## Fourier transform demodulation of pixelated phase-masked interferograms |

Optics Express, Vol. 18, Issue 15, pp. 16090-16095 (2010)

http://dx.doi.org/10.1364/OE.18.016090

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### Abstract

Recently a new type of spatial phase shifting interferometer was proposed that uses a phase-mask over the camera’s pixels. This new interferometer allows one to phase modulate each pixel independently by setting the angle of a linear polarizer built in contact over the camera’s CCD. In this way neighbor pixels may have any desired (however fixed) phase shift without cross taking. The standard manufacturing of these interferometers uses a 2x2 array with phase-shifts of 0, *π*/2, *π*, and 3*π*/2 radians. This 2x2 array is tiled all over the video camera’s CCD. In this paper we propose a new way to phase demodulate these phase-masked interferograms using the squeezing phase-shifting technique. A notable advantage of this squeezing technique is that it allows one the use of Fourier interferometry wiping out the detuning error that most phase shifting algorithms suffers. Finally we suggest the use of an alternative phase-mask to phase modulate the camera’s pixels using a linear spatial carrier along a given axis.

© 2010 OSA

## 1. Introduction

2. 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]

3. M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express **16**(13), 9276–9283 (2008). [CrossRef] [PubMed]

*x*axis) interferogram is,Where

*ϕ*(

*x,y*) is the modulating phase,

*a*(

*x,y*) is the background, and

*b*(

*x,y*) the contrast of the interferogram. Finally, the carrier frequency

*ω*

_{0}is in radians per spatial or temporal pixel.

### 1.1 Four steps phase-shift demodulation of 2x2 phase-masked interferograms

7. B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. **40**(19), 3211–3214 (2001). [CrossRef]

8. 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]

9. 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]

8. 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]

9. 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]

*π*/2 radians among them.

*ϕ*(

*x,y*). These 4 pixels are not only phase-shifted but also

*spatially-displaced*. Just as a comment, the spatial impulse response associated with the 4-steps PSI algorithm in Eq. (3) is:

*ϕ*(

*x,y*). We may graphically see this error

8. 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]

10. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express **17**(7), 5618–5623 (2009). [CrossRef] [PubMed]

*N*-point, class A averaging algorithm [11

11. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. **45**(19), 4554–4562 (2006). [CrossRef] [PubMed]

11. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. **45**(19), 4554–4562 (2006). [CrossRef] [PubMed]

11. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. **45**(19), 4554–4562 (2006). [CrossRef] [PubMed]

**45**(19), 4554–4562 (2006). [CrossRef] [PubMed]

## 2. Fourier Transform Demodulation (FTD) of 2x2 phase-masked interferograms

2. 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]

10. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express **17**(7), 5618–5623 (2009). [CrossRef] [PubMed]

3. M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express **16**(13), 9276–9283 (2008). [CrossRef] [PubMed]

*L*x

*L*phase shifted interferograms (at the right hand side of Eq. (5)) into the single 4

*LxL*linear carrier fringe pattern

*Ic*(

*x,y*) shown in Fig. 4 .

**5531**, 304–314 (2004). [CrossRef]

9. 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]

## 3. Suggested phase-mask to obtain a linear carrier interferogram

*x*direction (Fig. 4), at the expense of a non-continuous rearrangement of the spatial information in

*ϕ*(

*x,y*). That is, the quadrature filtering needed to demodulate the LCFI will smooth out the newly defined neighborhood of pixels shown in Fig. 6. Then the “backward” mapping (from panel 5(b) to panel 5(c)) generate small high-frequency phase jumps that where absent in the original CCD interferogram. These high frequency spurious phase jumps can be easily eliminated using a low pass filter over the demodulated phase. Alternatively one may low pass filter the two real-valued quadrature fringe patterns (mapped into the CCD’s geometry) obtained by the FTD technique before computing the phase in Fig. 5(c).

*x*axis) interferogram given by,

*ϕ*(

*x,y*) without accessory geometrical mapping, and without detuning. In other words, the detuning-error associated with the complex-valued 2x2 or 3x3 demodulating convolution masks [11

**45**(19), 4554–4562 (2006). [CrossRef] [PubMed]

*ε*a small real number. We do not think that the complex-valued 2x2 or 3x3 masks applied to demodulate the interferogram in Fig. 1 may have a faster phase response than the one obtained from this one-sided filter applied to an LCFI.

## 4. Conclusions

*x*axis) modulation. This new spatial interferogram’s re-arrangement allows one to Fourier demodulate it without detuning-error [2

2. 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]

3. M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express **16**(13), 9276–9283 (2008). [CrossRef] [PubMed]

**45**(19), 4554–4562 (2006). [CrossRef] [PubMed]

*x*axis) modulating phase-mask. The proposed mask in Fig. 7, allows one to phase demodulate the interferogram [8

**5531**, 304–314 (2004). [CrossRef]

**44**(32), 6861–6868 (2005). [CrossRef] [PubMed]

**72**(1), 156–160 (1982). [CrossRef]

## References and links

1. | D. Malacara, M. Servin, and Z. Malacara, |

2. | 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 |

3. | M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express |

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

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

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

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

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

9. | 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. |

10. | J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express |

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

**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: April 7, 2010

Revised Manuscript: July 2, 2010

Manuscript Accepted: July 5, 2010

Published: July 15, 2010

**Citation**

M. Servin, J. C. Estrada, and O. Medina, "Fourier transform demodulation of pixelated phase-masked interferograms," Opt. Express **18**, 16090-16095 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-16090

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### References

- D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2 ed., (Taylor & Francis Group, CRC Press, 2005).
- 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]
- M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express 16(13), 9276–9283 (2008). [CrossRef] [PubMed]
- R. Smithe 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. SPIE 1531, 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]
- 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]
- 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]
- J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). [CrossRef] [PubMed]
- B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006). [CrossRef] [PubMed]

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