## A polarized low-coherence interferometry demodulation algorithm by recovering the absolute phase of a selected monochromatic frequency |

Optics Express, Vol. 20, Issue 16, pp. 18117-18126 (2012)

http://dx.doi.org/10.1364/OE.20.018117

Acrobat PDF (1043 KB)

### Abstract

A demodulation algorithm based on absolute phase recovery of a selected monochromatic frequency is proposed for optical fiber Fabry-Perot pressure sensing system. The algorithm uses Fourier transform to get the relative phase and intercept of the unwrapped phase-frequency linear fit curve to identify its interference-order, which are then used to recover the absolute phase. A simplified mathematical model of the polarized low-coherence interference fringes was established to illustrate the principle of the proposed algorithm. Phase unwrapping and the selection of monochromatic frequency were discussed in detail. Pressure measurement experiment was carried out to verify the effectiveness of the proposed algorithm. Results showed that the demodulation precision by our algorithm could reach up to 0.15*kPa*, which has been improved by 13 times comparing with phase slope based algorithm.

© 2012 OSA

## 1. Introduction

*π*phase ambiguity and has an unlimited measurement range in principle [1

1. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. **13**(4), 832–843 (1996). [CrossRef]

2. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. **31**(7), 919–925 (1992). [CrossRef] [PubMed]

4. A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory,” Appl. Opt. **41**(23), 4876–4883 (2002). [CrossRef] [PubMed]

5. L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. **27**(7), 530–532 (2002). [CrossRef] [PubMed]

6. J. G. Kim, “Absolute temperature measurement using white light interferometry,” J. Opt. Soc. Kor. **4**(2), 89–93 (2000). [CrossRef]

8. S. H. Kim, S. H. Lee, J. I. Lim, and K. H. Kim, “Absolute refractive index measurement method over a broad wavelength region based on white-light interferometry,” Appl. Opt. **49**(5), 910–914 (2010). [CrossRef] [PubMed]

9. G. S. Kino and S. S. C. Chim, “Miraucorrelation microscope,” Appl. Opt. **29**(26), 3775–3783 (1990). [CrossRef] [PubMed]

11. P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett. **22**(14), 1065–1067 (1997). [CrossRef] [PubMed]

3. P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. **44**(3), 519–534 (1997). [CrossRef]

12. S. Chen, A. W. Palmer, K. T. V. Grattan, and B. T. Meggitt, “Digital signal-processing techniques for electronically scanned optical-fiber white-light interferometry,” Appl. Opt. **31**(28), 6003–6010 (1992). [CrossRef] [PubMed]

13. A. Pf rtner and J. Schwider, “Dispersion error in white-light linnik interferometers and its implications for evaluation procedures,” Appl. Opt. **40**(34), 6223–6228 (2001). [CrossRef] [PubMed]

*et al.*proposed spatial-frequency domain analysis (SFDA) algorithm [14

14. P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. **42**(2), 389–401 (1995). [CrossRef]

1. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. **13**(4), 832–843 (1996). [CrossRef]

16. 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]

17. A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. **39**(13), 2107–2115 (2000). [CrossRef] [PubMed]

3. P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. **44**(3), 519–534 (1997). [CrossRef]

*et al.*improved SFDA algorithm by using the phase gap between phase and coherence information to obtain fringe-order [18

18. P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. **41**(22), 4571–4578 (2002). [CrossRef] [PubMed]

*et al.*proposed a spectrally resolved phase algorithm [19

19. S. K. Debnath and M. P. Kothiyal, “Improved optical profiling using the spectral phase in spectrally resolved white-light interferometry,” Appl. Opt. **45**(27), 6965–6972 (2006). [CrossRef] [PubMed]

*π*through a temporal phase-shifting technique based on different scanning step to adapt for the limited focus depth of Mirau-type LCI. However, the two algorithms are complicated in the analysis of phase gap since they were both developed for three-dimensional surface profiling. In this paper we proposed a simplified demodulation algorithm for optical fiber Fabry-Perot pressure sensing system. Phase unwrapping and the selection of monochromatic frequency were discussed in detail, which is explored little in other literature, to our best knowledge. The algorithm is then successfully applied to the measurement of air pressure.

## 2. Experimental setup

20. R. Dändliker, E. Zimmermann, and G. Frosio, “Electronically scanned white-light interferometry: a novel noise-resistant signal processing,” Opt. Lett. **17**(9), 679–681 (1992). [CrossRef] [PubMed]

*d*is the cavity length change, Δ

*P*is the pressure change,

*r*is the effective radius of the silicon diaphragm,

*h*is its thickness,

*E*and

*υ*are the Young's modulus and Poisson's ratio of silicon, respectively. Equation (1) shows that the cavity length change is proportional to the air pressure.

*kPa*to 170

*kPa*at interval 1

*kPa*, which is applied by an air pressure chamber with accuracy of 0.02

*kPa*. For layout of the optical wedge and CCD in our experiment, the interferogram shifts from the right to the left on the CCD with the increase of air pressure. Figure 3(b) is a typical low-coherence interferogram acquired with F-P sensor under 100

*kPa*in experiment, which consists of 3000 discrete data points.

## 3. Theoretical analysis

*F*(Ω), which is also a Gaussian function, and Ω is the frequency. According to the time-shift and frequency-shift feature of Fourier transform, Eq. (2)’s Fourier transform can be written as:

*M*(Ω) and phase-frequency characteristic

*φ*(Ω) can be respectively written as:

*is the selected frequency of the interference fringes, then the absolute phase*

_{sf}*φ*(Ω,

*xₒ*) at the small frequency region centered on Ω

*can be written as -*

_{sf}*xₒ*Ω. If expressed with interference-order

*m*, it can also be written as:where Φ(Ω,

*xₒ*) is the relative phase of Ω, the value of

*m*depends on the measurand

*x*and the specific Ω.

_{0}*in continuous form should be replaced with a specific discrete monochromatic frequency Ω*

_{sf}*which has relative high amplitude, where subscript*

_{k}*k*is the DFT serial number. We unwrap the relative phase of the small frequency region centered on Ω

*with the relative phase Φ(Ω*

_{k}*,*

_{k}*xₒ*) as the reference phase, and unwrapped phase is denoted as Φ’(Ω

*,*

_{l}*xₒ*). Let

*n*be the interference-order of monochromatic frequency Ω

*, Eq. (8) is then changed to the following expression with discrete frequency being taken into account:where*

_{k}*k-p*and

*k + q*are the DFT serial number of starting frequency and ending frequency respectively. The unwrapped phase Φ’(Ω

*,*

_{l}*xₒ*) could be seen as pseudo-absolute-phase. There is an overall 2

*nπ*upwards translation from the theoretical absolute phase to the unwrapped phase. The intercept is then also moved from zero to 2

*nπ*in theory. Therefore, we can easily calculate the interference-order

*n*from the following formula:where T is the intercept obtained from the unwrapped phase linear fit curve with Ω

*as variable. The function floor() returns the largest integer which is smaller or equal to the parameter in the bracket. Once the interference-order*

_{l}*n*is identified, we could get the absolute phase

*φ*(Ω

*,*

_{k}*xₒ*) at the selected frequency Ω

*:*

_{k}*φ*(Ω

*,*

_{k}*xₒ*) is -

*xₒ*Ω

*and Ω*

_{k}*is fixed, the linear relationship between the measurand and the absolute phase at the selected frequency Ω*

_{k}*is therefore established and demodulation is realized.*

_{k}*φ*(Ω,

*xₒ*) curves with F-P sensor under different pressure, i.e. the F-P sensor has different

*xₒ*, e.g.

*x*,

_{0}^{1}*x*…

_{0}^{2}*x*. Straight curves ①’-⑥’ represent corresponding unwrapped Φ’(Ω

_{0}^{6}*,*

_{l}*xₒ*) curves after phase unwrapping process based on reference phase Φ(Ω

*,*

_{k}*xₒ*). Curves ①’-⑥’ are parallel to curves ①-⑥, respectively, and the vertical translation is 2

*nπ*. The unwrapped phase curves are divided into many clusters after phase unwrapping. Each cluster represents a specific interference-order

*n*depending on

*xₒ*and Ω

*. Phase curves belonging to the same cluster have a common intercept. Only two clusters are symbolically shown in Fig. 4(b). In other words, the change of interference-order*

_{k}*n*is reflected on the 2

*π*equi-spaced step change of phase intercept. Points A-F in Fig. 4(b) represent different absolute phase at Ω

*with the change of air pressure,*

_{k}*i.e. xₒ*, e.g.

*x*,

_{0}^{1}*x*…

_{0}^{2}*x*, while points A’-F’ represent corresponding wrapped relative phase. Figure 4(c) shows the relative phase at Ω

_{0}^{6}*obtained by DFT with*

_{k}*xₒ*as variable. The relative phase of a specific frequency changes periodically with the monotonous change of

*xₒ*. Combining the identified interference-order

*n*from Fig. 4(b) and the relative phase Φ(Ω

*,*

_{k}*xₒ*) obtained by DFT in Fig. 4(c), we could recover the absolute phase

*φ*(Ω

*,*

_{k}*xₒ*), as shown in Fig. 4(d). The absolute phase curve is what we will use for demodulation.

*n*through Eq. (9) and Eq. (10). We introduce a relative phase at a specific frequency Ω

*as the reference phase to carry out phase unwrapping, which is different from traditional process. The small frequency range centered on Ω*

_{k}*is divide into two parts: downside frequency range (Ω*

_{k}*, Ω*

_{k-p}*) and upside frequency range (Ω*

_{k}*, Ω*

_{k}*). For the downside frequency range (Ω*

_{k + q}*, Ω*

_{k-p}*), relative phase unwrapping recursive formula is,For the upside frequency range (Ω*

_{k}*, Ω*

_{k}*), relative phase unwrapping recursive formula is,*

_{k + q}*kPa*pressure. The selected monochromatic frequency Ω

*is Ω*

_{k}_{1558}, which has been marked out in Fig. 5(c) and its relative phase is chosen as the reference phase for phase unwrapping. The unwrapped phase presents a good monotonic and linear relationship with frequency.

_{1547}, Ω

_{1572}) is preliminarily determined. Next, we make a linearity quantitative comparison of the determined frequencies. The frequency with best linearity is the optimal frequency that we need for demodulation. Table 1 shows the comparative results. According to Table 1, we can see that the frequency Ω

_{1558}has best linearity. Therefore, it is selected as the optimal frequency in our system.

## 4. Pressure measure experiment results and discussion

*kPa*–170

*kPa*has been performed at room temperature using the configuration in Fig. 1. The performance of the proposed algorithm is compared with the phase-slope-based algorithm. Figure 7(a) and Fig. 7(b) show the change of phase slope and intercept of the unwrapped phase-frequency linear fit curve with the increase of air pressure from 30

*kPa*to 170

*kPa*at interval 1

*kPa*, respectively. Phase unwrapping is carried out for the small frequency range (Ω

_{1547}, Ω

_{1572}) using Eq. (12) and Eq. (13). The relative phase of Ω

_{1558}which is the selected optimal frequency is chosen as the reference phase for phase unwrapping. From Fig. 7(a) and Fig. 7(b), it can be seen that the phase slope varies continuously while the intercept varies by step accompanying with the change of air pressure, as we expected from the theoretical analysis. Phase slope keeps a linear relationship with the air pressure on the whole, but linear response is locally distorted, which eventually leads to low-precision demodulation results. The intercept-pressure curve has obvious ladder feature and guarantees the validity of interference-order identification with a strong anti-noise capability.

*kPa*or precision 0.11% full scale, while the tolerance of phase-slope-based algorithm is about 2

*kPa*or precision 1.43% full scale. The demodulation precision has been improved 13 times.

## 5. Conclusion

*. A theoretical model is established and we choose relative phase at a selected frequency Ω*

_{k}*as the reference phase to carry out phase unwrapping. The intercept of the unwrapped phase-frequency linear fit curve is used to identify the interference-order. The optimal monochromatic frequency is selected based on the linearity of relative-phase-measurand curve. Combining the relative phase obtained by DFT and the identified interference-order, the absolute phase is recovered. The algorithm is completely realized in frequency domain and needs no filtering process. Experimental results verified the effectiveness of our algorithm, and precision of pressure demodulation by our algorithm is improved by 13 times comparing with the direct phase-slope-based algorithm.*

_{k}## Acknowledgments

## References and links

1. | K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am. |

2. | T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. |

3. | P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt. |

4. | A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory,” Appl. Opt. |

5. | L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. |

6. | J. G. Kim, “Absolute temperature measurement using white light interferometry,” J. Opt. Soc. Kor. |

7. | L. M. Smith and C. C. Dobson, “Absolute displacement measurements using modulation of the spectrum of white light in a Michelson interferometer,” Appl. Opt. |

8. | S. H. Kim, S. H. Lee, J. I. Lim, and K. H. Kim, “Absolute refractive index measurement method over a broad wavelength region based on white-light interferometry,” Appl. Opt. |

9. | G. S. Kino and S. S. C. Chim, “Miraucorrelation microscope,” Appl. Opt. |

10. | S. S. C. Chim and G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. |

11. | P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett. |

12. | S. Chen, A. W. Palmer, K. T. V. Grattan, and B. T. Meggitt, “Digital signal-processing techniques for electronically scanned optical-fiber white-light interferometry,” Appl. Opt. |

13. | A. Pf rtner and J. Schwider, “Dispersion error in white-light linnik interferometers and its implications for evaluation procedures,” Appl. Opt. |

14. | P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. |

15. | P. Sandoz, “An algorithm for profilometry by white-light phase-shifting interferometry,” J. Mod. Opt. |

16. | 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 |

17. | A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. |

18. | P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. |

19. | S. K. Debnath and M. P. Kothiyal, “Improved optical profiling using the spectral phase in spectrally resolved white-light interferometry,” Appl. Opt. |

20. | R. Dändliker, E. Zimmermann, and G. Frosio, “Electronically scanned white-light interferometry: a novel noise-resistant signal processing,” Opt. Lett. |

21. | S. Timoshenko and S. Woinowsky-Krieger, |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(100.5070) Image processing : Phase retrieval

(110.4500) Imaging systems : Optical coherence tomography

(120.5475) Instrumentation, measurement, and metrology : Pressure measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 25, 2012

Revised Manuscript: July 18, 2012

Manuscript Accepted: July 19, 2012

Published: July 23, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Junfeng Jiang, Shaohua Wang, Tiegen Liu, Kun Liu, Jinde Yin, Xiange Meng, Yimo Zhang, Shuang Wang, Zunqi Qin, Fan Wu, and Dingjie Li, "A polarized low-coherence interferometry demodulation algorithm by recovering the absolute phase of a selected monochromatic frequency," Opt. Express **20**, 18117-18126 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18117

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### References

- K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white-light interferometry,” J. Opt. Soc. Am.13(4), 832–843 (1996). [CrossRef]
- T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt.31(7), 919–925 (1992). [CrossRef] [PubMed]
- P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in white-light phase-shifting interferometry,” J. Mod. Opt.44(3), 519–534 (1997). [CrossRef]
- A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory,” Appl. Opt.41(23), 4876–4883 (2002). [CrossRef] [PubMed]
- L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett.27(7), 530–532 (2002). [CrossRef] [PubMed]
- J. G. Kim, “Absolute temperature measurement using white light interferometry,” J. Opt. Soc. Kor.4(2), 89–93 (2000). [CrossRef]
- L. M. Smith and C. C. Dobson, “Absolute displacement measurements using modulation of the spectrum of white light in a Michelson interferometer,” Appl. Opt.28(16), 3339–3342 (1989). [CrossRef] [PubMed]
- S. H. Kim, S. H. Lee, J. I. Lim, and K. H. Kim, “Absolute refractive index measurement method over a broad wavelength region based on white-light interferometry,” Appl. Opt.49(5), 910–914 (2010). [CrossRef] [PubMed]
- G. S. Kino and S. S. C. Chim, “Miraucorrelation microscope,” Appl. Opt.29(26), 3775–3783 (1990). [CrossRef] [PubMed]
- S. S. C. Chim and G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt.31(14), 2550–2553 (1992). [CrossRef] [PubMed]
- P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett.22(14), 1065–1067 (1997). [CrossRef] [PubMed]
- S. Chen, A. W. Palmer, K. T. V. Grattan, and B. T. Meggitt, “Digital signal-processing techniques for electronically scanned optical-fiber white-light interferometry,” Appl. Opt.31(28), 6003–6010 (1992). [CrossRef] [PubMed]
- A. Pf rtner and J. Schwider, “Dispersion error in white-light linnik interferometers and its implications for evaluation procedures,” Appl. Opt.40(34), 6223–6228 (2001). [CrossRef] [PubMed]
- P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt.42(2), 389–401 (1995). [CrossRef]
- P. Sandoz, “An algorithm for profilometry by white-light phase-shifting interferometry,” J. Mod. Opt.43, 1545–1554 (1996).
- 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. A14(4), 918–930 (1997). [CrossRef]
- A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt.39(13), 2107–2115 (2000). [CrossRef] [PubMed]
- P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt.41(22), 4571–4578 (2002). [CrossRef] [PubMed]
- S. K. Debnath and M. P. Kothiyal, “Improved optical profiling using the spectral phase in spectrally resolved white-light interferometry,” Appl. Opt.45(27), 6965–6972 (2006). [CrossRef] [PubMed]
- R. Dändliker, E. Zimmermann, and G. Frosio, “Electronically scanned white-light interferometry: a novel noise-resistant signal processing,” Opt. Lett.17(9), 679–681 (1992). [CrossRef] [PubMed]
- S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1989).

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