Abstract

A simple and efficient transfer-matrix method based on a discrete coupling model is presented to analyze uniform and nonuniform fiber grating couplers between copropagating core and cladding modes. Uniform and piecewise-uniform long-period gratings were fabricated by a point-by-point arc discharge technique. Their measured transmission spectra were compared with the transmission spectra calculated by the presented method.

© 2012 Optical Society of America

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References

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  1. G. Humbert and A. Malki, “Electric-arc-induced gratings in non-hydrogenated fibres: fabrication and high-temperature characterizations,” J. Opt. A: Pure Appl. Opt. 4, 194–198 (2002).
    [CrossRef]
  2. K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).
  3. G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
    [CrossRef]
  4. K. Morishita and Y. Miyake, “Fabrication and resonance wavelengths of long-period gratings written in a pure-silica photonic crystal fiber by the glass structure change,” J. Lightwave Technol. 22, 625–630 (2004).
    [CrossRef]
  5. K. Morishita and A. Kaino, “Adjusting resonance wavelengths of long-period fiber gratings by the glass-structure change,” Appl. Opt. 44, 5018–5023 (2005).
    [CrossRef]
  6. F. Abrishamian and K. Morishita, “Broadening adjustable range on post-fabrication resonance wavelength trimming of long-period fiber gratings and the mechanisms of resonance wavelength shifts,” IEICE Trans. Electron. E94-C, 641–647 (2011).
    [CrossRef]
  7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  8. R. Feced and M. N. Zervas, “Efficient inverse scattering algorithm for the design of grating-assisted codirectional mode couplers,” J. Opt. Soc. Am. A 17, 1573–1582 (2000).
    [CrossRef]
  9. L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
    [CrossRef]
  10. J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave Technol. 21, 254–263 (2003).
    [CrossRef]
  11. J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
    [CrossRef]
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  13. H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998).
    [CrossRef]
  14. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209(1965).
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  15. R. Feced and M. N. Zervas, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” J. Lightwave Technol. 18, 90–101 (2000).
    [CrossRef]
  16. Y. Yamamoto and K. Morishita, “Resonance wavelength changes of arc-induced long-period fiber gratings caused by the discharge condition,” in Proceedings of 2011 Kansai-section Joint Convention of Institutes of Electrical Engineering (IEICE, 2011), paper 30P1-16 (in Japanese).

2011 (1)

F. Abrishamian and K. Morishita, “Broadening adjustable range on post-fabrication resonance wavelength trimming of long-period fiber gratings and the mechanisms of resonance wavelength shifts,” IEICE Trans. Electron. E94-C, 641–647 (2011).
[CrossRef]

2005 (1)

2004 (1)

2003 (4)

K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave Technol. 21, 254–263 (2003).
[CrossRef]

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

2002 (1)

G. Humbert and A. Malki, “Electric-arc-induced gratings in non-hydrogenated fibres: fabrication and high-temperature characterizations,” J. Opt. A: Pure Appl. Opt. 4, 194–198 (2002).
[CrossRef]

2001 (1)

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

2000 (2)

1998 (1)

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998).
[CrossRef]

1997 (1)

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

1987 (1)

1965 (1)

Abrishamian, F.

F. Abrishamian and K. Morishita, “Broadening adjustable range on post-fabrication resonance wavelength trimming of long-period fiber gratings and the mechanisms of resonance wavelength shifts,” IEICE Trans. Electron. E94-C, 641–647 (2011).
[CrossRef]

Brenne, J. K.

Cheng, X. P.

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

Chiang, K. S.

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998).
[CrossRef]

Erdogan, T.

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

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

Feced, R.

Février, S.

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

Fujihara, T.

K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).

Humbert, G.

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

G. Humbert and A. Malki, “Electric-arc-induced gratings in non-hydrogenated fibres: fabrication and high-temperature characterizations,” J. Opt. A: Pure Appl. Opt. 4, 194–198 (2002).
[CrossRef]

Kaino, A.

Ke, H.

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998).
[CrossRef]

Li, S. Y.

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

Malitson, I. H.

Malki, A.

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

G. Humbert and A. Malki, “Electric-arc-induced gratings in non-hydrogenated fibres: fabrication and high-temperature characterizations,” J. Opt. A: Pure Appl. Opt. 4, 194–198 (2002).
[CrossRef]

Miyake, Y.

K. Morishita and Y. Miyake, “Fabrication and resonance wavelengths of long-period gratings written in a pure-silica photonic crystal fiber by the glass structure change,” J. Lightwave Technol. 22, 625–630 (2004).
[CrossRef]

K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).

Morishita, K.

F. Abrishamian and K. Morishita, “Broadening adjustable range on post-fabrication resonance wavelength trimming of long-period fiber gratings and the mechanisms of resonance wavelength shifts,” IEICE Trans. Electron. E94-C, 641–647 (2011).
[CrossRef]

K. Morishita and A. Kaino, “Adjusting resonance wavelengths of long-period fiber gratings by the glass-structure change,” Appl. Opt. 44, 5018–5023 (2005).
[CrossRef]

K. Morishita and Y. Miyake, “Fabrication and resonance wavelengths of long-period gratings written in a pure-silica photonic crystal fiber by the glass structure change,” J. Lightwave Technol. 22, 625–630 (2004).
[CrossRef]

K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).

Y. Yamamoto and K. Morishita, “Resonance wavelength changes of arc-induced long-period fiber gratings caused by the discharge condition,” in Proceedings of 2011 Kansai-section Joint Convention of Institutes of Electrical Engineering (IEICE, 2011), paper 30P1-16 (in Japanese).

Ng, J. H.

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

Ngo, N. Q.

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

Pagnoux, D.

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

Peng, J. H.

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998).
[CrossRef]

Roy, P.

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

Sakuda, K.

Shum, P.

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

Skaar, J.

Wang, L.

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

Yamada, M.

Yamamoto, Y.

Y. Yamamoto and K. Morishita, “Resonance wavelength changes of arc-induced long-period fiber gratings caused by the discharge condition,” in Proceedings of 2011 Kansai-section Joint Convention of Institutes of Electrical Engineering (IEICE, 2011), paper 30P1-16 (in Japanese).

Yuan, S. F.

K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).

Zervas, M. N.

Zhang, J.

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (2)

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

G. Humbert, A. Malki, S. Février, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fibres,” Electron. Lett. 39, 349–350 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10, 1596–1598 (1998).
[CrossRef]

J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. 15, 1558–1560 (2003).
[CrossRef]

IEICE Trans. Electron. (2)

K. Morishita, S. F. Yuan, Y. Miyake, and T. Fujihara, “Refractive index variations and long-period fiber gratings made by the glass structure change,” IEICE Trans. Electron. E86-C, 1749–1758 (2003).

F. Abrishamian and K. Morishita, “Broadening adjustable range on post-fabrication resonance wavelength trimming of long-period fiber gratings and the mechanisms of resonance wavelength shifts,” IEICE Trans. Electron. E94-C, 641–647 (2011).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. A: Pure Appl. Opt. (1)

G. Humbert and A. Malki, “Electric-arc-induced gratings in non-hydrogenated fibres: fabrication and high-temperature characterizations,” J. Opt. A: Pure Appl. Opt. 4, 194–198 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Other (1)

Y. Yamamoto and K. Morishita, “Resonance wavelength changes of arc-induced long-period fiber gratings caused by the discharge condition,” in Proceedings of 2011 Kansai-section Joint Convention of Institutes of Electrical Engineering (IEICE, 2011), paper 30P1-16 (in Japanese).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of the discretized-coupling model of a codirectional grating coupler. The coupling between the copropagating core and cladding modes occurs in the narrow region within each section, and then the modes propagate the section without coupling until the next coupling region.

Fig. 2.
Fig. 2.

Calculated relationship between resonance wavelengths and grating periods. The squares, circles, and triangles are the measured resonance wavelengths of uniform LPFGs with the grating periods of 490 µm, 500 µm, and 510 µm, respectively.

Fig. 3.
Fig. 3.

The measured and calculated transmission spectra of the uniform LPFG with increasing numbers of discharge points, 20, 32, 38, and 42.

Fig. 4.
Fig. 4.

The measured and calculated transmission spectra of the piecewise-uniform LPFGs while increasing the number of discharge points: (a) 39, 61, and 70; and (b) 34, 41, and 48. The insets indicate discharge-point-interval versus discharge-number, and the grating period was changed (a) from 496 µm to 504 µm after the discharge number of 39 and (b) from 490 µm to 510 µm after the discharge number of 34.

Equations (13)

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dA(z)dz=jδA(z)jκejϕB(z)dB(z)dz=jδB(z)jκejϕA(z),
[AkBk]=TpTc[Ak1Bk1],
Tc=[cosκΔjexp(jϕ)sinκΔjexp(jϕ)sinκΔcosκΔ],
Tp=[exp(jδΔ)00exp(jδΔ)],
Tck=[cosCkjexp(jϕk)sinCkjexp(jϕk)sinCkcosCk],
Tpk=[exp(jβcoΔk)00exp(jβclΔk)]=exp(jβavΔk)[exp(jδΔk)00exp(jδΔk)],
[ANBN]=TcNΠk=1N1(TpkTck)[A0B0]=exp(jβavk=1N1Δk)[cosCNjexp(jϕN)sinCNjexp(jϕN)sinCNcosCN]×Πk=1N1([exp(jδΔk)00exp(jδΔk)][cosCkjexp(jϕk)sinCkjexp(jϕk)sinCkcosCk])[A0B0].
[ANBN]=exp(jβavk=1N1Δk)[T11exp(jϕ)T12exp(jϕ)T21T22][A0B0]
[T11T12T21T22]=[cosCNjsinCNjsinCNcosCN]×Πk=1N1([exp(jδΔk)00exp(jδΔk)][cosCkjsinCkjsinCkcosCk]).
|AN|2=|T11|2|A0|2,|BN|2=|T21|2|A0|2.
[ANBN]=exp(jβavk=1N1Δk)(1)N1[cos(k=1NCk)jexp(jϕ)sin(k=1NCk)jexp(jϕ)sin(k=1NCk)cos(k=1NCk)][A0B0].
|AN|2=cos2(k=1NCk)|A0|2,|BN|2=sin2(k=1NCk)|A0|2.
|AN|2=cos2(NC)|A0|2,|BN|2=sin2(NC)|A0|2.

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