Abstract

We propose to use a genetic algorithm to determine the physical parameters of Bragg gratings from their reflection spectra for both design purposes and fiber sensor applications. A real-coded genetic algorithm is used for inversion purposes, along with an F-matrix formalism for synthesis of uniform, chirped, and apodized gratings. An example of bandpass filter design is also studied. The method is easily applicable and shows promising results.

© 2001 Optical Society of America

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References

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  1. B. Ortega, L. Dong, and L. Reekie, “All-fiber optical add–drop multiplexer based on a selective fused coupler and a single fiber Bragg grating,” Appl. Opt. 37, 7712–7717 (1998).
    [CrossRef]
  2. S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
    [CrossRef]
  3. L. Y. Lo, “Using in-fiber Bragg grating sensors for measuring axial strain and temperature simultaneously on surfaces of structures,” Opt. Eng. 37, 2272–2276 (1998).
    [CrossRef]
  4. J. A. R. Williams and I. Bennion, “Applications of fiber gratings in microwave photonics,” Photonics Research Group, Aston University, http://benedick.aston.ac.uk/Photonics/publications/MR97/ (1997).
  5. K. A. Winick and J. E. Roman, “Design of corrugated waveguide filters by Fourier transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
    [CrossRef]
  6. G. H. Song and S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  7. E. Peral, J. Capmany, and 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]
  8. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  9. J. Skaar and K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
    [CrossRef]
  10. S. Huang, M. LeBlanc, M. N. Ohn, and R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003–5009 (1995).
    [CrossRef] [PubMed]
  11. J. H. Holland, Adaptation in Natural and Artificial Systems (U. Michigan Press, Ann Arbor, Mich., 1975).
  12. R. Boudreau and N. Turkkan, “Solving the forward kinematics of parallel manipulators with a genetic algorithm,” J. Rob. Syst. 13, 111–125 (1996).
    [CrossRef]
  13. A. Minato and N. Sugimoto, “Design of a four-element, hollow-cube corner retroreflector for satellites by use of a genetic algorithm,” Appl. Opt. 37, 438–442 (1998).
    [CrossRef]
  14. G. Cormier and R. Boudreau, “Genetic algorithm for ellipsometric data inversion of absorbing layers,” J. Opt. Soc. Am. A 17, 129–134 (2000).
    [CrossRef]
  15. A. H. Wright, “Genetic algorithms for real parameter optimization,” in Foundations of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufman, San Mateo, Calif., 1991), pp. 205–218.
  16. S. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  17. L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).
  18. Z. Michalewicz, Genetic Algorithms (Springer-Verlag, New York, 1992).
  19. T. Kuo and S.-H. Hwang, “Using disruptive selection to maintain diversity in genetic algorithms,” Appl. Intel. 7, 257–267 (1997).
    [CrossRef]

2000 (1)

1999 (1)

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998 (5)

1997 (1)

T. Kuo and S.-H. Hwang, “Using disruptive selection to maintain diversity in genetic algorithms,” Appl. Intel. 7, 257–267 (1997).
[CrossRef]

1996 (2)

E. Peral, J. Capmany, and 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]

R. Boudreau and N. Turkkan, “Solving the forward kinematics of parallel manipulators with a genetic algorithm,” J. Rob. Syst. 13, 111–125 (1996).
[CrossRef]

1995 (1)

1990 (1)

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

1985 (1)

Albert, J.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Béliveau, A.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Bilodeau, F.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Boudreau, R.

G. Cormier and R. Boudreau, “Genetic algorithm for ellipsometric data inversion of absorbing layers,” J. Opt. Soc. Am. A 17, 129–134 (2000).
[CrossRef]

R. Boudreau and N. Turkkan, “Solving the forward kinematics of parallel manipulators with a genetic algorithm,” J. Rob. Syst. 13, 111–125 (1996).
[CrossRef]

Capmany, J.

E. Peral, J. Capmany, and 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]

Cormier, G.

Dong, L.

Drouin, G.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Hill, K. O.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Huang, S.

Hwang, S.-H.

T. Kuo and S.-H. Hwang, “Using disruptive selection to maintain diversity in genetic algorithms,” Appl. Intel. 7, 257–267 (1997).
[CrossRef]

Johnson, D. C.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Kuo, T.

T. Kuo and S.-H. Hwang, “Using disruptive selection to maintain diversity in genetic algorithms,” Appl. Intel. 7, 257–267 (1997).
[CrossRef]

LeBlanc, M.

Lo, L. Y.

L. Y. Lo, “Using in-fiber Bragg grating sensors for measuring axial strain and temperature simultaneously on surfaces of structures,” Opt. Eng. 37, 2272–2276 (1998).
[CrossRef]

Marti, J.

E. Peral, J. Capmany, and 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.

Minato, A.

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Ohn, M. N.

Ortega, B.

Peral, E.

E. Peral, J. Capmany, and 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]

Reekie, L.

Risvik, K. M.

Roman, J. E.

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

Shin, S. Y.

Skaar, J.

Song, G. H.

Sugimoto, N.

Thériault, S.

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Turkkan, N.

R. Boudreau and N. Turkkan, “Solving the forward kinematics of parallel manipulators with a genetic algorithm,” J. Rob. Syst. 13, 111–125 (1996).
[CrossRef]

Winick, K. A.

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

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Appl. Intel. (1)

T. Kuo and S.-H. Hwang, “Using disruptive selection to maintain diversity in genetic algorithms,” Appl. Intel. 7, 257–267 (1997).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (3)

K. A. Winick and 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, and 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]

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (2)

J. Rob. Syst. (1)

R. Boudreau and N. Turkkan, “Solving the forward kinematics of parallel manipulators with a genetic algorithm,” J. Rob. Syst. 13, 111–125 (1996).
[CrossRef]

Opt. Eng. (1)

L. Y. Lo, “Using in-fiber Bragg grating sensors for measuring axial strain and temperature simultaneously on surfaces of structures,” Opt. Eng. 37, 2272–2276 (1998).
[CrossRef]

Opt. Rev. (1)

S. Thériault, K. O. Hill, F. Bilodeau, D. C. Johnson, J. Albert, G. Drouin, and A. Béliveau, “High-g accelerometer based on in-fiber Bragg grating,” Opt. Rev. 4, 145–147 (1998).
[CrossRef]

Other (6)

J. A. R. Williams and I. Bennion, “Applications of fiber gratings in microwave photonics,” Photonics Research Group, Aston University, http://benedick.aston.ac.uk/Photonics/publications/MR97/ (1997).

J. H. Holland, Adaptation in Natural and Artificial Systems (U. Michigan Press, Ann Arbor, Mich., 1975).

A. H. Wright, “Genetic algorithms for real parameter optimization,” in Foundations of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufman, San Mateo, Calif., 1991), pp. 205–218.

S. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).

Z. Michalewicz, Genetic Algorithms (Springer-Verlag, New York, 1992).

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Figures (8)

Fig. 1
Fig. 1

Bragg grating model.

Fig. 2
Fig. 2

Fitted and original spectra for a uniform grating.

Fig. 3
Fig. 3

Original and computed spectra for a chirped grating (ξ=0.02%) calculated with 32 sections.

Fig. 4
Fig. 4

Convergence of the genetic algorithm for a chirped grating (0.02%) for three different trials.

Fig. 5
Fig. 5

Real and computed spectra for an apodized grating calculated with eight sections.

Fig. 6
Fig. 6

Real and computed spectra for an apodized grating calculated with 100 sections.

Fig. 7
Fig. 7

Convergence of the genetic algorithm for an apodized grating calculated with 100 sections.

Fig. 8
Fig. 8

Desired and calculated spectra for the design of a bandpass filter. Parameters obtained are L=32 mm, Λ=534.01, and Δn=3.8×10-4.

Tables (4)

Tables Icon

Table 1 Optimized Parameters of the Genetic Algorithm

Tables Icon

Table 2 Average Parameters of 10 Trials for the Chirped Grating and 32 Sections

Tables Icon

Table 3 Results for an Apodized Grating Calculated and Eight Sections

Tables Icon

Table 4 Results for an Apodized Grating Calculated and 100 Sections

Equations (21)

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F=λ(Rm-Rc)2,
PRj=RWj(1/η)Σ RW,
P1, P2=O1=0.5P1+0.5P2O2=1.5P1-0.5P2 O3=-0.5P1+1.5P2,
Xi=Xi+δ [Max(Xi)-Xi](heads)Xi-δ [Xi-Min(Xi)](tails),
δ(y)=yr(1-t/T)B,
Ea(z,t)=A(z)exp[i(ωt+βz)],
Eb(z,t)=B(z)exp[i(ωt-βz)],
dA(z)dz=iκB(z)exp[-i2(Δβ)z],
dB(z)dz=-iκ¯A(z)exp[i2(Δβ)z],
κ=i(1-cos mπ)2mλ cos θ 2(n22-n12)n22+n12,
a(0)b(L)=S11S12S21S22a(L)b(0),
S11=S22=is exp(-iβ0l)-Δβ sinh(sl)+is cosh(sl),
S12=kk¯S21 exp(2iβ0l)=k sinh(sl)-Δβ sinh(sl)+is cosh(sl),
s=[(|κ|)2-(Δβ)2]1/2.
a(0)b(0)=I11I12I21I22a(l)b(l),
I11=I¯22=Δβ sinh(sl)+is cosh(sl)is exp(-iβ0l),
I12=I¯21=k sinh(sl)is exp(iβ0l).
Rj=I11I222.
[IL]=Il1Il2Ilm,
apo=12 cosπ(k-1-0.5Ns)Ns2+cosπ(k-0.5Ns)Ns2,
Rλ=11550.5 nm<λ<1550.7 nm0otherwise.

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