Abstract

The grating-period profile and length of an arbitrary fiber Bragg grating structure can be reconstructed from the structure’s reflection response by use of a time–frequency signal representation based on the well-known Wigner–Ville distribution and spectrogram. We present a detailed description of this synthesis technique. By means of numerical simulations, the technique is tested with several fiber grating structures. In general, our results show good agreement between exact and reconstructed functions. The technique’s advantages and limitations are discussed. We propose and demonstrate the application of the proposed synthesis technique to distributed mechanical strain or temperature sensing.

© 2000 Optical Society of America

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  1. A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Norwood, Mass., 1999).
  2. R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).
  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]
  4. G. H. Song, 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]
  5. 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]
  6. R. Feced, M. N. Zervas, 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]
  7. M. A. Muriel, J. Azaña, A. Carballar, “Fiber grating synthesis by use of time–frequency representations,” Opt. Lett. 23, 1526–1528 (1998).
    [CrossRef]
  8. M. A. Muriel, J. Azaña, “Signal processing techniques applied to fiber gratings synthesis,” in Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., to be published).
  9. S. Qian, D. Chen, Joint Time–Frequency Analysis: Methods and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  10. L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  11. P. J. Loughlin, special issue on time-frequency analysis, Proc. IEEE 84, 1195–1344 (1996).
    [CrossRef]
  12. S. Huang, M. LeBlanc, M. M. Ohn, R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003–5009 (1995).
    [CrossRef] [PubMed]
  13. S. Huang, M. M. Ohn, R. M. Measures, “Phase-based Bragg intragrating distributed strain sensor,” Appl. Opt. 35, 1135–1142 (1996).
    [CrossRef] [PubMed]
  14. M. Leblanc, S. Y. Huang, M. Ohn, R. M. Measures, A. Guemes, A. Othonos, “Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis,” Opt. Lett. 21, 1405–1407 (1996).
    [CrossRef] [PubMed]
  15. M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
    [CrossRef]
  16. M. M. Ohn, S. Y. Huang, R. M. Measures, J. Chwang, “Arbitrary strain profile measurement within fibre gratings using interferometric Fourier transform technique,” Electron. Lett. 33, 1242–1243 (1997).
    [CrossRef]
  17. S. Kadambe, F. Boudreaux-Bartels, “A comparison of the existence of  ‘cross-terms’ in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
    [CrossRef]
  18. S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
    [CrossRef]
  19. H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech Signal Process. 37, 862–871 (1989).
    [CrossRef]
  20. P. Faldrin, B. Vidalie, O. Rioul, “Fourier and wavelets spectrograms seen as smoothed Wigner–Ville distributions,” in Wavelets and Applications, Proceedings of International Conference, Marseille (France), May 1989, Y. Meyer, ed. (Masson/Springer-Verlag, Berlin, 1992), pp. 93–103.
  21. C. J. Brooks, G. L. Vossler, K. A. Winick, “Phase response measurement technique for waveguide grating filters,” Appl. Phys. Lett. 66, 2168–2170 (1995).
    [CrossRef]
  22. A. Carballar, M. A. Muriel, “Phase reconstruction from reflectivity in fiber Bragg gratings,” IEEE J. Lightwave Technol. 15, 1314–1322 (1997).
    [CrossRef]
  23. L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
    [CrossRef]
  24. J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.
  25. L. R. Chen, D. J. F. Copper, P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
    [CrossRef]
  26. L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
    [CrossRef]
  27. C. D. Butter, G. B. Hocker, “Fiber optics strain gauge,” Appl. Opt. 17, 2867–2869 (1978).
    [CrossRef] [PubMed]

1999 (1)

R. Feced, M. N. Zervas, 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 (3)

M. A. Muriel, J. Azaña, A. Carballar, “Fiber grating synthesis by use of time–frequency representations,” Opt. Lett. 23, 1526–1528 (1998).
[CrossRef]

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
[CrossRef]

L. R. Chen, D. J. F. Copper, P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[CrossRef]

1997 (2)

M. M. Ohn, S. Y. Huang, R. M. Measures, J. Chwang, “Arbitrary strain profile measurement within fibre gratings using interferometric Fourier transform technique,” Electron. Lett. 33, 1242–1243 (1997).
[CrossRef]

A. Carballar, M. A. Muriel, “Phase reconstruction from reflectivity in fiber Bragg gratings,” IEEE J. Lightwave Technol. 15, 1314–1322 (1997).
[CrossRef]

1996 (5)

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]

P. J. Loughlin, special issue on time-frequency analysis, Proc. IEEE 84, 1195–1344 (1996).
[CrossRef]

S. Huang, M. M. Ohn, R. M. Measures, “Phase-based Bragg intragrating distributed strain sensor,” Appl. Opt. 35, 1135–1142 (1996).
[CrossRef] [PubMed]

M. Leblanc, S. Y. Huang, M. Ohn, R. M. Measures, A. Guemes, A. Othonos, “Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis,” Opt. Lett. 21, 1405–1407 (1996).
[CrossRef] [PubMed]

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

1995 (3)

S. Huang, M. LeBlanc, M. M. Ohn, R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003–5009 (1995).
[CrossRef] [PubMed]

L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[CrossRef]

C. J. Brooks, G. L. Vossler, K. A. Winick, “Phase response measurement technique for waveguide grating filters,” Appl. Phys. Lett. 66, 2168–2170 (1995).
[CrossRef]

1992 (2)

S. Kadambe, F. Boudreaux-Bartels, “A comparison of the existence of  ‘cross-terms’ in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[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]

1989 (1)

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech Signal Process. 37, 862–871 (1989).
[CrossRef]

1985 (1)

1978 (1)

Azaña, J.

M. A. Muriel, J. Azaña, A. Carballar, “Fiber grating synthesis by use of time–frequency representations,” Opt. Lett. 23, 1526–1528 (1998).
[CrossRef]

M. A. Muriel, J. Azaña, “Signal processing techniques applied to fiber gratings synthesis,” in Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., to be published).

Benjamin, S. D.

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
[CrossRef]

Bennion, I.

L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[CrossRef]

Boudreaux-Bartels, F.

S. Kadambe, F. Boudreaux-Bartels, “A comparison of the existence of  ‘cross-terms’ in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

Brooks, C. J.

C. J. Brooks, G. L. Vossler, K. A. Winick, “Phase response measurement technique for waveguide grating filters,” Appl. Phys. Lett. 66, 2168–2170 (1995).
[CrossRef]

Butter, C. D.

Cai, J.-X.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.

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]

Carballar, A.

M. A. Muriel, J. Azaña, A. Carballar, “Fiber grating synthesis by use of time–frequency representations,” Opt. Lett. 23, 1526–1528 (1998).
[CrossRef]

A. Carballar, M. A. Muriel, “Phase reconstruction from reflectivity in fiber Bragg gratings,” IEEE J. Lightwave Technol. 15, 1314–1322 (1997).
[CrossRef]

Chen, D.

S. Qian, D. Chen, Joint Time–Frequency Analysis: Methods and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Chen, L. R.

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
[CrossRef]

L. R. Chen, D. J. F. Copper, P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[CrossRef]

Cheung, S.

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[CrossRef]

Choi, H. I.

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech Signal Process. 37, 862–871 (1989).
[CrossRef]

Chwang, J.

M. M. Ohn, S. Y. Huang, R. M. Measures, J. Chwang, “Arbitrary strain profile measurement within fibre gratings using interferometric Fourier transform technique,” Electron. Lett. 33, 1242–1243 (1997).
[CrossRef]

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Cole, M. J.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Copper, D. J. F.

L. R. Chen, D. J. F. Copper, P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[CrossRef]

Dakin, J. P.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Faldrin, P.

P. Faldrin, B. Vidalie, O. Rioul, “Fourier and wavelets spectrograms seen as smoothed Wigner–Ville distributions,” in Wavelets and Applications, Proceedings of International Conference, Marseille (France), May 1989, Y. Meyer, ed. (Masson/Springer-Verlag, Berlin, 1992), pp. 93–103.

Feced, R.

R. Feced, M. N. Zervas, 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]

Feinberg, J.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.

Feng, K.-M.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.

Geiger, H.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Grubsky, V.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.

Guemes, A.

Hocker, G. B.

Huang, S.

Huang, S. Y.

M. M. Ohn, S. Y. Huang, R. M. Measures, J. Chwang, “Arbitrary strain profile measurement within fibre gratings using interferometric Fourier transform technique,” Electron. Lett. 33, 1242–1243 (1997).
[CrossRef]

M. Leblanc, S. Y. Huang, M. Ohn, R. M. Measures, A. Guemes, A. Othonos, “Distributed strain measurement based on a fiber Bragg grating and its reflection spectrum analysis,” Opt. Lett. 21, 1405–1407 (1996).
[CrossRef] [PubMed]

Kadambe, S.

S. Kadambe, F. Boudreaux-Bartels, “A comparison of the existence of  ‘cross-terms’ in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

Kalli, K.

A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Norwood, Mass., 1999).

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).

Leblanc, M.

Lim, J. S.

S. Cheung, J. S. Lim, “Combined multiresolution (wide-band/narrow-band) spectrogram,” IEEE Trans. Signal Process. 40, 975–977 (1992).
[CrossRef]

Loughlin, P. J.

P. J. Loughlin, special issue on time-frequency analysis, Proc. IEEE 84, 1195–1344 (1996).
[CrossRef]

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.

Molony, A.

L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, 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]

M. A. Muriel, J. Azaña, A. Carballar, “Fiber grating synthesis by use of time–frequency representations,” Opt. Lett. 23, 1526–1528 (1998).
[CrossRef]

A. Carballar, M. A. Muriel, “Phase reconstruction from reflectivity in fiber Bragg gratings,” IEEE J. Lightwave Technol. 15, 1314–1322 (1997).
[CrossRef]

M. A. Muriel, J. Azaña, “Signal processing techniques applied to fiber gratings synthesis,” in Bragg Gratings, Photosensitivity and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., to be published).

Ohn, M.

Ohn, M. M.

M. M. Ohn, S. Y. Huang, R. M. Measures, J. Chwang, “Arbitrary strain profile measurement within fibre gratings using interferometric Fourier transform technique,” Electron. Lett. 33, 1242–1243 (1997).
[CrossRef]

S. Huang, M. M. Ohn, R. M. Measures, “Phase-based Bragg intragrating distributed strain sensor,” Appl. Opt. 35, 1135–1142 (1996).
[CrossRef] [PubMed]

S. Huang, M. LeBlanc, M. M. Ohn, R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003–5009 (1995).
[CrossRef] [PubMed]

Othonos, A.

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]

Qian, S.

S. Qian, D. Chen, Joint Time–Frequency Analysis: Methods and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Rioul, O.

P. Faldrin, B. Vidalie, O. Rioul, “Fourier and wavelets spectrograms seen as smoothed Wigner–Ville distributions,” in Wavelets and Applications, Proceedings of International Conference, Marseille (France), May 1989, Y. Meyer, ed. (Masson/Springer-Verlag, Berlin, 1992), pp. 93–103.

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]

Shin, S. Y.

Sipe, J. E.

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
[CrossRef]

Smith, P. W. E.

L. R. Chen, D. J. F. Copper, P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[CrossRef]

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
[CrossRef]

Song, G. H.

Starodubov, D. S.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.

Sugden, K.

L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[CrossRef]

Vidalie, B.

P. Faldrin, B. Vidalie, O. Rioul, “Fourier and wavelets spectrograms seen as smoothed Wigner–Ville distributions,” in Wavelets and Applications, Proceedings of International Conference, Marseille (France), May 1989, Y. Meyer, ed. (Masson/Springer-Verlag, Berlin, 1992), pp. 93–103.

Volanthen, M.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Vossler, G. L.

C. J. Brooks, G. L. Vossler, K. A. Winick, “Phase response measurement technique for waveguide grating filters,” Appl. Phys. Lett. 66, 2168–2170 (1995).
[CrossRef]

Williams, W. J.

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech Signal Process. 37, 862–871 (1989).
[CrossRef]

Willner, A. E.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, J. Feinberg, “Dynamic dispersion compensation in a 10-Gbit/s optical system using a novel nonlinearly-chirped fiber Bragg grating,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 365–367.

Winick, K. A.

C. J. Brooks, G. L. Vossler, K. A. Winick, “Phase response measurement technique for waveguide grating filters,” Appl. Phys. Lett. 66, 2168–2170 (1995).
[CrossRef]

K. A. Winick, 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, 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]

Zhang, L.

L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. J. Brooks, G. L. Vossler, K. A. Winick, “Phase response measurement technique for waveguide grating filters,” Appl. Phys. Lett. 66, 2168–2170 (1995).
[CrossRef]

Electron. Lett. (3)

L. Zhang, K. Sugden, I. Bennion, A. Molony, “Wide-stopband chirped fibre moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[CrossRef]

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

M. M. Ohn, S. Y. Huang, R. M. Measures, J. Chwang, “Arbitrary strain profile measurement within fibre gratings using interferometric Fourier transform technique,” Electron. Lett. 33, 1242–1243 (1997).
[CrossRef]

IEEE J. Lightwave Technol. (1)

A. Carballar, M. A. Muriel, “Phase reconstruction from reflectivity in fiber Bragg gratings,” IEEE J. Lightwave Technol. 15, 1314–1322 (1997).
[CrossRef]

IEEE J. Quantum Electron. (4)

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, “Applications of ultrashort pulse propagation in Bragg gratings for wavelength-division-multiplexing and code-division multiple access,” IEEE J. Quantum Electron. 34, 2117–2129 (1998).
[CrossRef]

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]

R. Feced, M. N. Zervas, 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]

IEEE Photon. Technol. Lett. (1)

L. R. Chen, D. J. F. Copper, P. W. E. Smith, “Transmission filters with multiple flattened passbands based on chirped moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
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Figures (15)

Fig. 1
Fig. 1

(a) WV analysis of a multicomponent signal that contains three frequency-modulated Gaussian functions. (b) Cross-term-free SP analysis of the same signal.

Fig. 2
Fig. 2

(a) WV distribution of the signal analyzed in Fig. 1. Dashed circles, 10% contour lines of the SP, used as masks to isolate the self-terms in WV representation. (b) Optimal WVS representation, obtained by direct multiplication of the WV distribution and the SP.

Fig. 3
Fig. 3

Block diagram of the FBG period reconstruction algorithm; notation defined in text.

Fig. 4
Fig. 4

FBG #1: Field reflection coefficient (reflectivity and group delay).

Fig. 5
Fig. 5

FBG #1: (a) WV analysis, (b) SP analysis, and (c) optimal WVS representation of the grating reflection response.

Fig. 6
Fig. 6

FBG #1: Reconstructed grating period along the perturbation length.

Fig. 7
Fig. 7

FBG #2: Field reflection coefficient (reflectivity and group delay).

Fig. 8
Fig. 8

FBG #2: WV analysis of the grating reflection response.

Fig. 9
Fig. 9

FBG #2: Reconstructed grating period (solid curve) and exact grating period (dashed curve) along the perturbation length.

Fig. 10
Fig. 10

FBG #3: Field reflection coefficient (reflectivity and group delay).

Fig. 11
Fig. 11

FBG #3: WVS analysis of the grating reflection response.

Fig. 12
Fig. 12

FBG #3: Reconstructed grating period (solid curves) and exact grating period (dashed curves) along the perturbation length.

Fig. 13
Fig. 13

Resonant FBG structure with strong coupling: WVS analysis of the grating reflection response.

Fig. 14
Fig. 14

FBG under a nonmonotonic strain distribution: WVS analysis of the reflection response.

Fig. 15
Fig. 15

Recovered strain distribution (solid curve) and exact strain distribution (dashed curve) along the grating length.

Equations (13)

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WV(t, f)=-+h(t+τ/2)h*(t-τ/2)×exp(-j2πfτ)dτ,
WV(t, f)=-+H*(f+ν/2)H(f-ν/2)exp(-j2πνt)dν,
SP(t, f)=|STFT(t, f)|2=-+h(τ)g(τ-t)exp(-j2πfτ)dτ2,
SP(t, f)=|SFFT(t, f)|2=-+H(ν)G(f-ν)exp(j2πνt)dν2,
n(z)=nav(z)+ΔnT(z)cos[φ(z)],0zL,
φ(z)=0z2πΛ(z)dz,
f¯(t)=-+fWVS(t, f)df-+WVS(t, f)df.
t=2c0znav(z)dz,
t=2navzc,
fB(z)=f¯t=2navzc.
Λ(z)=c2navfB(z)=c2navf¯t=2navzc.
fB(z)=c2nav(z)Λ(z)fB01+z(z)(1-nav02ξ).
z(z)[fB0/fB(z)]-11-nav02ξ.

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