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

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References

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  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]
  4. M. Prabhugoud and K. Peters, “Finite element model for embedded fiber Bragg grating sensor” Smart Mat. Struct. 15, 550 ( 2006).
    [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]
  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]
  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. 34, 2622 ( 2009).
    [Crossref] [PubMed]
  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]
  10. J. Gil, “Polarimetric characterization of light and media”, The European Physical Journal Applied Physics 40, 1 ( 2007).
    [Crossref]
  11. T. Erdogan, “Fiber Grating Spectra”, Journal of Lightwave Technology 15, 1277 ( 1997).
    [Crossref]
  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).

2009 (3)

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]

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]

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]

2008 (1)

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

2007 (1)

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

2006 (1)

M. Prabhugoud and K. Peters, “Finite element model for embedded fiber Bragg grating sensor” Smart Mat. Struct. 15, 550 ( 2006).
[Crossref]

1997 (3)

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

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

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

Barybin, A.

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

Buck, T. C.

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]

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]

Dmitriev, V.

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

El-Khozondar, H. J.

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]

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]

Erdogan, T.

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

Gil, J.

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

Haugse, E.

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]

Hill, K. O.

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

Hoffmann, L.

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]

Koch, A. W.

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]

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]

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]

Mawatari, T.

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

Meltz, G.

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

Müller, M. S.

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]

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]

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]

Narasimhamutry, T.

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

Nelson, D.

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

Othonos, A.

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

Peters, K.

M. Prabhugoud and K. Peters, “Finite element model for embedded fiber Bragg grating sensor” Smart Mat. Struct. 15, 550 ( 2006).
[Crossref]

Prabhugoud, M.

M. Prabhugoud and K. Peters, “Finite element model for embedded fiber Bragg grating sensor” Smart Mat. Struct. 15, 550 ( 2006).
[Crossref]

Sandmair, A.

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]

Schulz, W.

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]

Seim, J.

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]

Udd, E.

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]

Yariv, A.

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

Yeh, P.

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

J. Lightwave Technol. (1)

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]

J. of Lightwave Technol. (1)

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

J. Quantum Electron. (1)

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]

Journal of Lightwave Technology (1)

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

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

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

Smart Mat. Struct. (2)

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]

The European Physical Journal Applied Physics (1)

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

Other (4)

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]

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

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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Figures (4)

Fig. 1.
Fig. 1.

Fiber Bragg grating in polarization maintaining fiber loaded by an arbitrary strain tensor ē, leading to changes in impermeability tensor B̄. The grating is illuminated by light represented by a coherency matrix Φ. The reflected light is split into the average intensity components 〈Ip -〉 and 〈Is -〉.

Fig. 2.
Fig. 2.

Reflected intensities from a shear strained FBG with a beat length of 7.5 mm, with exy =1·10-3 for light linearly polarized at 45° to the principal axes and unpolarized light.

Fig. 3.
Fig. 3.

Left: Intensities corresponding to figure 3 with normal strains exx =-2000 µm/m, eyy =800 µm/m and exy =2000 µm/m. Right: Reflected intensities according to numerical and analytical description. The loads are exx =eyy =0 µm/m, exy =2000 µm/m.

Fig. 4.
Fig. 4.

Change of the V-parameter depending on the shear strain exy for different values of normal strain exx ={-800 µm/m, …, 1000 µm/m} and a constant value of eyy =0.

Equations (14)

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Φ ij = ε i ( t ) ε j ( t ) , X ( t ) = = lim T 1 T 0 T X ( t ) d t
lim T 1 T 0 T A p A s d t = lim T 1 T 0 T A s A p d t = 0 .
A = Q ̱ ρ ̱ Q ̱ 1 . A + .
I p / s = A p / s = A p / s = [ Q ̅ ρ ̅ Q ̅ 1 A + . { Q ̅ ρ ̅ Q ̅ 1 A + } ] p / s .
I p / s = 1 T 0 T A p / s A p / s d t .
Q 11 Q 22 , Q 12 Q 21 or Q 11 Q 22 , Q 12 Q 21
I p = ( Q 11 2 ρ p 2 + Q 12 2 ρ s 2 ) I 0 +
I s = ( Q 12 2 ρ p 2 + Q 11 2 ρ s 2 ) I 0 + .
B x x = 1 / ( n eff , p 0 ) 2 + p 11 e xx + p 12 e yy + p 12 e zz
B yy = 1 / ( n eff , s 0 ) 2 + p 12 e xx + p 11 e yy + p 12 e zz , B xy = ( p 11 p 12 ) e xy / 2 .
Q 11 2 = 2 B xy 2 b 2 b b 2 + 4 B xy 2 + 4 B xy 2
Q 12 2 = b 2 b b 2 + 4 B xy 2 + 2 B xy 2 b 2 b b 2 + 4 B xy 2 + 4 B xy 2 .
n eff , p / s 2 = ( B xx + B yy ) ± ( B xx B yy ) 2 + 4 B xy 2 2 , Λ ( e ̅ ) = Λ 0 ( 1 + e zz ) ,
V = 2 B xy 2 b 2 b b 2 + 4 B xy 2 + 2 B xy 2 .

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