Abstract

When a fiber-optic intracore Bragg grating is subject to an appreciable strain gradient, its reflective spectrum will not only be shifted but also be distorted because of the chirp of the grating. We employed the T-matrix formalism to calculate the influence of different strain gradients on the reflective spectra of Bragg gratings and have undertaken experiments to test these calculations. The results of these experiments have confirmed that intracore Bragg gratings can be used to evaluate strain gradients and can be thought of as quasi-distributed strain sensors. This adds a new dimension to structural sensing, permitting measurements in any situation where strain gradients exist. It also provides a warning of any sensor/host debonding.

© 1995 Optical Society of America

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References

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  1. W. W. Morey, G. Meltz, W. H. Glenn, “Fibre optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. De Paula, E. Udd, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1169, 98–107 (1989).
  2. S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
    [CrossRef]
  3. A. D. Kersey, T. A. Berkoff, W. W. Morey, “High-resolution fiber-grating based strain sensor with interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992).
    [CrossRef]
  4. A. D. Kersey, T. A. Berkoff, “Fiber-optic Bragg-grating differential-temperature sensors,” IEEE Photon. Technol. Lett. 4, 1183–1185 (1992).
    [CrossRef]
  5. R. M. Measures, “Smart composite structures with embedded sensors,” Composites Eng. 2, 597–618 (1992).
    [CrossRef]
  6. K. O. Hill, “Aperiodic distributed-parameter waveguides for integrated optics,” Appl. Opt. 13, 1853–1856 (1974).
    [CrossRef] [PubMed]
  7. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1975).
  8. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E. 48, 4758–4767 (1993).
    [CrossRef]
  9. D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
    [CrossRef]
  10. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A 2, 863–871 (1985).
    [CrossRef]
  11. M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguide via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  12. H. A. Haus, “Mirrors and interferometers,” in Waves and Fields in Optoelectronics, N. Holonyak, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 3, p. 55–80.
  13. A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE Quantum Electron. 4, 233–253 (1977).
    [CrossRef]
  14. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: a comparison of the results of Rouard’s method and coupled-mode theory,” J. Opt. Soc. Am. A 4, 60–65 (1987).
    [CrossRef]
  15. M. Matsuhara, K. O. Hill, “Optical-waveguide band-rejection filters: design,” Appl. Opt. 13, 2886–2888 (1974).
    [CrossRef] [PubMed]
  16. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase grating,” IEEE J. Lightwave Technol. LT-11, 1513–1517 (1993).
    [CrossRef]

1993

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E. 48, 4758–4767 (1993).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase grating,” IEEE J. Lightwave Technol. LT-11, 1513–1517 (1993).
[CrossRef]

1992

A. D. Kersey, T. A. Berkoff, W. W. Morey, “High-resolution fiber-grating based strain sensor with interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992).
[CrossRef]

A. D. Kersey, T. A. Berkoff, “Fiber-optic Bragg-grating differential-temperature sensors,” IEEE Photon. Technol. Lett. 4, 1183–1185 (1992).
[CrossRef]

R. M. Measures, “Smart composite structures with embedded sensors,” Composites Eng. 2, 597–618 (1992).
[CrossRef]

1987

1985

1977

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE Quantum Electron. 4, 233–253 (1977).
[CrossRef]

1975

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1975).

1974

1969

Alavie, A. T.

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

Berkoff, T. A.

A. D. Kersey, T. A. Berkoff, W. W. Morey, “High-resolution fiber-grating based strain sensor with interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992).
[CrossRef]

A. D. Kersey, T. A. Berkoff, “Fiber-optic Bragg-grating differential-temperature sensors,” IEEE Photon. Technol. Lett. 4, 1183–1185 (1992).
[CrossRef]

Coroy, T.

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

Glenn, W. H.

W. W. Morey, G. Meltz, W. H. Glenn, “Fibre optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. De Paula, E. Udd, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1169, 98–107 (1989).

Hall, D. G.

Haus, H. A.

H. A. Haus, “Mirrors and interferometers,” in Waves and Fields in Optoelectronics, N. Holonyak, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 3, p. 55–80.

Hill, K. O.

Karr, S.

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

Kermisch, D.

Kersey, A. D.

A. D. Kersey, T. A. Berkoff, “Fiber-optic Bragg-grating differential-temperature sensors,” IEEE Photon. Technol. Lett. 4, 1183–1185 (1992).
[CrossRef]

A. D. Kersey, T. A. Berkoff, W. W. Morey, “High-resolution fiber-grating based strain sensor with interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1975).

Liu, K.

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

Matsuhara, M.

Measures, R. M.

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

R. M. Measures, “Smart composite structures with embedded sensors,” Composites Eng. 2, 597–618 (1992).
[CrossRef]

Melle, S. M.

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

Meltz, G.

W. W. Morey, G. Meltz, W. H. Glenn, “Fibre optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. De Paula, E. Udd, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1169, 98–107 (1989).

Mizrahi, V.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase grating,” IEEE J. Lightwave Technol. LT-11, 1513–1517 (1993).
[CrossRef]

Morey, W. W.

A. D. Kersey, T. A. Berkoff, W. W. Morey, “High-resolution fiber-grating based strain sensor with interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992).
[CrossRef]

W. W. Morey, G. Meltz, W. H. Glenn, “Fibre optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. De Paula, E. Udd, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1169, 98–107 (1989).

Nakamura, M.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE Quantum Electron. 4, 233–253 (1977).
[CrossRef]

Poladian, L.

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E. 48, 4758–4767 (1993).
[CrossRef]

Sakuda, K.

Sipe, J. E.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase grating,” IEEE J. Lightwave Technol. LT-11, 1513–1517 (1993).
[CrossRef]

Weller-Brophy, L. A.

Yamada, M.

Yariv, A.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE Quantum Electron. 4, 233–253 (1977).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1975).

Composites Eng.

R. M. Measures, “Smart composite structures with embedded sensors,” Composites Eng. 2, 597–618 (1992).
[CrossRef]

Electron. Lett.

A. D. Kersey, T. A. Berkoff, W. W. Morey, “High-resolution fiber-grating based strain sensor with interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992).
[CrossRef]

IEEE J. Lightwave Technol.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase grating,” IEEE J. Lightwave Technol. LT-11, 1513–1517 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

A. D. Kersey, T. A. Berkoff, “Fiber-optic Bragg-grating differential-temperature sensors,” IEEE Photon. Technol. Lett. 4, 1183–1185 (1992).
[CrossRef]

S. M. Melle, A. T. Alavie, S. Karr, T. Coroy, K. Liu, R. M. Measures, “A Bragg grating-tuned fiber laser strain sensor system,” IEEE Photon. Technol. Lett. 5, 263–266 (1993).
[CrossRef]

IEEE Quantum Electron

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE Quantum Electron. 4, 233–253 (1977).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev. E.

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E. 48, 4758–4767 (1993).
[CrossRef]

Other

H. A. Haus, “Mirrors and interferometers,” in Waves and Fields in Optoelectronics, N. Holonyak, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 3, p. 55–80.

W. W. Morey, G. Meltz, W. H. Glenn, “Fibre optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. De Paula, E. Udd, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1169, 98–107 (1989).

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Figures (10)

Fig. 1
Fig. 1

T-matrix model for (a) a uniform grating and (b) a nonuniform grating.

Fig. 2
Fig. 2

Spectral response of the Bragg grating with Gaussian-distributed Δn0(z) as well as the constant pitch length Λ and constant n ¯. The four lower spectral curves, (a), (b), (c), and (d), correspond to the four Δn0(z) distributions shown in the upper figure.

Fig. 3
Fig. 3

Spectral response of the Bragg grating with Gaussian-distributed Δ n0(z) = 1.28 × 10−4 exp{−10[(zL/2)/L]2} and n ¯ (z) = 1.5 + γ × 1.28 × 10−4 exp{−10[(zL/2)/L]2} with (a) γ = 0, (b) γ = 1, (c) γ = 2, and (d) γ = 3.

Fig. 4
Fig. 4

Spectral response of the Bragg grating with a uniform Δn0(z) = 7 × 10−5 and linearly chirped pitch length. Curves, a, b, c, d, correspond to δΛ = 0, 1, 3, and 5 × 10−5 nm, respectively.

Fig. 5
Fig. 5

Spectral response of Bragg grating with Gaussian-distributed Δ n0(z) = 1.28 × 10−4 exp{−10[(zL/2)/L]2}. n ¯ (z) = 1.5 + γ × 1.28 × 10−4 exp{−10[(zL/2)/L]2}, and linearly chirped pitch length. Curves a, b, c, d, correspond to δΛ = 0, 1, 3, and 5 ×10−5 nm, respectively.

Fig. 6
Fig. 6

Specially designed cantilever beam (upper section, side view; lower section, top view). Strain gradients can be induced near the neck part of the beam on application of an end deflection. The fiber grating is embedded in the groove on the left side of the neck so that the grating is subject to an almost linear strain gradient.

Fig. 7
Fig. 7

Calculated strain profile along the grating produced by deflecting the special loading beam shown in Fig. 6 for eight different deflection conditions.

Fig. 8
Fig. 8

Measured reflection spectra (solid curves) in the strain conditions of Fig. 7 and their theoretical curve fits (dotted curves). The fitting curves were obtained with the T-matrix formulation and smoothed to match the 0.1-nm spectral resolution of the spectrum analyzer used in the experiment. Reflectivity is in percentages.

Fig. 9
Fig. 9

Assumed Gaussian distribution of the index modulation depth Δn0(z) in the grating under test.

Fig. 10
Fig. 10

Evolution of the grating spectrum with increasing temperature. Reflectivity is in percentages.

Equations (15)

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{ d A ( z ) d z = i κ B ( z ) exp [ - i 2 ( Δ β ) z ] , d B ( z ) d z = - i κ * A ( z ) exp [ i 2 ( Δ β ) z ] ,             ( 0 z l )
n ( z ) = n ¯ + Δ n ( z ) = n ¯ + Δ n 0 cos ( 2 π z / Λ ) ,
κ = π Δ n 0 λ ,
[ a ( 0 ) b ( 1 ) ] = [ S 11 S 12 S 21 S 22 ] [ a ( l ) b ( 0 ) ] ,
S 11 = S 22 = i s exp ( - i β 0 l ) - Δ β sinh ( s l ) + i s cosh ( s l ) , S 12 = κ κ * S 21 exp ( 2 i β 0 l ) = κ sinh ( s l ) - Δ β sinh ( s l ) + i s cosh ( s l ) ,
s = ( κ 2 - Δ β 2 ) 1 / 2 .
[ a ( 0 ) b ( 0 ) ] = [ T 11 T 12 T 21 T 22 ] [ a ( l ) b ( l ) ] ,
T 11 = T 22 * = Δ β sinh ( s l ) + i s cosh ( s l ) i s exp ( - i β 0 l ) , T 12 = T 21 * = κ sinh ( s l ) i s exp ( i β 0 l ) .
[ T L ] = [ T l 1 ] [ T l 2 ] [ T l m ] ,
[ a ( 0 ) b ( 0 ) ] = [ T M ] [ T M - 1 ] [ T j ] [ T 1 ] [ a ( L ) b ( L ) ] .
Δ n 0 ( z ) = Δ n 0 c exp { - a [ ( z - L / 2 ) / L ] 2 } ,
n ¯ ( z ) = n d + γ × 1.28 × 10 - 4 exp { - 10 [ ( z - L / 2 ) / L ] 2 } ,
Λ ( z ) = Λ 0 + ( δ Λ / Λ 0 ) z ,
n ¯ ( z ) = n d + 1.7 × 10 - 4 exp { - 4.41 × [ z - L / 2 ) / L ] 2 } - ½ n d 3 [ p 12 - ν ( p 11 + p 12 ) ] z ( z ) ,
Δ λ λ 0 g L ,

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