Abstract

Reflection spectra and index structures created by the growth of fiber Bragg gratings (FBGs) are modeled using a modified piecewise-uniform approach that can accommodate realistic index growth behavior. Because grating formation generally involves nonlinear index growth, models that assume sinusoidal modulation shapes do not accurately predict the evolution of the grating spectra during the writing exposure. The authors first present a generally applicable treatment of arbitrarily shaped index modulations such that their reflection spectra can be accurately treated with an established modeling technique. This approach examines the actual photoinduced index modulation shape at each subregion of the grating and identifies the ac and dc coupling coefficients (from coupled-mode theory) of an equivalent sinusoidal modulation at the fundamental Bragg resonance. These derived coupling coefficients are then used to compute the grating spectrum via the fundamental matrix (F-matrix) method. Given an accurate description of index at each point along the grating, the modified F-matrix method can efficiently model grating spectra that result from complex exposure schemes including scanned exposures, various apodization profiles, chirp, and postexposures with fringeless light. Additionally, this paper presents a method for determining the detailed index profiles formed by arbitrary exposures. To obtain realistic index modulation profiles, a new index growth model consisting of a three-dimensional (3-D) surface of induced index (versus exposure time and intensity) and a rule for linking complex sequences of index growth under differing intensities is introduced. Using the index growth surface, the compound growth rule, and the modified F-matrix technique, the spectra of weak FBGs similar to those found in distributed fiber sensor systems are numerically synthesized.

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  1. G. Meltz, W. W. Morey and W. H. Glenn, "Formation of Bragg gratings in optical fibers by a transverse holographic method", Opt. Lett., vol. 14, no. 15, pp. 823-825, Aug. 1989.
  2. D. Z. Anderson, V. Mizrahi, T. Erdogan and A. E. White, "Production of in-fibre gratings using a diffractive optical element", Electron. Lett., vol. 29, no. 6, pp. 566-568, Mar. 1993.
  3. R. Kashyap, P. F. Mckee and D. Armes, "UV written reflection grating structures in photosensitive optical fibres using phase-shifted phase masks", Electron. Lett., vol. 30, no. 23, pp. 1977-1978, Nov. 1994.
  4. Y. Wang, J. Grant, A. Sharma and G. Myers, "Modified Talbot interferometer for fabrication of fiber-optic grating filter over a wide range of Bragg wavelength and bandwidth using a single phase mask", J. Lightw. Technol., vol. 19, no. 10, pp. 1569-1573, Oct. 2001.
  5. A. Yariv, "Coupled-mode theory for guided-wave optics", IEEE J. Quantum Electron., vol. QE-9, no. 9, pp. 919-933, Sep. 1973.
  6. M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach", Appl. Opt., vol. 26, no. 16, pp. 3474-3478, Aug. 1987.
  7. T. Erdogan, "Fiber grating spectra", J. Lightw. Technol., vol. 15, no. 8, pp. 1277-1294, Aug. 1997.
  8. L. A. Weller-Brophy and D. G. Hall, "Analysis of waveguide gratings: Application of Rouard's method", J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 2, no. 6, pp. 863-871, Jun. 1985.
  9. T. A. Strasser, "Photosensitivity in phosphorus fibers", in Tech. Dig. OFC, 1996, Paper TuO1,. pp. 81-82.
  10. B. Leconte, W. Xie, M. Douay, P. Bernage, P. Niay, J. F. Bayon, E. Delevaque and H. Poignant, "Analysis of color-center-related contribution to Bragg grating formation in Ge:SiO2 fiber based on a local Kramers-Kronig transformation of excess loss spectra", Appl. Opt., vol. 36, no. 24, pp. 5923-5930, Aug. 1997.
  11. J. Echevarria, A. Quintela, C. Jauregui, A. Cobo and J. M. Lopez-Higuera, "Efficient temperature and strain discrimination with a single type I fiber Bragg grating transducer", in Tech. Dig. LEOS, vol. 2, 2000,Paper WH2,. pp. 458-459.
  12. A. Carballar and M. A. Muriel, "Growth modeling of fiber gratings: A numerical investigation", Fiber Integr. Opt., vol. 21, no. 6, pp. 451-463, Nov. 2002.
  13. R. M. Atkins, V. Mizrahi and T. Erdogan, "248 nm induced vacuum UV spectral changes in optical fibre preform cores: Support for a colour centre model of photosensitivity", Electron. Lett., vol. 29, no. 4, pp. 385-387, Feb. 1993.
  14. D. Z. Anderson, V. Mizrahi, T. Erdogan and A. E. White, "Production of in-fibre gratings using a diffractive optical element", Electron. Lett., vol. 29, no. 6, pp. 566-568, Mar. 1993.
  15. H. Patrick and S. L. Gilbert, "Growth of Bragg gratings produced by continuous-wave ultraviolet light in optical fiber", Opt. Lett., vol. 18, no. 18, pp. 1484-1486, Sep. 1993.
  16. G. M. Williams, M. A. Putnam, T. E. Tsai, C. G Askins and E. J. Friebele, "Growth dynamics of fiber Bragg gratings written with a KrF excimer laser", in Tech. Dig. BGPP, 1995, Paper SuA5,. pp. 82-85.
  17. P. L. Swart, M. G. Shlyagin, A. A. Chtcherbakov and V. V. Spirin, "Photosensitivity measurement in optical fibre with Bragg grating interferometers", Electron. Lett., vol. 38, no. 24, pp. 1508-1510, Nov. 2002.
  18. G. A. Miller, C. G. Askins, P. Skeath, C. C. Wang and E. J. Friebele, "Fabricating fiber Bragg gratings with tailored spectral properties for strain sensor arrays using a post-exposure rescan technique", in Tech. Dig. OFS, 2002, Postdeadline Paper PD1,. pp. 1-4.
  19. P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, F. Cochet and B. Leuenberger, "Tension increase correlated to refractive-index change in fibers containing UV-written Bragg gratings", Opt. Lett., vol. 20, no. 11, pp. 1346-1348, Jun. 1995.
  20. H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé and F. Cochet, "Compaction-and photoelastic-induced index changes in fiber Bragg gratings", Appl. Phys. Lett., vol. 68, no. 22, pp. 3069-3071, May 1996.
  21. B. Poumellec, P. Niay, M. Douay and J. F. Bayon, "The UV-induced refractive index grating in Ge:SiO2 preforms: Additional CW experiments and the macroscopic origin of the change in index", J. Phys. D, Appl. Phys., vol. 29, no. 7, pp. 1842-18556, Jul. 1996.
  22. M. Douay, W. X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, B. Poumellec, L. Dong, J. F. Bayon, H. Poignant and E. Delevaque, "Densification involved in the UV-based photosensitivity of silica glasses and optical fibers", J. Lightw. Technol., vol. 15, no. 8, pp. 1329-1342, Aug. 1996.
  23. C. G. Askins, "Periodic UV-induced modulations in doped-silica optical fibers: Formation and properties of the fiber Bragg grating," in Defects in SiO2 and Related Dielectrics: Science and Technology, G. Pacchioni, L. Skuja, and D. L. Griscom, Eds. Boston, MA: Kluwer, 2000.
  24. J. A. Besley, L. Reekie, C. Weeks, T. Wang and C. Murphy, "Grating writing model for materials with nonlinear photosensitive response", J. Lightw. Technol., vol. 21, no. 10, pp. 2421-2428, Oct. 2003.

Other (24)

G. Meltz, W. W. Morey and W. H. Glenn, "Formation of Bragg gratings in optical fibers by a transverse holographic method", Opt. Lett., vol. 14, no. 15, pp. 823-825, Aug. 1989.

D. Z. Anderson, V. Mizrahi, T. Erdogan and A. E. White, "Production of in-fibre gratings using a diffractive optical element", Electron. Lett., vol. 29, no. 6, pp. 566-568, Mar. 1993.

R. Kashyap, P. F. Mckee and D. Armes, "UV written reflection grating structures in photosensitive optical fibres using phase-shifted phase masks", Electron. Lett., vol. 30, no. 23, pp. 1977-1978, Nov. 1994.

Y. Wang, J. Grant, A. Sharma and G. Myers, "Modified Talbot interferometer for fabrication of fiber-optic grating filter over a wide range of Bragg wavelength and bandwidth using a single phase mask", J. Lightw. Technol., vol. 19, no. 10, pp. 1569-1573, Oct. 2001.

A. Yariv, "Coupled-mode theory for guided-wave optics", IEEE J. Quantum Electron., vol. QE-9, no. 9, pp. 919-933, Sep. 1973.

M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach", Appl. Opt., vol. 26, no. 16, pp. 3474-3478, Aug. 1987.

T. Erdogan, "Fiber grating spectra", J. Lightw. Technol., vol. 15, no. 8, pp. 1277-1294, Aug. 1997.

L. A. Weller-Brophy and D. G. Hall, "Analysis of waveguide gratings: Application of Rouard's method", J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 2, no. 6, pp. 863-871, Jun. 1985.

T. A. Strasser, "Photosensitivity in phosphorus fibers", in Tech. Dig. OFC, 1996, Paper TuO1,. pp. 81-82.

B. Leconte, W. Xie, M. Douay, P. Bernage, P. Niay, J. F. Bayon, E. Delevaque and H. Poignant, "Analysis of color-center-related contribution to Bragg grating formation in Ge:SiO2 fiber based on a local Kramers-Kronig transformation of excess loss spectra", Appl. Opt., vol. 36, no. 24, pp. 5923-5930, Aug. 1997.

J. Echevarria, A. Quintela, C. Jauregui, A. Cobo and J. M. Lopez-Higuera, "Efficient temperature and strain discrimination with a single type I fiber Bragg grating transducer", in Tech. Dig. LEOS, vol. 2, 2000,Paper WH2,. pp. 458-459.

A. Carballar and M. A. Muriel, "Growth modeling of fiber gratings: A numerical investigation", Fiber Integr. Opt., vol. 21, no. 6, pp. 451-463, Nov. 2002.

R. M. Atkins, V. Mizrahi and T. Erdogan, "248 nm induced vacuum UV spectral changes in optical fibre preform cores: Support for a colour centre model of photosensitivity", Electron. Lett., vol. 29, no. 4, pp. 385-387, Feb. 1993.

D. Z. Anderson, V. Mizrahi, T. Erdogan and A. E. White, "Production of in-fibre gratings using a diffractive optical element", Electron. Lett., vol. 29, no. 6, pp. 566-568, Mar. 1993.

H. Patrick and S. L. Gilbert, "Growth of Bragg gratings produced by continuous-wave ultraviolet light in optical fiber", Opt. Lett., vol. 18, no. 18, pp. 1484-1486, Sep. 1993.

G. M. Williams, M. A. Putnam, T. E. Tsai, C. G Askins and E. J. Friebele, "Growth dynamics of fiber Bragg gratings written with a KrF excimer laser", in Tech. Dig. BGPP, 1995, Paper SuA5,. pp. 82-85.

P. L. Swart, M. G. Shlyagin, A. A. Chtcherbakov and V. V. Spirin, "Photosensitivity measurement in optical fibre with Bragg grating interferometers", Electron. Lett., vol. 38, no. 24, pp. 1508-1510, Nov. 2002.

G. A. Miller, C. G. Askins, P. Skeath, C. C. Wang and E. J. Friebele, "Fabricating fiber Bragg gratings with tailored spectral properties for strain sensor arrays using a post-exposure rescan technique", in Tech. Dig. OFS, 2002, Postdeadline Paper PD1,. pp. 1-4.

P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, F. Cochet and B. Leuenberger, "Tension increase correlated to refractive-index change in fibers containing UV-written Bragg gratings", Opt. Lett., vol. 20, no. 11, pp. 1346-1348, Jun. 1995.

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé and F. Cochet, "Compaction-and photoelastic-induced index changes in fiber Bragg gratings", Appl. Phys. Lett., vol. 68, no. 22, pp. 3069-3071, May 1996.

B. Poumellec, P. Niay, M. Douay and J. F. Bayon, "The UV-induced refractive index grating in Ge:SiO2 preforms: Additional CW experiments and the macroscopic origin of the change in index", J. Phys. D, Appl. Phys., vol. 29, no. 7, pp. 1842-18556, Jul. 1996.

M. Douay, W. X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, B. Poumellec, L. Dong, J. F. Bayon, H. Poignant and E. Delevaque, "Densification involved in the UV-based photosensitivity of silica glasses and optical fibers", J. Lightw. Technol., vol. 15, no. 8, pp. 1329-1342, Aug. 1996.

C. G. Askins, "Periodic UV-induced modulations in doped-silica optical fibers: Formation and properties of the fiber Bragg grating," in Defects in SiO2 and Related Dielectrics: Science and Technology, G. Pacchioni, L. Skuja, and D. L. Griscom, Eds. Boston, MA: Kluwer, 2000.

J. A. Besley, L. Reekie, C. Weeks, T. Wang and C. Murphy, "Grating writing model for materials with nonlinear photosensitive response", J. Lightw. Technol., vol. 21, no. 10, pp. 2421-2428, Oct. 2003.

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