## Harmonics rejection in pixelated interferograms using spatio-temporal demodulation |

Optics Express, Vol. 19, Issue 20, pp. 19508-19513 (2011)

http://dx.doi.org/10.1364/OE.19.019508

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### Abstract

Pixelated phase-mask interferograms have become an industry standard in spatial phase-shifting interferometry. These pixelated interferograms allow full wavefront encoding using a single interferogram. This allows the study of fast dynamic events in hostile mechanical environments. Recently an error-free demodulation method for ideal pixelated interferograms was proposed. However, non-ideal conditions in interferometry may arise due to non-linear response of the CCD camera, multiple light paths in the interferometer, etc. These conditions generate non-sinusoidal fringes containing harmonics which degrade the phase estimation. Here we show that two-dimensional Fourier demodulation of pixelated interferograms rejects most harmonics except the complex ones at {-3^{rd}, +5^{th}, −7^{th}, +9^{th}, −11^{th},…}. We propose temporal phase-shifting to remove these remaining harmonics. In particular, a 2-step phase-shifting algorithm is used to eliminate the −3^{rd} and +5^{th} complex harmonics, while a 3-step one is used to remove the −3^{rd}, +5^{th}, −7^{th} and +9^{th} complex harmonics.

© 2011 OSA

## 1. Introduction

1. 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]

*x*2 circular orientation”, illustrated in Fig. 1 , however the “stacked” 2x2 spatial configuration [6

6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE **5531**, 304–314 (2004). [CrossRef]

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

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

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

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

**18**(17), 18492–18497 (2010). [CrossRef] [PubMed]

## 3 Fourier demodulation with distorted-fringe pixelated interferograms

*n*-th harmonics, and

*x*2 fundamental cell to obtain the spectral shift for every harmonic as:

*k*labels the harmonic carriers in Eq. (7). From Eq. (8), it is clear that for higher than the 4th harmonics the carrier is chosen from the above 4 possibilities using

^{st}row do not undergo any frequency shift; they are all located at the spectral origin along with

^{nd}row are multiplied by

^{nd}and 3

^{rd}rows in Eq. (9) are supressed:

*A*is a proportionality constant and

*not-rejected*by the low-pass filtering

^{rd}, +5

^{th}, −7

^{th}, +9

^{th}, −11

^{th},...}. Please note that, these harmonics coincides with those

*not-rejected*in a standard 4-steps temporal least-squares PSA [10

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

## 4. Spatio-temporal demodulation of non-sinusoidal pixelated interferograms

^{rd}, 5

^{th}, −7

^{th}, 9

^{th},…}, but we can use temporal quadrature filters (PSAs) to reject them [1

1. 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]

11. A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, “Design of phase-shifting algorithms by fine-tuning spectral shaping,” Opt. Express **19**(11), 10692–10697 (2011). [CrossRef] [PubMed]

^{rd}and 5

^{th}complex harmonics:

^{rd}, 5

^{th}, −7

^{th}, and 9

^{th}complex harmonics:

*A*is a proportionality constant. The convolutions order in Eq. (18) may be interchanged, but for numerical efficiency we suggest to first apply the temporal convolution.

*x*512 pixels, but higher resolution provides greater spectral separation allowing better phase-demodulation. Today, pixelated phase-mask interferograms have resolutions almost sixteen times higher (4 mega-pixels).

## 5 Conclusions

*x*2 pixelated interferogram leaves the complex harmonics {-3

^{rd}, +5

^{th}, −7

^{th}, +9

^{th}, −11

^{th},...} overlapping with the fundamental signal

^{rd}, and +5

^{th}complex harmonics, and a 3-step one to remove (at least) the −3

^{rd}, +5

^{th}, −7

^{th}, and +9

^{th}complex harmonics. Note that temporal demodulation of

*N*-step, 2x2 pixelated interferograms (

*N*=1,2,3,…) rejects the same harmonics as a temporal 4

*N*-step least-squares PSA. In other words,

*N*-steps temporal interferometry of 2

*x*2 pixelated fringe patterns is 4 times faster, preserving the harmonics rejection robustness of a 4

*N*-step least-squares PSA.

## Acknowledgements

## References and Links

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

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

3. | O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. |

4. | C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE |

5. | B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. |

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

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

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

9. | 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 |

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

11. | A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, “Design of phase-shifting algorithms by fine-tuning spectral shaping,” 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: July 13, 2011

Revised Manuscript: August 12, 2011

Manuscript Accepted: August 15, 2011

Published: September 22, 2011

**Citation**

J. M. Padilla, M. Servin, and J. C. Estrada, "Harmonics rejection in pixelated interferograms using spatio-temporal demodulation," Opt. Express **19**, 19508-19513 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19508

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### References

- M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express17(24), 21867–21881 (2009). [CrossRef] [PubMed]
- R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng.23, 361–364 (1984).
- O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett.9(2), 59–61 (1984). [CrossRef] [PubMed]
- C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE1531, 119–127 (1992). [CrossRef]
- B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt.40(19), 3211–3214 (2001). [CrossRef] [PubMed]
- 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]
- M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express18(17), 18492–18497 (2010). [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, 1–12 (2010).
- 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]
- Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt.35(1), 51–60 (1996). [CrossRef] [PubMed]
- A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, “Design of phase-shifting algorithms by fine-tuning spectral shaping,” Opt. Express19(11), 10692–10697 (2011). [CrossRef] [PubMed]

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