Abstract

A simple, accurate, and fast method to synthesize the physical parameters of a fiber Bragg grating numerically from its reflectivity is proposed and demonstrated. Our program uses the transfer matrix method and is based on a Nelder–Mead simplex optimization algorithm. It can be applied to both uniform and nonuniform (apodized and chirped) fiber Bragg gratings. The method is then used to synthesize a uniform Bragg grating from its reflectivity taken at different temperatures. It gives a good estimate of the thermal expansion coefficient and the thermo-optic coefficient of the fiber.

© 2005 Optical Society of America

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References

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  1. R. Kashyap, Fiber Bragg Gratings (Academic, London, 1999).
  2. K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
    [CrossRef]
  3. E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
    [CrossRef]
  4. J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [CrossRef]
  5. F. Casagrande, P. Crespi, A. M. Grassi, A. Lulli, R. P. Kenny, M. P. 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]
  6. P. Dong, J. Azana, A. G. Kirk, “Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm,” Opt. Commun. 228, 303–308 (2003).
    [CrossRef]
  7. M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  8. S. Y. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
    [CrossRef]
  9. P. L. Swart, A. P. Kotze, B. M. Lacquet, “Effects of the nature of the starting population on the properties of Rugate filters designed with the genetic algorithm,” J. Lightwave Technol. 18, 853–859 (2000).
    [CrossRef]
  10. J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
    [CrossRef]
  11. S. Yin, “Distributed fiber optic sensors,” in Fiber Optic Sensors, F. T. S. Yu, S. Yin, eds. (Marcel Dekker, New York, 2002), pp. 183–233.
  12. Y.-J. Kim, U.-C. Paek, B. H. Lee, “Measurement of refractive-index variation with temperature by use of long-period fiber gratings,” Opt. Lett. 27, 1297–1299 (2002).
    [CrossRef]
  13. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  14. B. H. Lee, J. Nishii, “Dependence of fringe spacing on the grating separation in a long-period fiber grating pair,” Appl. Opt. 38, 3450–3459 (1999).
    [CrossRef]

2003 (1)

P. Dong, J. Azana, A. G. Kirk, “Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm,” Opt. Commun. 228, 303–308 (2003).
[CrossRef]

2002 (2)

2001 (1)

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (2)

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

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

1996 (1)

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

1990 (1)

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

1987 (1)

Azana, J.

P. Dong, J. Azana, A. G. Kirk, “Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm,” Opt. Commun. 228, 303–308 (2003).
[CrossRef]

Capmany, J.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Casagrande, F.

Crespi, P.

Dong, P.

P. Dong, J. Azana, A. G. Kirk, “Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm,” Opt. Commun. 228, 303–308 (2003).
[CrossRef]

Erdogan, T.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Grassi, A. M.

Huang, S. Y.

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

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, London, 1999).

Kenny, R. P.

Kim, Y.-J.

Kirk, A. G.

P. Dong, J. Azana, A. G. Kirk, “Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm,” Opt. Commun. 228, 303–308 (2003).
[CrossRef]

Kotze, A. P.

Lacquet, B. M.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

LeBlanc, M.

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

Lee, B. H.

Lulli, A.

Marti, J.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Measures, R. M.

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

Nishii, J.

Ohn, M. M.

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

Paek, U.-C.

Peral, E.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Roman, J. E.

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

Sakuda, K.

Skaar, J.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Swart, P. L.

Wang, L.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Whelan, M. P.

Winick, K. A.

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Yamada, M.

Yin, S.

S. Yin, “Distributed fiber optic sensors,” in Fiber Optic Sensors, F. T. S. Yu, S. Yin, eds. (Marcel Dekker, New York, 2002), pp. 183–233.

Appl. Opt. (3)

IEEE J. Quantum Electron. (3)

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Commun. (1)

P. Dong, J. Azana, A. G. Kirk, “Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm,” Opt. Commun. 228, 303–308 (2003).
[CrossRef]

Opt. Lett. (1)

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Smart Mater. Struct. (1)

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

Other (2)

S. Yin, “Distributed fiber optic sensors,” in Fiber Optic Sensors, F. T. S. Yu, S. Yin, eds. (Marcel Dekker, New York, 2002), pp. 183–233.

R. Kashyap, Fiber Bragg Gratings (Academic, London, 1999).

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Figures (6)

Fig. 1
Fig. 1

Flow chart showing the steps of the optimization procedure.

Fig. 2
Fig. 2

Synthesis of a uniform Bragg grating from the experimental reflected spectrum: target (squares) and reconstructed (circles) spectra.

Fig. 3
Fig. 3

Synthesis of a chirped Bragg grating from the experimental reflected spectrum: target (squares) and reconstructed (circles) spectra.

Fig. 4
Fig. 4

Synthesis of a numerically simulated Gaussian apodized Bragg grating from the reflected spectrum: target (squares) and reconstructed (circles) spectra.

Fig. 5
Fig. 5

Experimental setup to obtain the reflected spectrum of a Bragg grating at different temperatures.

Fig. 6
Fig. 6

Synthesis of a uniform grating taken at two temperatures: target (squares and diamonds) and reconstructed (circles and triangles) spectra at 25 °C and 80 °C, respectively.

Tables (1)

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Table 1 Evolution of Parameter Values of a Uniform Bragg Grating in Response to a Temperature Change

Equations (7)

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ρ = - κ sinh ( κ 2 - σ ^ 2 L ) σ ^ sinh ( κ 2 - σ ^ 2 L ) + i κ 2 - σ ^ 2 cosh ( κ 2 - σ ^ 2 L ) ,
σ ^ = 2 π n eff ( 1 λ - 1 λ Bragg ) + 2 π λ δ n ,
κ = π λ v δ n .
Λ ( z ) = Λ 0 + C z ,
δ n ( z ) = δ n exp [ - ( z - z c σ ) 2 ] ,
Δ λ Bragg = λ Bragg ( 1 Λ Λ T + 1 n eff n eff T ) Δ T ,
Δ λ Bragg = λ Bragg ( α Λ + α n ) Δ T ,

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