## Sampled Fiber Bragg Grating spectral synthesis |

Optics Express, Vol. 20, Issue 20, pp. 22429-22441 (2012)

http://dx.doi.org/10.1364/OE.20.022429

Acrobat PDF (1261 KB)

### Abstract

In this paper, a technique to estimate the deformation profile of a Sampled Fiber Bragg Grating (SFBG) is proposed and experimentally verified. From the SFBG intensity reflection spectrum, any arbitrary longitudinal axis deformation profile applied to a SFBG is estimated. The synthesis algorithm combines a custom defined error metric to compare the measured and the synthetic spectra and the Particle Swarm Optimization technique to get the deformation profile. Using controlled deformation profiles, the proposed method has been successfully checked by means of simulated and experimental tests. The results obtained under different controlled cases show a remarkable repetitiveness (< 50 *με*) and good spatial accuracy (< 1 *mm*).

© 2012 OSA

## 1. Introduction

1. K. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. **15**, 1263–1276 (1997). [CrossRef]

3. Z. Zang, “Numerical analysis of optical bistability based on fiber Bragg grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun. **285**, 521–526 (2011). [CrossRef]

4. Z. Zang and Y. Zhang, “Low-switching power (< 45 mw) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt. **59**, 161–165 (2012). [CrossRef]

5. M. LeBlanc, S. Huang, M. Ohn, A. Guemes, and A. Othonos, “Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis,” Opt. Lett. **21**, 1405–1407 (1996). [CrossRef] [PubMed]

6. S. Huang, M. M. Ohn, M. LeBlanc, and R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Sruct. **7**, 248–256 (1998). [CrossRef]

7. J. Azaa and M. Muriel, “Reconstructing arbitrary strain distributions within fiber gratings by timefrequency signal analysis,” Opt. Lett. **25**, 698–700 (2000). [CrossRef]

8. X. Chapeleau, P. Casari, D. Leduc, Y. Scudeller, C. Lupi, R. Ny, and C. Boisrobert, “Determination of strain distribution and temperature gradient profiles from phase measurements of embedded fiber Bragg gratings,” J. Opt. A-Pure Appl. Op. **8**, 775 (2006). [CrossRef]

9. F. Casagrande, P. Crespi, A. Grassi, A. Lulli, R. Kenny, and M. Whelan, “From the reflected spectrum to the properties of a fiber Bragg grating: a genetic algorithm approach with application to distributed strain sensing,” Appl. Opt. **41**, 5238–5244 (2002). [CrossRef] [PubMed]

10. C. Cheng, Y. Lo, W. Li, C. Kuo, and H. Cheng, “Estimations of fiber Bragg grating parameters and strain gauge factor using optical spectrum and strain distribution information,” Appl. Opt. **46**, 4555–4562 (2007). [CrossRef] [PubMed]

12. Z. Wu, Q. Qiao, F. Wu, and L. Cai, “Research on fiber Bragg grating spectral optimization with particle swarm optimization algorithm,” Appl. Mech. Mater. **128**, 690–693 (2012). [CrossRef]

## 2. Theoretical model

13. B. Eggleton, P. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. **30**, 1620–1622 (1994). [CrossRef]

**s**(z) and

**a**(z) are the sampling and the apodization functions respectively,

**A**(z) is the grating apodization profile, Λ is the FBG period and

*ϕ*(

*z*) is the phase. Currently,the period (P) of the sampling function is much larger than the period of the grating (Λ), causing a double modulation in the structure: a rapidly varying component with a period Λ and a slowly varying sampled envelope with a period P. Based on Fourier theory, the periodic sampling function of period P can be expressed as a comb function modulated by complex coefficients

*F*: being

_{m}*m*the Fourier order. For each comb (

*m*) of the Fourier equivalent function, one

*ghost*grating appears on the resulting spectrum. The spectral separation between two

*ghost*gratings 1 is inversely proportional to the sampling period:

*P*∝ 1

*mm*), a clear spatial division is created on the grating structure and more information (in comparison to an uniform FBG) can be obtained by analyzing the SFBG spectral response with the same span. Besides the main spectral contribution of the

*m*=0 order, some

*ghost*gratings can be also analyzed giving extra points of view of the fiber deformation structure.

### 2.1. Metrics for spectral comparison

**Spectra correlation:**The correlation between the two central wavelengths of both spectra is calculated. This metric should be enough if there had not been noise and measurement errors in the spectra, consequently this value should be complemented with other parameters in a real situation.**Peak position:**The difference between the wavelength of the most relevant peaks of the synthetic and measured spectrum is calculated. This value can indicate an offset deformation between the two compared spectra.**Peak value:**The difference between the reflection value of the most relevant peaks of the synthetic and measured spectra is calculated. When a different strain profile is applied to a SFBG, the peak reflection values also change depending on the grating deformed length matching each grating period (wavelengths).**Peak width:**The width (measured in nm) of the most relevant peaks of the synthetic and measured spectra is calculated. This error value will penalize when the most reflective peaks are not the same in different spectra.**Peak ratio:**The ratio between the two most relevant peaks is calculated. The peak ratio should remain unaltered despite the intensity noise.**Number of peaks:**The number of peaks above a specific threshold of both spectra are also subtracted. As this parameter increases, it indicates that both spectra are in very different deformation cases.**Peak Kurtosis:**The Kurtosis of each of the most relevant peaks is computed in a pre-fixed lambda span. This value indicates the*sharpness*of a peak but it is also very noise sensitive.

### 2.2. Optimization method: particle swarm optimization

- Initialize a population array of particles (SFBG strain vector) with random positions and velocities in the search space of N dimensions (number of SFBG sections).
- Evaluate the error function for each particle (comparing the synthetic spectrum of each particle with the desired one as explained in Section 2.1).
- Compare the latest error evaluation of the current particle with its “previous best” error value:
*p*. If the current error value is better, then_{best}*p*will be updated, and_{best}*p*(previous best position) will be updated to the current location_{i}*x*._{i} - Determine the particle within the swarm with the best error value (
*g*) and assign its location to_{best}*p*._{g} - Change velocity and position of each particle within the swarm according to the following expression:Where
*w*is the inertia weight,*c*_{1}and*c*_{2}are positive constants, typically defined as learning rates, and*r*_{1}and*r*_{2}are random functions in the range [0,1]. - If the stopping condition is met then exit with the best result so far; otherwise repeat from point 2.

*X*and velocity

_{i}*V*within the N-dimensional search space, where:

_{i}*X*) represents a possible strain vector applied to the SFBG model and the velocity (

_{i}*V*) is a vector that represents how the particle evolves within the search space. With the proposed error metric, the optimization goal is to find the deformation profile that causes the desired spectrum so, the best particle position found during the PSO run should match the physical strain profile of the SFBG. Due to the stochastic nature of the PSO, the obtained strain vector (

_{i}*X*), corresponding with the best error value found in the swarm (

_{p}*g*), may be also a local minimum of the error function. By using a high (> 100) number of starting particles should be enough to reduce the local minimum convergence. However, wrong solutions caused by a local minimum can be detected just by comparing the obtained best error value (

_{best}*g*) to a specific threshold. If a wrong solution is detected, the PSO algorithm should be run again with a higher number of starting particles.

_{best}*N*= 5 section SFBG with a total length of

*L*= 4

*mm*and with a linear decreasing apodization profile with a decreasing factor

*M*= 0.25. For each strain vector, the synthetic spectrum is calculated using the Transfer Matrix method (with

*N*= 5 sections) and compared to the desired one with the defined error function. The spatial resolution of the employed structure is

*P*/2 = 0.8

*mm*. In the following section, the response of the algorithm to simulated spectra is analyzed under ideal and realistic situations.

## 3. Theoretical simulations

*P*/2 = 0.8

*mm*). On the other simulated cases, the deformation value applied to each section is non-uniform, trying to reproduce a continuum strain profile. All the artificial spectra are generated using the Transfer Matrix method [15

15. M. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” Phot. Tech. Lett. IEEE **9**, 955–957 (1997). [CrossRef]

*N*= 5, matching the SBFG structure. For the non-uniform deformation cases, the number of employed sections was set to

*N*= 50, having

*N*= 10 different deformation values within each sampling section.

_{s}*ε*= ±15

*με*. On the right side (Fig. 3), the synthetic spectrum of the best achieved solution is compared to its artificial (desired) spectrum. Both spectra match each other almost perfectly, giving rise to an error value lower than 0.25 that has been considered the error stop condition. Although these tests recreate situations where the spatial deformation variations perfectly match the proposed sensing structure (

*P*/2 = 0.8

*mm*), they are helpful to validate the algorithm resolution and repeatability.

*P*/2 = 0.8

*mm*). A two-stages linear varying deformation profile is used to generate the artificial spectrum. In Fig. 4 (left), five obtained strain profiles are compared to the strain profile used to generate the artificial spectrum. As the resolution of the sensing principle is limited to the section length, the obtained strain for each section is the averaged deformation of the correspondent length. The resulting strain profiles qualitatively follow the deformation profile applied to get the artificial spectrum. In these tests, the achieved range that contains the correct solution is Δ

*ε*= ±30

*με*. This range is worse than the already commented but it still is sufficiently small to be considered as highly accurate. In Fig. 4 (right), the synthetic spectrum of the best obtained solution is compared to the artificial one. The two depicted spectra are in good agreement, having a small error value (lower than the established threshold of 0.25 for simulated scenarios). The synthetic spectrum exhibits the same characteristic shape as the artificial one: two smooth peaks at lower wavelengths followed by four sharper peaks at higher wavelengths. There are slight mismatches due to the higher order spectral components of the simulated spectra. These components are more sensitive to slight mechanical deformations introduced, in this scenario, by the spatial resolution of the employed model. This second case is more realistic, thus demonstrating the good algorithm response even under non-ideal conditions.

## 4. Experimental demonstration

*N*= 5 sections of

*P*/2 = 0.8

*mm*linearly apodized with a decreasing factor of

*M*= 0.25 as shown in Fig. 5. This SFBG was embedded into a epoxy resin block with a specific shape to cause a non-uniform strain profile on the SBFG. The resin block has been also mechanically simulated using Finite Element Analysis to qualitatively obtain the applied deformation profile.

*με*) than the first ones, so further deformation profiles have to be compensated using the obtained residual strain profile as a reference. This compensation step is required since the entire block is stretched, not just the written SFBG, so the reference deformation profile is the packaged one. To obtain the strain profile transfered to the SFBG by the resin block, a Finite Element Analysis (FEA) has been carried out and it is detailed in the following.

### 4.1. Finite element analysis

*h*= 0.8

*mm*) resin block where an optical fiber is embedded in the longitudinal axis. The Young’s modulus was set to 3.5 GPa for epoxy resin and 74 GPa for optical fiber silica. The employed SFBG of

*L*= 4

*mm*length is longitudinally centered within the resin block as shown in Fig. 7 with a tolerance of ±0.1

*mm*.

### 4.2. Experimental setup

*N*= 20 runs for each of the five captured spectra to delimit the algorithm convergence range. The stop condition for the maximum allowable error has been set to 0.75 based on previous runs. If a particular PSO run obtains a final error higher than the threshold value (less than 10% of total runs), the algorithm is executed again with a higher number of starting particles. The achieved convergence range for real spectra was ±50

_{i}*με*. The spectrum captured with the higher strain value is detailed in Fig. 9, where the obtained results for the resin block subjected to a high load are shown. On the right side, measured and synthetic spectra of the highest deformation case are presented (an increase of Δ

*L*≈ 25

*μm*over

*L*= 6

*mm*). Both spectra show a very good agreement (the depicted synthetic spectrum has an error metric of 0.667) exhibiting the same characteristic shape as the measured one (four sharp peaks at lower wavelengths followed by two smoother ones at higher wavelengths). However there are still small differences mainly caused to the higher order spectral components of both spectra such as the incorrect location and width of the last lobe, the value of the less reflective lobes These mismatches are created by small contributions of several factors that will be discussed in the next section. Also in Fig. 9, the deformation profile of the synthetic spectrum is compared to the normalized deformation profile simulated with FEA. The obtained profile is also compensated for the residual strain caused during the resin block. Both profiles are also interpolated for viewing purposes. The compensated profile remarkably agrees with the simulated one, thus demonstrating the correct response of the proposed algorithm.

## 5. Discussion

*N*= 5 SFBG sections. Results show a perfect match between the desired and synthetic spectra with a great repeatability under this ideal environment. Once the error metric and the PSO algorithm are validated, a more realistic deformation profile where each SFBG section is deformed with a non-uniform strain profile is tested. The achieved results match almost perfectly the applied deformation profile proving the ability of the algorithm to work with non-uniform deformation cases.

*L*= 4

*mm*was embedded into a resin block designed to apply a non-uniform deformation profile to the SBFG when it is stretched. The obtained deformation profiles are compared with the theoretical deformation profile obtained using FEA which is applied to the SFBG. Due to the fabrication process, a residual strain profile is created into the SFBG prior to stretching the block so the obtained deformation profiles have to be compensated with the residual strain profile. Even after the compensation, the obtained deformation profile matches the FEA simulations remarkably, having also a very good agreement between measured and synthetic spectra. Under experimental conditions the algorithm exhibits an excellent performance, but it also shows some disadvantages.

*με*for the experimental case). This convergence process has to be controlled by evaluating the obtained error and re-running the algorithm when it is required. In addition to the own algorithm nature, some extra factors reduce its final performance when it deals with real spectra: the simplified optical model selected to reduce the computation time does not perfectly replicate a real SFBG structure (slight misalignments during the SBFG fabrication or the apodization effect due to the laser spot width). The interpolation technique and the resolution of the Optical Spectrum Analyzer (60 pm) reduces the correlation with the synthetic spectrum (not interpolated). The limitation of the optical model, in addition to the difficulty of accurately stretching small pieces, even an incorrect gluing process may lead to less accurate results.

*N*= 5 sections of

*P*/2 = 0.8

*mm*with a linear apodization, but the algorithm is applicable to any SFBG structure. By increasing the number of sections (

*N*), the computation time is also increased. All the performed test were run in a custom implementation made in Matlab (The MathWorks, Inc.) and each run of the whole algorithm takes a few minutes. By using a more efficient implementation, this computation time can be reduced to a few seconds for each spectrum, allowing the proposed algorithm to work in quasi-on-line applications.

## 6. Conclusion

*mm*) of the SFBG. The error metric is a critical point to lead the PSO algorithm to the right deformation profile. Simulated tests have shown the good accuracy and repetitiveness of the proposed algorithm. An experimental demonstration has been also performed by embedding a SFBG into a resin block with a particular shape. This block shape has been simulated using Finite Element Analysis to get the deformation profile and it has been compared to the achieved deformation profile. Mechanical simulations and the obtained deformation profile show an excellent agreement, demonstrating the validity of the proposed algorithm to accurately get any deformation profile dealing with any SFBG structure with a good spatial resolution, just from the reflection spectrum intensity. The proposed technique can be very useful for distributed sensing applications based on FBG technology.

## Acknowledgments

## References and links

1. | K. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. |

2. | J. Lopez-Higuera, |

3. | Z. Zang, “Numerical analysis of optical bistability based on fiber Bragg grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun. |

4. | Z. Zang and Y. Zhang, “Low-switching power (< 45 mw) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt. |

5. | M. LeBlanc, S. Huang, M. Ohn, A. Guemes, and A. Othonos, “Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis,” Opt. Lett. |

6. | S. Huang, M. M. Ohn, M. LeBlanc, and R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Sruct. |

7. | J. Azaa and M. Muriel, “Reconstructing arbitrary strain distributions within fiber gratings by timefrequency signal analysis,” Opt. Lett. |

8. | X. Chapeleau, P. Casari, D. Leduc, Y. Scudeller, C. Lupi, R. Ny, and C. Boisrobert, “Determination of strain distribution and temperature gradient profiles from phase measurements of embedded fiber Bragg gratings,” J. Opt. A-Pure Appl. Op. |

9. | F. Casagrande, P. Crespi, A. Grassi, A. Lulli, R. Kenny, and M. Whelan, “From the reflected spectrum to the properties of a fiber Bragg grating: a genetic algorithm approach with application to distributed strain sensing,” Appl. Opt. |

10. | C. Cheng, Y. Lo, W. Li, C. Kuo, and H. Cheng, “Estimations of fiber Bragg grating parameters and strain gauge factor using optical spectrum and strain distribution information,” Appl. Opt. |

11. | F. Teng, W. Yin, F. Wu, Z. Li, and T. Wu, “Analysis of a fiber Bragg grating sensing system with transverse uniform press by using genetic algorithm,” Opto-electron. Lett. |

12. | Z. Wu, Q. Qiao, F. Wu, and L. Cai, “Research on fiber Bragg grating spectral optimization with particle swarm optimization algorithm,” Appl. Mech. Mater. |

13. | B. Eggleton, P. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. |

14. | J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of International Conference on Neural Networks, ed. (IEEE, 1995), vol. 4, pp. 1942–1948. |

15. | M. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” Phot. Tech. Lett. IEEE |

**OCIS Codes**

(070.4790) Fourier optics and signal processing : Spectrum analysis

(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 12, 2012

Revised Manuscript: August 14, 2012

Manuscript Accepted: August 17, 2012

Published: September 17, 2012

**Citation**

L. Rodriguez-Cobo, A. Cobo, and J. M. Lopez-Higuera, "Sampled Fiber Bragg Grating spectral synthesis," Opt. Express **20**, 22429-22441 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22429

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### References

- K. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol.15, 1263–1276 (1997). [CrossRef]
- J. Lopez-Higuera, Handbook of Optical Fibre Sensing Technology (John Wiley and Sons Inc, 2002).
- Z. Zang, “Numerical analysis of optical bistability based on fiber Bragg grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun.285, 521–526 (2011). [CrossRef]
- Z. Zang and Y. Zhang, “Low-switching power (< 45 mw) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” J. Mod. Opt.59, 161–165 (2012). [CrossRef]
- M. LeBlanc, S. Huang, M. Ohn, A. Guemes, and A. Othonos, “Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis,” Opt. Lett.21, 1405–1407 (1996). [CrossRef] [PubMed]
- S. Huang, M. M. Ohn, M. LeBlanc, and R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Sruct.7, 248–256 (1998). [CrossRef]
- J. Azaa and M. Muriel, “Reconstructing arbitrary strain distributions within fiber gratings by timefrequency signal analysis,” Opt. Lett.25, 698–700 (2000). [CrossRef]
- X. Chapeleau, P. Casari, D. Leduc, Y. Scudeller, C. Lupi, R. Ny, and C. Boisrobert, “Determination of strain distribution and temperature gradient profiles from phase measurements of embedded fiber Bragg gratings,” J. Opt. A-Pure Appl. Op.8, 775 (2006). [CrossRef]
- F. Casagrande, P. Crespi, A. Grassi, A. Lulli, R. Kenny, and M. Whelan, “From the reflected spectrum to the properties of a fiber Bragg grating: a genetic algorithm approach with application to distributed strain sensing,” Appl. Opt.41, 5238–5244 (2002). [CrossRef] [PubMed]
- C. Cheng, Y. Lo, W. Li, C. Kuo, and H. Cheng, “Estimations of fiber Bragg grating parameters and strain gauge factor using optical spectrum and strain distribution information,” Appl. Opt.46, 4555–4562 (2007). [CrossRef] [PubMed]
- F. Teng, W. Yin, F. Wu, Z. Li, and T. Wu, “Analysis of a fiber Bragg grating sensing system with transverse uniform press by using genetic algorithm,” Opto-electron. Lett.4, 121–125 (2008).
- Z. Wu, Q. Qiao, F. Wu, and L. Cai, “Research on fiber Bragg grating spectral optimization with particle swarm optimization algorithm,” Appl. Mech. Mater.128, 690–693 (2012). [CrossRef]
- B. Eggleton, P. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett.30, 1620–1622 (1994). [CrossRef]
- J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of International Conference on Neural Networks, ed. (IEEE, 1995), vol. 4, pp. 1942–1948.
- M. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” Phot. Tech. Lett. IEEE9, 955–957 (1997). [CrossRef]

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