Abstract

We present a new method for fabricating structural long-period gratings (LPGs) in photonic-crystal fibers (PCFs). The method is based on periodically drilling holes into the PCF cladding along the length of the fiber by use of a focused femtosecond infrared laser. A very short LPG with only 9 periods and a grating length of < 4 mm exhibited resonance strength of over 20 dB and a polarization dependent loss of 25 dB. The high resonance strength is attributed to the strong modulated mode-field profile caused by the significant perturbation of the fiber geometry. The mechanism of LPG formation is discussed based on coupled local-mode theory.

© 2010 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photon. Technol. Lett. 18(21), 2266–2268 (2006).
    [CrossRef]
  14. S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2008

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

C. L. Zhao, L. Xiao, J. Ju, M. S. Demokan, and W. Jin, “Strain and temperature characteristics of a long-period grating written in a photonic crystal fiber and its application as a temperature-insensitive strain sensor,” J. Lightwave Technol. 26(2), 220–227 (2008).
[CrossRef]

2007

2006

H. Dobb, K. Kalli, and D. J. Webb, “Measured sensitivity of long period gratings in photonic crystal fibre,” Opt. Commun. 260, 184–191 (2006).
[CrossRef]

Y. P. Wang, D. N. Wang, W. Jin, Y. J. Rao, and G. D. Peng, “Asymmetric long period fiber gratings fabricated by use of CO2 laser to carve periodic grooves on the optical fiber,” Appl. Phys. Lett. 89(15), 151105 (2006).
[CrossRef]

Y. P. Wang, L. M. Xiao, D. N. Wang, and W. Jin, “Highly sensitive long-period fiber-grating strain sensor with low temperature sensitivity,” Opt. Lett. 31(23), 3414–3416 (2006).
[CrossRef] [PubMed]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photon. Technol. Lett. 18(21), 2266–2268 (2006).
[CrossRef]

T. Matsumura, T. Nakatani, and T. Yagi, “Deep drilling on a silicon plate with a femtosecond laser: experiment and model analysis,” Appl. Phys., A Mater. Sci. Process. 86(1), 107–114 (2006).
[CrossRef]

2005

2004

2003

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14(5), R49–R61 (2003).
[CrossRef]

C. Kerbage and B. J. Eggleton, “Tunable microfluidic optical fiber gratings,” Appl. Phys. Lett. 82(9), 1338–1340 (2003).
[CrossRef]

B. L. Bachim and T. K. Gaylord, “Polarization-dependent loss and birefringence in long-period fiber gratings,” Appl. Opt. 42(34), 6816–6823 (2003).
[CrossRef] [PubMed]

2002

2001

G. Rego, O. Okhotnikov, E. Dianov, and V. Sulimov, “High-temperature stability of long-period fiber gratings produced using an electric arc,” J. Lightwave Technol. 19(10), 1574–1579 (2001).
[CrossRef]

C. B. Schaffer, A. Brodeur, and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses,” Meas. Sci. Technol. 12(11), 1784–1794 (2001).
[CrossRef]

2000

1999

1996

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(1), 58–65 (1996).
[CrossRef]

Allsop, T.

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

Bachim, B. L.

Bay, H.-W.

Bennion, I.

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

J. S. Petrovic, H. Dobb, V. K. Mezentsev, K. Kalli, D. J. Webb, and I. Bennion, “Sensitivity of LPGs in PCFs fabricated by an electric arc to temperature, strain, and external refractive index,” J. Lightwave Technol. 25(5), 1306–1312 (2007).
[CrossRef]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photon. Technol. Lett. 18(21), 2266–2268 (2006).
[CrossRef]

Bhatia, V.

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(1), 58–65 (1996).
[CrossRef]

Birks, T. A.

Broaddus, D. H.

Brodeur, A.

C. B. Schaffer, A. Brodeur, and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses,” Meas. Sci. Technol. 12(11), 1784–1794 (2001).
[CrossRef]

Demokan, M. S.

Dianov, E.

Digonnet, M. J. F.

Ding, H.

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

S. J. Mihailov, C. W. Smelser, D. Grobnic, R. B. Walker, P. Lu, H. Ding, and J. Unruh, “Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800-nm femtosecond radiation and a phase mask,” J. Lightwave Technol. 22(1), 94–100 (2004).
[CrossRef]

Dobb, H.

J. S. Petrovic, H. Dobb, V. K. Mezentsev, K. Kalli, D. J. Webb, and I. Bennion, “Sensitivity of LPGs in PCFs fabricated by an electric arc to temperature, strain, and external refractive index,” J. Lightwave Technol. 25(5), 1306–1312 (2007).
[CrossRef]

H. Dobb, K. Kalli, and D. J. Webb, “Measured sensitivity of long period gratings in photonic crystal fibre,” Opt. Commun. 260, 184–191 (2006).
[CrossRef]

H. Dobb, K. Kalli, and D. J. Webb, “Temperature insensitive long period grating sensors in photonic crystal fibre,” Electron. Lett. 40(11), 657–658 (2004).
[CrossRef]

Dubov, M.

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photon. Technol. Lett. 18(21), 2266–2268 (2006).
[CrossRef]

Eggleton, B. J.

C. Kerbage and B. J. Eggleton, “Tunable microfluidic optical fiber gratings,” Appl. Phys. Lett. 82(9), 1338–1340 (2003).
[CrossRef]

Erdogan, T.

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(1), 58–65 (1996).
[CrossRef]

Février, S.

Gaeta, A. L.

Gaylord, T. K.

Grivas, C.

Grobnic, D.

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

S. J. Mihailov, C. W. Smelser, D. Grobnic, R. B. Walker, P. Lu, H. Ding, and J. Unruh, “Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800-nm femtosecond radiation and a phase mask,” J. Lightwave Technol. 22(1), 94–100 (2004).
[CrossRef]

Hao, J.

Hensley, C. J.

Hirao, K.

Hu, J.

Humbert, G.

James, S. W.

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14(5), R49–R61 (2003).
[CrossRef]

Jin, W.

Ju, J.

Judkins, J. B.

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(1), 58–65 (1996).
[CrossRef]

Kakarantzas, G.

Kalli, K.

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

J. S. Petrovic, H. Dobb, V. K. Mezentsev, K. Kalli, D. J. Webb, and I. Bennion, “Sensitivity of LPGs in PCFs fabricated by an electric arc to temperature, strain, and external refractive index,” J. Lightwave Technol. 25(5), 1306–1312 (2007).
[CrossRef]

H. Dobb, K. Kalli, and D. J. Webb, “Measured sensitivity of long period gratings in photonic crystal fibre,” Opt. Commun. 260, 184–191 (2006).
[CrossRef]

H. Dobb, K. Kalli, and D. J. Webb, “Temperature insensitive long period grating sensors in photonic crystal fibre,” Electron. Lett. 40(11), 657–658 (2004).
[CrossRef]

Kazansky, P. G.

Kerbage, C.

C. Kerbage and B. J. Eggleton, “Tunable microfluidic optical fiber gratings,” Appl. Phys. Lett. 82(9), 1338–1340 (2003).
[CrossRef]

Khrushchev, I.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photon. Technol. Lett. 18(21), 2266–2268 (2006).
[CrossRef]

Kino, G. S.

Komodromos, M.

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

Kondo, Y.

Lai, Y.

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

Laia, Y.

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

Lemaire, P. J.

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(1), 58–65 (1996).
[CrossRef]

Lu, C.

Lu, P.

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

S. J. Mihailov, C. W. Smelser, D. Grobnic, R. B. Walker, P. Lu, H. Ding, and J. Unruh, “Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800-nm femtosecond radiation and a phase mask,” J. Lightwave Technol. 22(1), 94–100 (2004).
[CrossRef]

Malki, A.

Martinez, A.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photon. Technol. Lett. 18(21), 2266–2268 (2006).
[CrossRef]

Matsumura, T.

T. Matsumura, T. Nakatani, and T. Yagi, “Deep drilling on a silicon plate with a femtosecond laser: experiment and model analysis,” Appl. Phys., A Mater. Sci. Process. 86(1), 107–114 (2006).
[CrossRef]

Mazur, E.

C. B. Schaffer, A. Brodeur, and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses,” Meas. Sci. Technol. 12(11), 1784–1794 (2001).
[CrossRef]

Mezentsev, V. K.

Mihailov, S. J.

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

S. J. Mihailov, C. W. Smelser, D. Grobnic, R. B. Walker, P. Lu, H. Ding, and J. Unruh, “Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800-nm femtosecond radiation and a phase mask,” J. Lightwave Technol. 22(1), 94–100 (2004).
[CrossRef]

Mitsuyu, T.

Nakatani, T.

T. Matsumura, T. Nakatani, and T. Yagi, “Deep drilling on a silicon plate with a femtosecond laser: experiment and model analysis,” Appl. Phys., A Mater. Sci. Process. 86(1), 107–114 (2006).
[CrossRef]

Nouchi, K.

Okhotnikov, O.

Pagnoux, D.

Peng, G. D.

Y. P. Wang, D. N. Wang, W. Jin, Y. J. Rao, and G. D. Peng, “Asymmetric long period fiber gratings fabricated by use of CO2 laser to carve periodic grooves on the optical fiber,” Appl. Phys. Lett. 89(15), 151105 (2006).
[CrossRef]

Petrovic, J. S.

Petrovich, M. N.

Rao, Y. J.

Y. P. Wang, D. N. Wang, W. Jin, Y. J. Rao, and G. D. Peng, “Asymmetric long period fiber gratings fabricated by use of CO2 laser to carve periodic grooves on the optical fiber,” Appl. Phys. Lett. 89(15), 151105 (2006).
[CrossRef]

Rego, G.

Richardson, D. J.

Roy, P.

Russell, P. St. J.

Savin, S.

Schaffer, C. B.

C. J. Hensley, D. H. Broaddus, C. B. Schaffer, and A. L. Gaeta, “Photonic band-gap fiber gas cell fabricated using femtosecond micromachining,” Opt. Express 15(11), 6690–6695 (2007).
[CrossRef] [PubMed]

C. B. Schaffer, A. Brodeur, and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses,” Meas. Sci. Technol. 12(11), 1784–1794 (2001).
[CrossRef]

Shaw, H. J.

Shum, P.

Sipe, J. E.

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(1), 58–65 (1996).
[CrossRef]

Smelser, C. W.

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

S. J. Mihailov, C. W. Smelser, D. Grobnic, R. B. Walker, P. Lu, H. Ding, and J. Unruh, “Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800-nm femtosecond radiation and a phase mask,” J. Lightwave Technol. 22(1), 94–100 (2004).
[CrossRef]

Smith, G.

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

Sugden, K.

T. Allsop, K. Kalli, K. Zhou, G. Smith, Y. Lai, M. Dubov, K. Sugden, D. Webb, I. Bennion, and M. Komodromos, “Characterisation of femtosecond laser inscribed long period gratings in photonic crystal fibre,” Proc. SPIE 6990, 1–13 (2008).

Sulimov, V.

Tatam, R. P.

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14(5), R49–R61 (2003).
[CrossRef]

Unruh, J.

van Brakel, A.

Vengsarkar, A. M.

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(1), 58–65 (1996).
[CrossRef]

Walker, R. B.

S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008).
[CrossRef]

S. J. Mihailov, C. W. Smelser, D. Grobnic, R. B. Walker, P. Lu, H. Ding, and J. Unruh, “Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800-nm femtosecond radiation and a phase mask,” J. Lightwave Technol. 22(1), 94–100 (2004).
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[CrossRef]

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Y. P. Wang, D. N. Wang, W. Jin, Y. J. Rao, and G. D. Peng, “Asymmetric long period fiber gratings fabricated by use of CO2 laser to carve periodic grooves on the optical fiber,” Appl. Phys. Lett. 89(15), 151105 (2006).
[CrossRef]

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[CrossRef] [PubMed]

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Webb, D. J.

T. Allsop, K. Kalli, K. Zhou, Y. Laia, G. Smith, M. Dubov, D. J. Webb, and I. Bennion, “Long period gratings written into a photonic crystal fibre by a femtosecond laser as directional bend sensors,” Opt. Commun. 281(20), 5092–5096 (2008).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Appl. Phys., A Mater. Sci. Process.

T. Matsumura, T. Nakatani, and T. Yagi, “Deep drilling on a silicon plate with a femtosecond laser: experiment and model analysis,” Appl. Phys., A Mater. Sci. Process. 86(1), 107–114 (2006).
[CrossRef]

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[CrossRef]

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H. Dobb, K. Kalli, and D. J. Webb, “Measured sensitivity of long period gratings in photonic crystal fibre,” Opt. Commun. 260, 184–191 (2006).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Experiment setup for LPG fabrication.

Fig. 2
Fig. 2

(a) Top view of a section of the LPG in PCF. (b) SEM cross-sectional image of the drilled region, where the PCF is cut at the A–A plane in (a). (c) Side view of the drilled region.

Fig. 3
Fig. 3

Evolution of transmission spectra and the measured PDL profile of a LPG in PCF.

Fig. 4
Fig. 4

Near-field intensity profiles at the resonant wavelength. (a) Measured at the output of the LPG by use of an infrared camera; (b) Calculated LP11 mode profile for the perturbed structure at location F as shown in Fig. 6(a).

Fig. 5
Fig. 5

The relationship between the LP01-LP11 resonant coupling wavelength and the grating period. The parameters used for numerical calculation are: diameter of the air hole d = 3.38 μm, hole to hole spacing D = 7.31 μm, grating period Λ= 420 μm.

Fig. 6
Fig. 6

(a) Side view of a single-hole model for the calculation of coupling coefficient and the calculated longitudinal variation of LP01-LP11 coupling coefficient around the hole. (b) Calculated evolution of transmission spectrum with increasing number of drilled-holes. The parameters used in the calculation are: diameter of the air hole d = 3.38 μm, hole to hole spacing D = 7.31 μm, grating period Λ= 420 μm. Position F is located between B and C and is represented by z = −0.75 μm plane.

Equations (2)

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d b co d z i β co ( z ) b co = C ( z ) b cl d b cl d z i β cl ( z ) b cl = C ( z ) b co
C = 1 4 A { h co × e cl z e co × h cl z } · z d A

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