Abstract

To analyze the various LPFGs with thermal changes, we present how makes the kernel function to translate the information of thermal change into the coupling coefficient and detuning factor changed by temperature. We propose the extended fundamental matrix model with the proposed kernel function. To verify the validity of the proposed model experimentally, we have manufactured the LPFG structures with the thermal changes using the divided coil heater. We have observed that the transmission spectra calculated using the proposed model are close to the corresponding measured spectra in the wavelength band of interest.

© 2004 Optical Society of America

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References

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  1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, �??Long-Period Fiber Gratings as Band-Rejection Filters,�?? J. Lightwave Technol. 14, 58-64 (1996).
    [CrossRef]
  2. J. Bae, J. Chun, and S. B. Lee, �??Two methods for synthesizing the long period fiber gratings with the inverted Erbium gain spectrum,�?? Jpn. J. Appl. Phys. Part 2, 38, L819-L822 (1999).
    [CrossRef]
  3. Y. Liu, J. A. R.Williams, L. Zhang, and I. Bennion, �??Phase shifted and cascaded long-period fiber gratings,�?? Opt. Commun. 164, 27-31 (1999).
    [CrossRef]
  4. M. Harumoto, M. Shigehara, and H. Suganuma, �??Gain-flattening filter using lonp-period fiber gratings,�?? J. Lightwave Technol. 21, 1027-1033 (2002).
    [CrossRef]
  5. X. Gu, �??Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,�?? Opt. Lett. 23, 509-590 (1998).
    [CrossRef]
  6. A. Othonos and K. Kalli, Fiber Bragg Gratings - Fundamentals and Applications in Telecommunications and Sensing, (Artech House, Boston, 1999).
  7. Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, �??Performance enhancement of long period fiber gratings for strain and temperature sensing,�?? IEICE Trans. Electron. E83-C, 1-6 (2000).
  8. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, �??Fiber Grating Sensors,�?? J. Lightwave Technol. 15, 1442-1463 (1997).
    [CrossRef]
  9. J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M Jeong, �??Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters,�?? IEEE Photon. Technol. Lett. 15, 407-409 (2003).
    [CrossRef]
  10. S. Matsumoto, T. Ohira, M. Takabayashi, and K. Yoshiara, �??Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmissions,�?? IEEE Photon. Technol. 13, 827-829 (2001).
    [CrossRef]
  11. T. Erdogan, �??Fiber Grating Spectra,�?? J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  12. T. Erdogan, �??Cladding-mode resonances in short- and long-period fiber grating filters,�?? J. Opt. Soc. Am. A 14, 1760-1773 (1997).
    [CrossRef]
  13. M. Yamada and K. Sakuda, �??Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,�?? Appl. Opt. 26, 3474-3478 (1987).
    [CrossRef] [PubMed]
  14. J. Bae, J. Chun, and S. B. Lee, �??Multiport Lattice Filter Model for Long-Period Fiber Gratings,�?? Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
    [CrossRef]
  15. X. Shu, T. Allsop, B. Gwandu, and L. Zhang, �??High-temperature Sensitivity of Long-Period Grating in B-Ge Codoped Fiber,�?? IEEE Photon. Technol. Lett. 13, 818-820 (2001).
    [CrossRef]
  16. I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, �??Generalized simulated annealing for function optimzation,�?? Technometrics 28, 209-217 (1986).
    [CrossRef]
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2th ed. (Cambridge, New York, 1992).
  18. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, (Springer-Verlag, New York, 1980).
  19. X. Shu, L. Zhang, and I, Bennion �??Sensitivity characteristics of long-period Fiber gratings,�?? J. Lightwave Technol. 20, 255-266 (2002).
    [CrossRef]
  20. H. Kim, J. Bae, J.W. Lee, J. Chun, and S. B. Lee, �??Analysis of Concatenated Long Period Fiber Gratings Having Phase-Shifted and Cascaded Effects,�?? Jpn. J. Appl. Phys. Part 1 42, 5098-5101 (2003).
    [CrossRef]

Appl. Opt. (1)

M. Yamada and K. Sakuda, �??Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,�?? Appl. Opt. 26, 3474-3478 (1987).
[CrossRef] [PubMed]

IEEE Photon. Technol. (1)

S. Matsumoto, T. Ohira, M. Takabayashi, and K. Yoshiara, �??Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmissions,�?? IEEE Photon. Technol. 13, 827-829 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, �??High-temperature Sensitivity of Long-Period Grating in B-Ge Codoped Fiber,�?? IEEE Photon. Technol. Lett. 13, 818-820 (2001).
[CrossRef]

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M Jeong, �??Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters,�?? IEEE Photon. Technol. Lett. 15, 407-409 (2003).
[CrossRef]

IEICE Trans. Electron. (1)

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, �??Performance enhancement of long period fiber gratings for strain and temperature sensing,�?? IEICE Trans. Electron. E83-C, 1-6 (2000).

J. Lightwave Technol. (5)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, �??Fiber Grating Sensors,�?? J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

T. Erdogan, �??Fiber Grating Spectra,�?? J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, �??Long-Period Fiber Gratings as Band-Rejection Filters,�?? J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

M. Harumoto, M. Shigehara, and H. Suganuma, �??Gain-flattening filter using lonp-period fiber gratings,�?? J. Lightwave Technol. 21, 1027-1033 (2002).
[CrossRef]

X. Shu, L. Zhang, and I, Bennion �??Sensitivity characteristics of long-period Fiber gratings,�?? J. Lightwave Technol. 20, 255-266 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

T. Erdogan, �??Cladding-mode resonances in short- and long-period fiber grating filters,�?? J. Opt. Soc. Am. A 14, 1760-1773 (1997).
[CrossRef]

Jpn. J. Appl. Phys. Part 1 (2)

H. Kim, J. Bae, J.W. Lee, J. Chun, and S. B. Lee, �??Analysis of Concatenated Long Period Fiber Gratings Having Phase-Shifted and Cascaded Effects,�?? Jpn. J. Appl. Phys. Part 1 42, 5098-5101 (2003).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, �??Multiport Lattice Filter Model for Long-Period Fiber Gratings,�?? Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
[CrossRef]

Jpn. J. Appl. Phys. part 2 (1)

J. Bae, J. Chun, and S. B. Lee, �??Two methods for synthesizing the long period fiber gratings with the inverted Erbium gain spectrum,�?? Jpn. J. Appl. Phys. Part 2, 38, L819-L822 (1999).
[CrossRef]

Opt. Commun. (1)

Y. Liu, J. A. R.Williams, L. Zhang, and I. Bennion, �??Phase shifted and cascaded long-period fiber gratings,�?? Opt. Commun. 164, 27-31 (1999).
[CrossRef]

Opt. Lett. (1)

X. Gu, �??Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,�?? Opt. Lett. 23, 509-590 (1998).
[CrossRef]

Technometrics (1)

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, �??Generalized simulated annealing for function optimzation,�?? Technometrics 28, 209-217 (1986).
[CrossRef]

Other (3)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2th ed. (Cambridge, New York, 1992).

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, (Springer-Verlag, New York, 1980).

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

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Figures (17)

Fig. 1.
Fig. 1.

Block diagram for the coupled mode equations with the kernel function.

Fig. 2.
Fig. 2.

(a) Transmission spectra with thermal changes. (b) X and δ for 24.9 °C.

Fig. 3.
Fig. 3.

(a) a with thermal changes (To detailed display, we have subtracted 0.9999 from a.). (b) Attained coefficients for a. (c) b with thermal changes. (d) Attained coefficients for b. (e) c with thermal changes. (f) Attained coefficients for c.

Fig. 4.
Fig. 4.

Block diagram of the extended fundamental matrix model for the LPFGs with the thermal changes.

Fig. 5.
Fig. 5.

(a) δ 24.9 for Λ=415 µm. (b) Transmission spectra for Nx . (c) Transmission spectrum for Nx =100. (d) Transmission spectrum for Nx =300.

Fig. 6.
Fig. 6.

(a) δ 24.9 for Λ=427.85 µm. (b) Transmission spectra for Nx . (c) Transmission spectrum for Nx =100. (d) Transmission spectrum for Nx =300.

Fig. 7.
Fig. 7.

(a) Transmission spectra for thermal changes (Λ=415 µm). (b) Transmission spectra for thermal changes (Λ=427.85 µm).

Fig. 8.
Fig. 8.

(a) Our experimental setup. (b) Equivalent block diagram of (a). (c) Prototype of a tunable LPFGs controlled by temeprature. (d) Ni-Cr coil and control circuit board.

Fig. 9.
Fig. 9.

(a) Transmission spectrum for 24.9 °C. (b) X and δ for 24.9 °C.

Fig. 10.
Fig. 10.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 11.
Fig. 11.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 12.
Fig. 12.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 13.
Fig. 13.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 14.
Fig. 14.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 15.
Fig. 15.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 16.
Fig. 16.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Fig. 17.
Fig. 17.

(a) Transmission spectrum with thermal changes. (b) Thermal changes for (a).

Equations (20)

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n ( z ) = n co + Δ n ( 1 + cos ( 2 π Λ z ) ) , 0 z L ,
d A ( z ) dz = j δ A ( z ) + j κ B ( z ) ,
d B ( z ) dz = j δ B ( z ) + j κ * A ( z ) ,
A ( z ) = cos ( κ 2 + δ 2 z ) + j δ κ 2 + δ 2 sin ( κ 2 + δ 2 z ) ,
B ( z ) = j κ κ 2 + δ 2 sin ( κ 2 + δ 2 z ) .
n c o = 1.45248 , n c l = 1.44532 , n a i r = 1 , r c o = 2.815 μ m , r c l = 62.51 μ m ,
κ = π Δ n λ C 1 λ X κ 24.9 ,
δ = 2 π λ n eff co 2 π λ n eff cl π Λ δ 24.9 ,
κ T = π λ ( Δ n + d Δ n T ) ( C + d C T )
= π λ [ ( 1 + d C T C + d Δ n T Δ n ) Δ n C + d Δ n T d C T ]
1 λ ( a X + b ) ,
δ T = 2 π λ ( n eff co + d n eff , T co ) 2 π λ ( n eff cl + d n eff , T cl ) π Λ
δ + c ,
a a 0 T m + a 2 T m 1 + + a m 1 T 1 + a m ,
b b 0 T m + b 2 T m 1 + + b m 1 T 1 + b m ,
c c 0 T m + c 2 T m 1 + + c m 1 T 1 + c m .
[ A i B i ] = F i [ A i 1 B i 1 ] .
F i = [ cos ( ϒ i L i ) + j δ T i ϒ i sin ( ϒ i L i ) j κ T i ϒ i sin ( ϒ i L i ) j κ T i ϒ i sin ( ϒ i L i ) cos ( ϒ i L i ) j δ T i ϒ i sin ( ϒ i L i ) ] ,
[ A M B M ] = F [ A 0 B 0 ] ,
F F M F M 1 F 1 [ F 11 F 12 F 21 F 22 ] .

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