## Synchronous phase-demodulation and harmonic rejection of 9-step pixelated dynamic interferograms |

Optics Express, Vol. 20, Issue 11, pp. 11734-11739 (2012)

http://dx.doi.org/10.1364/OE.20.011734

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### Abstract

We propose a novel synchronous phase-demodulation of pixelated interferograms using squared 3x3 phase-shifted unit-cells. This 3x3 unit-cell is tiled over the CCD image sensor to create a two-dimensional (2D) pixelated carrier. Our synchronous phase-demodulation uses this 2D carrier to demodulate the pixelated interferogram as in the standard 2x2 unit-cell case. The main motivation behind the use of a 3x3 pixelated carrier (instead of the usual 2x2) is its higher harmonic robustness, allowing one to demodulate intensity-distorted fringe patterns. The harmonic rejection robustness of our spatial 3x3 configuration equals the robustness of the temporal least-squares 9-step phase-shifting algorithm (PSA). In other words, extending from the usual 2x2 phase-shifting unit-cell to 3x3 unit-cells, one extends the harmonic rejection of the demodulation algorithm. Finally we also prove that our proposed 9-step, 3x3 pixelated carrier uses the 2D available spectral space more efficiently than using these 9-steps in a linear spatial-carrier configuration.

© 2012 OSA

## 1. Introduction

## 2. Synchronous phase-demodulation of pixelated phase-mask interferograms

7. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE **7790**, 77900K, 77900K-12 (2010). [CrossRef]

9. J. M. Padilla, M. Servin, and J. C. Estrada, “Harmonics rejection in pixelated interferograms using spatio-temporal demodulation,” Opt. Express **19**(20), 19508–19513 (2011). [CrossRef] [PubMed]

11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A **72**(1), 156–160 (1982). [CrossRef]

7. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE **7790**, 77900K, 77900K-12 (2010). [CrossRef]

8. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express **18**(17), 18492–18497 (2010). [CrossRef] [PubMed]

7. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE **7790**, 77900K, 77900K-12 (2010). [CrossRef]

8. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express **18**(17), 18492–18497 (2010). [CrossRef] [PubMed]

12. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express **19**(6), 5126–5133 (2011). [CrossRef] [PubMed]

## 3. Synchronous-Fourier demodulation of intensity-distorted pixelated interferograms

1. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in *Optical Shop Testing*, D. Malacara, ed. (Wiley, NJ, 2007), Chap. 14, doi: 10.1002/9780470135976.ch14 [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]

*n*-th harmonic; all others terms remain as previously defined. Following our synchronous 3x3 demodulation method, one forms the product

9. J. M. Padilla, M. Servin, and J. C. Estrada, “Harmonics rejection in pixelated interferograms using spatio-temporal demodulation,” Opt. Express **19**(20), 19508–19513 (2011). [CrossRef] [PubMed]

*A*is a proportionality constant. Usually, the energy contribution in Eq. (11) of the

*n*-th harmonic decreases very fast

1. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in *Optical Shop Testing*, D. Malacara, ed. (Wiley, NJ, 2007), Chap. 14, doi: 10.1002/9780470135976.ch14 [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]

10. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

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. Harmonic rejection of the 2x2, the 3x3, and the linear 9-step pixelated carriers

9. J. M. Padilla, M. Servin, and J. C. Estrada, “Harmonics rejection in pixelated interferograms using spatio-temporal demodulation,” Opt. Express **19**(20), 19508–19513 (2011). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgment

## References and Links

1. | H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in |

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

3. | Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. |

4. | J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE |

5. | B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. |

6. | 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. |

7. | B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE |

8. | M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express |

9. | J. M. Padilla, M. Servin, and J. C. Estrada, “Harmonics rejection in pixelated interferograms using spatio-temporal demodulation,” Opt. Express |

10. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

11. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A |

12. | J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express |

**OCIS Codes**

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

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 6, 2012

Revised Manuscript: April 23, 2012

Manuscript Accepted: April 30, 2012

Published: May 9, 2012

**Citation**

J. M. Padilla, M. Servin, and J. C. Estrada, "Synchronous phase-demodulation and harmonic rejection of 9-step pixelated dynamic interferograms," Opt. Express **20**, 11734-11739 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11734

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### References

- H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, NJ, 2007), Chap. 14, doi: 10.1002/9780470135976.ch14 [CrossRef]
- M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express17(24), 21867–21881 (2009). [CrossRef] [PubMed]
- Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett.85(6), 1069–1071 (2004). [CrossRef]
- J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE5531, 304–314 (2004). [CrossRef]
- B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt.45(19), 4554–4562 (2006). [CrossRef] [PubMed]
- 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]
- B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE7790, 77900K, 77900K-12 (2010). [CrossRef]
- M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express18(17), 18492–18497 (2010). [CrossRef] [PubMed]
- J. M. Padilla, M. Servin, and J. C. Estrada, “Harmonics rejection in pixelated interferograms using spatio-temporal demodulation,” Opt. Express19(20), 19508–19513 (2011). [CrossRef] [PubMed]
- Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt.35(1), 51–60 (1996). [CrossRef] [PubMed]
- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72(1), 156–160 (1982). [CrossRef]
- J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express19(6), 5126–5133 (2011). [CrossRef] [PubMed]

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