Abstract

A strong resonance and extremely short length long-period grating (LPG) has been fabricated in a large-mode-area photonic crystal fiber (PCF) by use of a CO2 laser heat source. We believe that such a long-period grating in pure silica PCF is the first example of a point-by-point technique. The fabrication method is simple and repeatable. The resulting LPG has been developed with eight periodic collapses within a 2.8-mm-long period of the fiber, which gives the strong resonance of core-cladding mode coupling. The lowest mode of LP01 is at a 1529.2-nm wavelength with a full width at half-maximum of ~0.7 nm and a resonance strength of -31.5 dB. The principal advantages of this LPG are that (1) it is temperature insensitive and stable, (2) the device is compact when it is packaged, and (3) it provides practical, low-cost all-fiber filters and PCF-based devices for optical fiber communications and sensing systems.

© 2003 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [CrossRef] [PubMed]
  4. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
    [CrossRef]
  5. T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
    [CrossRef]
  6. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air-silica microstructure optical fiber,” Opt. Lett. 25, 796–798 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2003 (4)

2002 (1)

2001 (4)

C. Y. Lin and L. A. Wang, “A wavelength- and loss- tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
[CrossRef]

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

R. Holzwarth, M. Zimmermann, T. Udem, T. W. Hansch, P. Russbuldt, K. Gabel, R. Poprawe, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light frequency comb generation with a diode-pumped Cr:LiSAF laser,” Opt. Lett. 26, 1376–1378 (2001).
[CrossRef]

B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698
[CrossRef] [PubMed]

2000 (3)

1999 (4)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4, 457–466 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-11-457
[CrossRef] [PubMed]

B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spalter, and T. A. Strasser, “Grating resonances in air-silica microstructured optical fibers,” Opt. Lett. 24, 1460–1462 (1999).
[CrossRef]

1998 (1)

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

1997 (1)

1996 (3)

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Atkin, D. M.

Bagget, J. C.

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Belardi, W.

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Bhatia, V.

V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4, 457–466 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-11-457
[CrossRef] [PubMed]

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, 58–65 (1996).
[CrossRef]

Birks, T. A.

G. Kakarantzas, A. Ortigosa-Mlanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
[CrossRef] [PubMed]

G. Kakarantzas, T. A. Birks, and P. St. J. Russell, “Structural long-period gratings in photonic crystal fibers,” Opt. Lett. 27, 1013–1015 (2002).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[CrossRef] [PubMed]

Broderick, N. G. R.

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Buckley, E.

Burdge, G. L.

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windelar, and G. L. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Tech. 18, 1084–1100 (2000).
[CrossRef]

Canning, J.

Couny, F.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

de Sandro, J.-P.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

Eggleton, B. J.

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, 58–65 (1996).
[CrossRef]

Farr, L.

Fevrier, S.

G. Humbert, A. Malki, S. Fevrier, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fiber,” Electron. Lett. 4, 349–350 (2003).
[CrossRef]

Folkenberg, J. R.

Furusawa, K.

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Gabel, K.

Groothoff, N.

Hale, A.

Hansch, T. W.

Holzwarth, R.

Humbert, G.

G. Humbert, A. Malki, S. Fevrier, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fiber,” Electron. Lett. 4, 349–350 (2003).
[CrossRef]

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, 58–65 (1996).
[CrossRef]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

Kakarantzas, G.

Kerbage, C.

Knight, J. C.

R. Holzwarth, M. Zimmermann, T. Udem, T. W. Hansch, P. Russbuldt, K. Gabel, R. Poprawe, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light frequency comb generation with a diode-pumped Cr:LiSAF laser,” Opt. Lett. 26, 1376–1378 (2001).
[CrossRef]

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[CrossRef] [PubMed]

Lemaire, P. J.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

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, 58–65 (1996).
[CrossRef]

Lin, C. Y.

C. Y. Lin and L. A. Wang, “A wavelength- and loss- tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
[CrossRef]

Lyttikainen, K.

Malki, A.

G. Humbert, A. Malki, S. Fevrier, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fiber,” Electron. Lett. 4, 349–350 (2003).
[CrossRef]

Mangan, B. J.

G. Kakarantzas, A. Ortigosa-Mlanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Mogilevstev, D.

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

Monro, T. M.

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Nielsen, M. D.

Ortigosa-Mlanch, A.

Pagnoux, D.

G. Humbert, A. Malki, S. Fevrier, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fiber,” Electron. Lett. 4, 349–350 (2003).
[CrossRef]

Pedrazzani, J. R.

Poprawe, R.

Ranka, J. K.

Richardson, D. J.

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Roy, P.

G. Humbert, A. Malki, S. Fevrier, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fiber,” Electron. Lett. 4, 349–350 (2003).
[CrossRef]

Russbuldt, P.

Russell, P. St. J.

G. Kakarantzas, A. Ortigosa-Mlanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
[CrossRef] [PubMed]

G. Kakarantzas, T. A. Birks, and P. St. J. Russell, “Structural long-period gratings in photonic crystal fibers,” Opt. Lett. 27, 1013–1015 (2002).
[CrossRef]

R. Holzwarth, M. Zimmermann, T. Udem, T. W. Hansch, P. Russbuldt, K. Gabel, R. Poprawe, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light frequency comb generation with a diode-pumped Cr:LiSAF laser,” Opt. Lett. 26, 1376–1378 (2001).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[CrossRef] [PubMed]

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, 58–65 (1996).
[CrossRef]

Spalter, S.

Stentz, A. J.

Strasser, T. A.

Udem, T.

Vengsarkar, A. M.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

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, 58–65 (1996).
[CrossRef]

Vienne, G.

Wadsworth, W. J.

Wang, L. A.

C. Y. Lin and L. A. Wang, “A wavelength- and loss- tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
[CrossRef]

Westbrook, P. S.

White, C. A.

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windelar, and G. L. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Tech. 18, 1084–1100 (2000).
[CrossRef]

Windelar, R. S.

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windelar, and G. L. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Tech. 18, 1084–1100 (2000).
[CrossRef]

Windeler, R. S.

Zagari, J.

Zimmermann, M.

Electron. Lett. (2)

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. Lett. 13, 1347–1348 (1998).
[CrossRef]

G. Humbert, A. Malki, S. Fevrier, P. Roy, and D. Pagnoux, “Electric arc-induced long-period gratings in Ge-free air-silica microstructure fiber,” Electron. Lett. 4, 349–350 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. St. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

C. Y. Lin and L. A. Wang, “A wavelength- and loss- tunable band-rejection filter based on corrugated long-period fiber grating,” IEEE Photon. Technol. Lett. 13, 332–334 (2001).
[CrossRef]

J. Lightwave Tech. (1)

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windelar, and G. L. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Tech. 18, 1084–1100 (2000).
[CrossRef]

J. Lightwave Technol. (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, 58–65 (1996).
[CrossRef]

Meas. Sci. Technol. (1)

T. M. Monro, W. Belardi, K. Furusawa, J. C. Bagget, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibers,” Meas. Sci. Technol. 12, 854–858 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (10)

R. Holzwarth, M. Zimmermann, T. Udem, T. W. Hansch, P. Russbuldt, K. Gabel, R. Poprawe, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light frequency comb generation with a diode-pumped Cr:LiSAF laser,” Opt. Lett. 26, 1376–1378 (2001).
[CrossRef]

G. Kakarantzas, T. A. Birks, and P. St. J. Russell, “Structural long-period gratings in photonic crystal fibers,” Opt. Lett. 27, 1013–1015 (2002).
[CrossRef]

G. Kakarantzas, A. Ortigosa-Mlanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
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N. Groothoff, J. Canning, E. Buckley, K. Lyttikainen, and J. Zagari, “Bragg gratings in air-silica structured fibers,” Opt. Lett. 28, 233–235 (2003).
[CrossRef] [PubMed]

M. D. Nielsen, G. Vienne, and J. R. Folkenberg, “Investigation of microdeformation-induced attenuation spectra in a photonic crystal fiber,” Opt. Lett. 28, 236–238 (2003).
[CrossRef] [PubMed]

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[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spalter, and T. A. Strasser, “Grating resonances in air-silica microstructured optical fibers,” Opt. Lett. 24, 1460–1462 (1999).
[CrossRef]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
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[CrossRef] [PubMed]

Science (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1.
Fig. 1.

Scanning electron micrograph (SEM) of a cross-section of the large-mode-area PCF used in the fabrication of the LPG.

Fig. 2.
Fig. 2.

Schematic diagram of the experimental setup: CO2, laser; PCF, photonic crystal fiber; CCD, camera; OM, optical micrograph; SLD, super-luminescent diode; OSA, optical spectrum analyzer.

Fig. 3.
Fig. 3.

Optical micrograph of a section of LPG after the PCF is treated by the heat source of a CO2 laser, Λ is 350 µm of the spacing of one period.

Fig. 4.
Fig. 4.

Transmission spectra of the LPG in large-mode-area PCF with 6, 7, and 8 periods.

Equations (3)

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λ i = ( n core n cladding i ) Λ
δn ( z ) = Δ n [ 1 2 + 2 π m = 1 1 m sin ( 2 ) cos ( 2 πm Λ z ) ]
( n silica n air ) f silica n air 2 + f air n silica 2 n eff f silica n silica 2 + f air n air 2

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