## Analytical coherency matrix treatment of shear strained fiber Bragg gratings

Optics Express, Vol. 17, Issue 25, pp. 22624-22631 (2009)

http://dx.doi.org/10.1364/OE.17.022624

Acrobat PDF (167 KB)

### Abstract

Reconstruction of the strain tensor at the position of an embedded fiber Bragg grating sensor has been the goal of recent research. However, ambiguities in the measurand - the polarization resolved reflected intensity spectrum - upon occurrence of shear strain hinder its achievement due to lack of an invertible model. In this work, we derive such a model using coherency matrix properties of unpolarized light. We deduce simplified sensor parameters for the ambiguous shear strain loading case, which possibly lead to a practical inversion of the problem.

© 2009 Optical Society of America

## 1. Introduction

1. K. O. Hill and G. Meltz, “Fiber Bragg Grating Technology Fundamentals and Overview” J. of Lightwave Technol. **15**, 1263 (
1997). [CrossRef]

2. A. Othonos, “Fiber Bragg gratings” Rev. Sci. Instrum. **68**, 4309 (
1997). [CrossRef]

3. T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement” Smart Mat. Struct. **17**, 19 (
2008). [CrossRef]

*λ*. By using PMFs two Bragg wavelengths

_{B}*λ*

_{B,p}and

*λ*

_{B,s}for

*p*- and

*s*-polarized modes are obtained. The sensing principle of these devices is based on the change in the Bragg wavelength as a function of the measurand to be detected. In the case of the embedded sensors, these influences can be strain or stress and temperature. The physics behind the opto-mechanic interaction lies in the photoelasticity of the material and the change in geometry upon an applied strain field. This is represented by the position dependent strain tensor

*e*̄(

*x*,

*y*,

*z*). Figure 1 shows the described scenario. The strain field is applied at the position of the sensor. The fiber, guiding the light to the position of the sensor, is considered unperturbed, although parts of this fiber may be subjected to the strain field as well. To be able to treat the problem analytically, the strain field is assumed to be homogeneous along the length of the sensor

*e*̄(

*x*,

*y*,

*z*)=

*e*̄(

*z*). This is a reasonable assumption in many cases as the length of the sensor is small, typically a few millimeters. Electromagnetic properties of the material change in a linear approximation according to Pockels law of photoelasticity. It linearly connects the change in the impermeability tensor 0394;

*B*̄=

*B*̄-

*B*̄

^{0}to the strains by

*B*=

_{ij}*p*, where the

_{ijkl}e_{kl}*p*are the photoelastic constants.

_{ijkl}*e*being nonzero. Thus, Δ

_{i j}*B*̄ can take an arbitrary form as well. The question is, which components of the strain tensor or impermeability tensor influence the optical response of the FBG. We demonstrated, that only the components

*B*,

_{xx}*B*and

_{yy}*B*, or in an isotropic material the strains

_{xy}*e*,

_{xx}*e*,

_{yy}*e*and

_{zz}*e*exhibit a noteworthy influence [5

_{xy}5. M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, “Full strain tensor treatment of fiber Bragg grating sensors” J. Quantum Electron. **45**, 547 (
2009). [CrossRef]

6. E. Udd, W. Schulz, J. Seim, and E. Haugse, in “Multidimensional strain field measurements using fiber optic grating sensors”, *Proceedings of SPIE*, 3986, 254–262 (
2000). [CrossRef]

3. T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement” Smart Mat. Struct. **17**, 19 (
2008). [CrossRef]

5. M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, “Full strain tensor treatment of fiber Bragg grating sensors” J. Quantum Electron. **45**, 547 (
2009). [CrossRef]

8. M. S. Müller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, “Analytical Solution of Four-Mode Coupling in Shear Strain Loaded Fiber-Bragg-Grating Sensors”, Opt. Lett. **34**, 2622 (
2009). [CrossRef] [PubMed]

*B*=(

_{xy}*p*

_{11}-

*p*

_{12})

*e*/2=0 is fulfilled, corresponding to an absence of shear strain. When shear strain is present, the simple reflection of one polarization mode of appropriate wavelength at the Bragg grating and its backward propagation is no longer the only effect to be considered. A coupling among the polarization modes is also observed, superimposed on the forward-backward coupling of the grating. Since both models, CMT and the analytical use monochromatic waves as a basis, only coherent effects may be described in a straight forward manner. This limits the model to using fully polarized light, viz. light with a full correlation between the fields in the two polarization modes, such as experimentally obtained by using tuneable lasers. Thus an interference effect arises, when light from one polarization mode is coherently coupled to the other mode, and in turn to complicated spectral responses, such as depicted in figure 2. We recently described a numerical model to incorporate incoherent effects such as unpolarized light within the CMT framework [9

_{xy}9. M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch, “Shear-Strain Influence on Fiber Bragg Grating Measurement Systems”, J. Lightwave Technol. **27**, 1–7 (
2009). [CrossRef]

8. M. S. Müller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, “Analytical Solution of Four-Mode Coupling in Shear Strain Loaded Fiber-Bragg-Grating Sensors”, Opt. Lett. **34**, 2622 (
2009). [CrossRef] [PubMed]

*e*and is thus not seamlessly capable of modeling unpolarized light. Our approach is to use the properties of the coherency matrix representation of light for the simplification of the model. We derive a simplification for the transformation matrix and give an approximation for the derived coefficients describing the reflected spectrum.

^{iωt}## 2. Model

10. J. Gil, “Polarimetric characterization of light and media”, The European Physical Journal Applied Physics **40**, 1 (
2007). [CrossRef]

*ε*are the components of the instantaneous Jones vector. The Jones vector entries are given by the complex amplitudes of the two polarization modes

_{i}*ε*={

*A*,

_{p}*A*}

_{s}*, if we neglect an absolute phase and only consider the relative phase of the modes [10*

^{T}10. J. Gil, “Polarimetric characterization of light and media”, The European Physical Journal Applied Physics **40**, 1 (
2007). [CrossRef]

**A**

_{+}={

*A*

_{p+},

*A*

_{s+}}

*for illumination of the sensor. The reflected amplitudes are given by*

^{T}**A**-={

*A*

_{p-},

*A*

_{s-}}

*. The monochromatic wave model connects the four amplitudes by [8*

^{T}8. M. S. Müller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, “Analytical Solution of Four-Mode Coupling in Shear Strain Loaded Fiber-Bragg-Grating Sensors”, Opt. Lett. **34**, 2622 (
2009). [CrossRef] [PubMed]

*Q*̱ is the transformation matrix from the unperturbed mode system into the strained mode system and

*ρ*̄=diag{

*ρ*,

_{p}*ρ*} summarizes the reflectivities

_{s}*ρ*and

_{p}*ρ*[8

_{s}**34**, 2622 (
2009). [CrossRef] [PubMed]

11. T. Erdogan, “Fiber Grating Spectra”, Journal of Lightwave Technology **15**, 1277 (
1997). [CrossRef]

*Q*̄ is non-diagonal, the reflected amplitude of one polarization mode depends upon the amplitudes of both illuminating polarization directions.

*A*

_{p/s}

*A*

^{*}

_{p/s}and inhomogeneous elements

*A*

_{p/s}

*A*

^{*}

_{s/p}. To simplify this equation we consider the time interval

*T*in which the intensities are measured

*Q*̱ given in [8

**34**, 2622 (
2009). [CrossRef] [PubMed]

*B*=

_{xx}*B*.

_{yy}^{-3}, the arising inequality can be neglected. With the requirement for unpolarized light, i.e. 〈|

*A*

_{p+}|

^{2}〉=〈|

*A*

_{s+}|

^{2}〉=〈

*I*

_{0+}〉 and using

*Q*

_{11}

^{2}+

*Q*

_{12}

^{2}=1, we obtain

*Q*

_{11}|

^{2}and |

*Q*

_{12}|

^{2}are unconvenient expressions, that require simplification for practical use. The following definitions are used for their simplification. The effective refractive index of the fundamentalmodes of the unperturbed fiber is given by

*n*

^{0}

_{eff,p}and

*n*

^{0}

_{eff,s}, with the birefringence Δ

*n*=

*n*

^{0}

_{eff,s}-

*n*

^{0}

_{eff,p}which is usually given by the beat length

*L*=2

_{B}*πλ*/Δ

*n*. Since only the normal strains

*e*and the

_{ii}*e*shear strain component show an influence on the optical response in an isotropic material, the impermeability tensor reads [12]

_{xy}*b*=

*B*-

_{xx}*B*and

_{yy}*B*is approximately 10

_{xy}^{-3}or less, compared to

*B*~1. Therefore we may neglect terms of the order

_{ii}*b*

^{2},

*B*

^{2}

*or higher and find for |*

_{xy}*Q*

_{11}|

^{2}

*Q*

_{12}|

^{2}

*ρ*

_{p/s}|

^{2}are real positive functions of the grating period Λ, the effective refractive index of the corresponding mode

*n*

_{eff,p/s}and the wavelength (|

*ρ*|

^{2}

_{p/s}=|

*ρ*(

*λ*,

*λ*(Λ,

_{B}*n*

_{eff}))|

^{2}). Both, the Λ and

*n*

_{eff,p/s}are well known functions of the impermeability tensor [13] reading

*B*are functions of strains given in equation (9) to (10). Thus, for unpolarized light, the reflection spectrum of the shear strained fiber Bragg grating in each polarization direction includes a linear combination of the two standard intensity reflection spectra of the unloaded FBG, shifted in wavelength by some amount.

_{ij}*V*=|

*Q*

_{11}|

^{2}/|

*Q*

_{12}|

^{2}of the two Bragg peaks in one polarization mode. This parameter takes the manageable form

*V*-parameter. Performing this at a strongly different wavelength, as suggested by Udd et. al.[6

6. E. Udd, W. Schulz, J. Seim, and E. Haugse, in “Multidimensional strain field measurements using fiber optic grating sensors”, *Proceedings of SPIE*, 3986, 254–262 (
2000). [CrossRef]

## 3. Results

5. M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, “Full strain tensor treatment of fiber Bragg grating sensors” J. Quantum Electron. **45**, 547 (
2009). [CrossRef]

9. M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch, “Shear-Strain Influence on Fiber Bragg Grating Measurement Systems”, J. Lightwave Technol. **27**, 1–7 (
2009). [CrossRef]

11. T. Erdogan, “Fiber Grating Spectra”, Journal of Lightwave Technology **15**, 1277 (
1997). [CrossRef]

*L*=5 mm at 1550 nm,

_{b}*λ*

^{0}

_{B,x}=1550 nm and a grating length of 5 mm. For clarity of presentation, we use strongly apodized gratings since they show virtually no sidelobes. The

*ρ*

_{p/s}are thus found numerically.

*Q*

_{12}|

^{2}and thus to a peak at the position of the orthogonally polarized Bragg peak. The

*V*-parameters are 3.76 and 3.74 in the analytical and the numerical model respectively, and are thus in close agreement. When normal strains are applied, the position of the Bragg peaks shift due to the change in effective refractive index and grating length, according to equations (13). This changes the difference in the impermeability tensor entries

*B*and

_{xx}*B*and hence

_{yy}*b*is changed, leading to an influence in the polarization mode coupling as well. Figure 3 right shows this change in the polarization mode coupling in comparison to the numerical solution of the CMT. The cross coupled intensity almost matches that of the backward coupled intensity. Again, both results are in close agreement.

*p*-spectrum plus the two in the s-spectrum), the parameters

*λ*

_{B,x},

*λ*

_{B,y}and

*V*can be extracted. Assuming a load case were

*e*≠0,

_{xx}*e*≠0 and

_{yy}*e*≠0, we can reconstruct the full load case using the derived equations. Figure 4 shows the change in

_{xy}*V*-parameter upon loading with shear strain for different levels of normal strain

*e*. The normal strain dependency in the

_{xx}*V*-parameter can be explained by the separation of the Bragg peaks by the normal strain and thus a reduced mode coupling.

*V*-parameter in figure 4, derived assuming the light to be unpolarized, only depends on the strains and temperature of the FBG sensor. Trying to define a comparable parameter using polarized light, the relative phase of the two polarization modes will have to be included, because the relative phase changes the direction of polarization mode coupling and thus the magnitude of such a parameter. Since the relative phase of the two modes is not preserved by the polarization maintaining fiber and may thus change during measurement, it is not suitable for determining the shear strain, since it introduces the phase-difference as additional parameter. Hence, the

*V*-parameter is stable against changes in the experimental setup, or movement of the fiber and can be considered as robust a parameter as the wavelengths’ of the Bragg peaks.

*e*≠0 and an additional change in temperature, the method of Udd et. al. [6

_{zz}6. E. Udd, W. Schulz, J. Seim, and E. Haugse, in “Multidimensional strain field measurements using fiber optic grating sensors”, *Proceedings of SPIE*, 3986, 254–262 (
2000). [CrossRef]

## 4. Conclusions

*λ*

_{Bp/s}, a method of reconstructing the shear strain

*e*together with two normals strains may be obtained. Furthermore, when combined with the method of Udd et. al. [6

_{xy}*Proceedings of SPIE*, 3986, 254–262 (
2000). [CrossRef]

## References and links

1. | K. O. Hill and G. Meltz, “Fiber Bragg Grating Technology Fundamentals and Overview” J. of Lightwave Technol. |

2. | A. Othonos, “Fiber Bragg gratings” Rev. Sci. Instrum. |

3. | T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement” Smart Mat. Struct. |

4. | M. Prabhugoud and K. Peters, “Finite element model for embedded fiber Bragg grating sensor” Smart Mat. Struct. |

5. | M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, “Full strain tensor treatment of fiber Bragg grating sensors” J. Quantum Electron. |

6. | E. Udd, W. Schulz, J. Seim, and E. Haugse, in “Multidimensional strain field measurements using fiber optic grating sensors”, |

7. | A. Barybin and V. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory, (Rinton Press, 2002). |

8. | M. S. Müller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, “Analytical Solution of Four-Mode Coupling in Shear Strain Loaded Fiber-Bragg-Grating Sensors”, Opt. Lett. |

9. | M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch, “Shear-Strain Influence on Fiber Bragg Grating Measurement Systems”, J. Lightwave Technol. |

10. | J. Gil, “Polarimetric characterization of light and media”, The European Physical Journal Applied Physics |

11. | T. Erdogan, “Fiber Grating Spectra”, Journal of Lightwave Technology |

12. | T. Narasimhamutry, “Photoelastic and Electro-Optic Properties of Crystals”, (Plenum Press, 1981). |

13. | A. Yariv and P. Yeh, “Optical Waves in Crystals”, (Wiley, 1984). |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2300) Fiber optics and optical communications : Fiber measurements

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: September 21, 2009

Revised Manuscript: November 4, 2009

Manuscript Accepted: November 4, 2009

Published: November 25, 2009

**Citation**

Mathias S. Müller and Christoph D. A. Schnarr, "Analytical coherency matrix treatment of shear strained fiber Bragg gratings," Opt. Express **17**, 22624-22631 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22624

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### References

- K. O. Hill and G. Meltz, "Fiber Bragg Grating Technology Fundamentals and Overview," J. Lightwave Technol. 15, 1263 (1997). [CrossRef]
- A. Othonos, "Fiber Bragg gratings," Rev. Sci. Instrum. 68, 4309 (1997). [CrossRef]
- T. Mawatari and D. Nelson, "A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement," Smart Mat. Struct. 17, 19 (2008). [CrossRef]
- M. Prabhugoud and K. Peters, "Finite element model for embedded fiber Bragg grating sensor," Smart Mat. Struct. 15, 550 (2006). [CrossRef]
- M. S. Muller, L. Hoffmann, A. Sandmair, and A. W. Koch, "Full strain tensor treatment of fiber Bragg grating sensors," J. Quantum Electron. 45, 547 (2009). [CrossRef]
- E. Udd, W. Schulz, J. Seim, and E. Haugse, in "Multidimensional strain field measurements using fiber optic grating sensors,"Proc. SPIE 3986, 254-262 (2000). [CrossRef]
- A. Barybin and V. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory, (Rinton Press, 2002).
- M. S. Muller, H. J. El-Khozondar, T. C. Buck, and A. W. Koch, "Analytical Solution of Four-Mode Coupling in Shear Strain Loaded Fiber-Bragg-Grating Sensors," Opt. Lett. 34, 2622 (2009). [CrossRef] [PubMed]
- M. S. Muller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch, "Shear-Strain Influence on Fiber Bragg Grating Measurement Systems," J. Lightwave Technol. 27, 1-7 (2009). [CrossRef]
- J. Gil, "Polarimetric characterization of light and media," The European Phys. J. Appl. Phys. 40, 1 (2007). [CrossRef]
- T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277 (1997). [CrossRef]
- T. Narasimhamutry, Photoelastic and Electro-Optic Properties of Crystals, (Plenum Press, 1981).
- A. Yariv and P. Yeh, Optical Waves in Crystals, (Wiley, 1984).

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