). [CrossRef] [PubMed]

## 4. Experiment

## 5. Conclusion

*N*= 50,

*SNR*= 60dB,0.001<ε<0.03 rad and the bit of CCD is 8.This method is well implemented in existing interferometers simply by incorporating a high-speed camera and choosing a proper preset phase shift. It has potential application for test and measurement of large-aperture optical elements.

## Acknowledgments

## References and links

1. | L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. |

2. | A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. |

3. | J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World |

4. | 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. |

5. | 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 |

6. | Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. |

7. | P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

8. | P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. |

9. | K. G. Larkin and B. F. Oreb, “Design and assessment of Symmetrical Phase-Shifting Algorithms,” J. Opt. Soc. Am. A |

10. | Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. |

11. | Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. |

12. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A |

13. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A |

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

15. | G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. |

16. | B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. |

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

18. | A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. |

19. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. |

20. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. |

21. | L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. |

22. | 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. |

23. | X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. |

24. | X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. |

25. | 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. |

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

27. | X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A |

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

29. | Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” 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 13, 2010

Revised Manuscript: October 21, 2010

Manuscript Accepted: October 21, 2010

Published: November 5, 2010

**Citation**

Jiancheng Xu, Yong Li, Hui Wang, Liqun Chai, and Qiao Xu, "Phase-shift extraction for phase-shifting interferometry by histogram of phase difference," Opt. Express **18**, 24368-24378 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24368

Sort: Year | Journal | Reset

### References

- L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009). [CrossRef] [PubMed]
- A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. 20(6), 635–637 (1995). [CrossRef] [PubMed]
- J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World 38, 109–116 (2002).
- 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/abstract.cfm?URI=oe-16-11-7806 . [CrossRef] [PubMed]
- Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985). [CrossRef] [PubMed]
- 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]
- P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29(22), 3271–3279 (1990). [CrossRef] [PubMed]
- K. G. Larkin and B. F. Oreb, “Design and assessment of Symmetrical Phase-Shifting Algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]
- Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef] [PubMed]
- Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. 20(18), 1883–1885 (1995). [CrossRef] [PubMed]
- K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12(4), 761–768 (1995). [CrossRef]
- K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997). [CrossRef]
- 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]
- G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994). [CrossRef] [PubMed]
- 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]
- 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]
- A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997). [CrossRef]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998). [CrossRef]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998). [CrossRef]
- L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (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]
- X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]
- X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009). [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]
- X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]
- X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A 17(11), 2061–2066 (2000). [CrossRef]
- J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

OSA is a member of CrossRef.