## Phase-shifting interferometry based on induced vibrations |

Optics Express, Vol. 19, Issue 2, pp. 584-596 (2011)

http://dx.doi.org/10.1364/OE.19.000584

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### Abstract

The presence of uncontrolled mechanical vibrations is typically the main precision-limiting factor of a phase-shifting interferometer. We present a method that instead of trying to insolate vibrations; it takes advantage of their presence to produce the different phase-steps. The method is based on spatial and time domain processing techniques to compute first the different unknown phase-steps and then reconstruct the phase from these tilt-shifted interferograms. In order to compensate the camera movement, it is needed to perform an affine registration process between the different interferograms. Simulated and experimental results demonstrate the effectiveness of the proposed technique without the use of any phase-shifter device.

© 2011 OSA

## 1. Introduction

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39**(22), 3894–3898 (2000). [CrossRef]

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE **4416**, 152–157 (2001). [CrossRef]

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39**(22), 3894–3898 (2000). [CrossRef]

6. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. **41**(13), 2435–2439 (2002). [CrossRef] [PubMed]

3. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. **40**(17), 2886–2894 (2001). [CrossRef]

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE **4416**, 152–157 (2001). [CrossRef]

*N*interferograms to obtain the modulating phase by a TPI technique is more accurate that obtaining the phase from a single interferogram by the Fourier transform method. On one hand, the TPI method reduces errors in phase calculations when noisy interferograms are involved [2]. Additionally, the Fourier method suffers from border effects that introduce additional errors in the recovered phase. Our method makes possible to obtain interferometric measurements in very hostile environments as thermal-vacuum chambers, where severe shock vibrations with a typical frequency of 0.5 Hz and induced displacements in the range of millimeters appear caused by the cryogenic pump. Additionally, the proposed method also permits to acquire interferometric measures without the need of any phase-shifter device.

## 2. Proposed method

### 2.1 Tilt-shift determination by the Fourier transform method

*n*

^{th}interferogram can be represented as,where,

*x*and

*y*directions and

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39**(22), 3894–3898 (2000). [CrossRef]

### 2.2 Affine registration between interferograms

*k*equal to

*x*or

*y*, denotes the

*x*or

*y*component of

*n*respectively. The binarization process is obtained assigning the pixels with positive local orientation a value equal to true and to negative pixels a value equal to false. In the binarized map, the critical points —points where the gradient of

*x*and

*y*axis respectively, a rotation of 0.5 rad and a scale change of 1.2 and 0.98 pixels in the

*x*and

*y*axis respectively. In Figs. 2(c) and 2(d) it is shown the corresponding binarized maps of Figs. 2(a) and 2(b) respectively and the detected critical points by the corner detector. These points are shown as red rectangles. Finally, in Figs. 2(e) and 2(f) are shown the phases with the detected control points.

^{th}interferogram and the first one (used as reference) can be obtained by at least three control-point correspondences [8

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE **4416**, 152–157 (2001). [CrossRef]

### 2.3 General temporal demodulation

*S*using the least-squares criterion that imposes

*N*the total number of interferograms. From expressions (13) and (14) we finally have,and,

## 3. Simulations

*pv*(peak to valley) of 5.3 rad and

*rms*(root mean square) of 0.7 rad. The background is

*rms*(root mean square error) of the difference between the phases after affine rectification

*err*) of the different affine rectificated phases obtained from the division between the computed

*rms*, shown in Table 3, and the

*pv*of the actual phase map that is 5.3 rad. As can be seen from Table 3, the mean of the

*rms*and relative errors are 0.096 rad and 0.28% respectively. After determining the different affine transformations, the different interferograms and tilt-shifts are affine registered with respect to the first one.

*rms*(root mean square error) of the difference between both phases is of about 0.054 rad. As can be seen from Table 3, the

*rms*values obtained from the Fourier transform demodulating method are in all cases larger than the

*rms*error recovered from the temporal demodulation method and using all interferograms.

## 4. Experimental results

*rms*(root mean square) and

*pv*(peak to valley) of the wave-front error measured by the Zygo interferometer using a four step phase-shifting algorithm and with the Mach-Zenhder interferometer using the proposed method with ten interferograms is shown in Table 5 .

*rms*measurement is of about 0.0009 waves. On the other hand, the difference in the

*pv*measurement is 0.026 waves.

## 5. Conclusions

## References and links

1. | K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. |

2. | D. Malacara, M. Servín, and Z. Malacara, |

3. | K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. |

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

5. | M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. |

6. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. |

7. | J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. |

8. | Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE |

9. | G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. |

10. | R. Hartley, and A. Zisserman, |

11. | C. Harris, and M. J. Stephens, “A combined corner and edge detector,” |

12. | J. C. Wyant, and K. Creath, “ |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(120.3180) Instrumentation, measurement, and metrology : Interferometry

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 18, 2010

Revised Manuscript: August 2, 2010

Manuscript Accepted: August 4, 2010

Published: January 5, 2011

**Citation**

J. Vargas, J. Antonio. Quiroga, A. Álvarez-Herrero, and T. Belenguer, "Phase-shifting interferometry based on induced vibrations," Opt. Express **19**, 584-596 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-584

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### References

- K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).
- D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)
- K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [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]
- M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef] [PubMed]
- J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008). [CrossRef]
- Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]
- G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008). [CrossRef] [PubMed]
- R. Hartley, and A. Zisserman, Multiple View Geometry in Computer Vision, (Cambridge University Press, 2004).
- C. Harris, and M. J. Stephens, “A combined corner and edge detector,” in Alvey Vision Conference, pp. 147–152 (1988).
- J. C. Wyant, and K. Creath, “Basic Wavefront Aberation Theory of Optical Metrology,” in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992).

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