Abstract

We study the modal properties of feasible hollow-core photonic bandgap fibers (HC-PBGFs) with cores formed by omitting either 7 or 19 central unit-cells. Firstly, we analyze fibers with thin core surrounds and demonstrate that even for large cores the proposed structures are optimum for broad-band transmission. We compare these optimized structures with fibers which incorporate antiresonant core surrounds which are known to have low-loss. Trade-offs between loss and useful bandwidth are presented. Finally, we study the effects that small modifications to the core surround have on the fiber’s group velocity dispersion, showing the possibility of engineering the dispersion in hollow-core photonic bandgap fibers.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507.
    [CrossRef] [PubMed]

2007

2006

2005

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236-244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236.
[CrossRef] [PubMed]

C. J. S. de Matos, and J. R. Taylor, "Chirped pulse Raman amplification with compression in air-core photonic bandgap fiber," Opt. Express 13,2828-2834 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2828.
[CrossRef] [PubMed]

J. D. Shephard, W. N. MacPherson, R. R. J. Maier, J. D. C. Jones, M. Mohebbi, A. K. George, P. J. Roberts, and J. C. Knight, "Single-mode mid-IR guidance in a hollow-core photonic crystal fiber," Opt. Express 13, 7139-7144 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-7139.
[CrossRef] [PubMed]

P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround," Opt. Express 13,8277-8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277.
[CrossRef] [PubMed]

F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, "The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers," Opt. Express 13, 9115-9124 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-9115.
[CrossRef] [PubMed]

2004

2003

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. Martijn de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1243.
[CrossRef] [PubMed]

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," Opt. Express 11, 1310-1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310
[CrossRef] [PubMed]

J. Loegsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, "Material effects in airguiding photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2046-2051 (2003).
[CrossRef]

C. J. S. de Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, "All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber," Opt. Express 11, 2832-2837 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2832.
[CrossRef] [PubMed]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, "All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber," Opt. Express 11, 3332-3337 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3332.
[CrossRef] [PubMed]

2002

1999

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 band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Abeeluck, A. K.

Ahmad, F. R.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Allan, D. C.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12,1485-1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485.
[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 band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Amezcua-Correa, R.

Benabid, F.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, "Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber," Science 298, 399-402 (2002).
[CrossRef] [PubMed]

Bird, D. M.

Birks, T. A.

Bjarklev, A.

Borrelli, N. F.

Broderick, N. G.

Broderick, N. G. R.

Broeng, 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 band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

de Matos, C. J. S.

Digonnet, M. J. F.

Dunn, S. C.

Eggleton, B. J.

Fan, S.

Farr, L.

Gaeta, A. L.

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507.
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Gallagher, M. T.

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507.
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

George, A. K.

Guobin, R.

Hansen, K. P.

Hansen, T. P.

Headley, C.

Hensley, C. J.

Jones, J. D. C.

Kim, H. K.

Kino, G. S.

Knight, J. C.

Koch, K. W.

Koshiba, M.

Limpert, J.

Litchinitser, N. M.

Loegsgaard, J.

MacPherson, W. N.

Maier, R. R. J.

Mangan, B. J.

Martijn de Sterke, C.

Martijnde Sterke, C.

McPhedran, R. C.

Mohebbi, M.

Monro, T. M.

Mortensen, N. A

Mortensen, N. A.

Muller, D.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Nielsen, M. D.

Nolte, S.

Ouzounov, D. G.

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507.
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Petrovich, M. N.

Poletti, F.

Richardson, D. J.

Riishede, J.

Roberts, P. J.

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14,7329-7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329.
[CrossRef] [PubMed]

P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround," Opt. Express 13,8277-8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277.
[CrossRef] [PubMed]

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236-244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236.
[CrossRef] [PubMed]

J. D. Shephard, W. N. MacPherson, R. R. J. Maier, J. D. C. Jones, M. Mohebbi, A. K. George, P. J. Roberts, and J. C. Knight, "Single-mode mid-IR guidance in a hollow-core photonic crystal fiber," Opt. Express 13, 7139-7144 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-7139.
[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 band gap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Russell, P. St. J.

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14,7329-7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329.
[CrossRef] [PubMed]

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, "Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber," Science 298, 399-402 (2002).
[CrossRef] [PubMed]

Russell, P. St.J.

Sabert, H.

Saitoh, K.

Schreiber, T.

Shephard, J. D.

Shin, J.

Shuisheng, J.

Shuqin, L.

Silcox, J.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Smith, C. M.

Taylor, J. R.

Thomas, M. G.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Tunnermann, A.

Usner, B.

Venkataraman, N.

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507.
[CrossRef] [PubMed]

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003).
[CrossRef] [PubMed]

Wadsworth, W. J.

West, J. A.

White, P.

White, T. P.

Williams, D. P.

Willliams, D. P.

Zellmer, H.

Zhi, W.

IEEE J. Quantum Electron.

H. K. Kim, J. Shin, S. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Electron. 40, 551-556 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," Opt. Express 11, 1310-1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310
[CrossRef] [PubMed]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, "All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber," Opt. Express 11, 3332-3337 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3332.
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C. J. S. de Matos, and J. R. Taylor, "Chirped pulse Raman amplification with compression in air-core photonic bandgap fiber," Opt. Express 13,2828-2834 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2828.
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C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507.
[CrossRef] [PubMed]

M. J. F. Digonnet, H. K. Kim, J. Shin, S. Fan, and G. S. Kino, "Simple geometric criterion to predict the existence of surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1864-1872 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864.
[CrossRef] [PubMed]

F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, "The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers," Opt. Express 13, 9115-9124 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-9115.
[CrossRef] [PubMed]

C. J. S. de Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, "All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber," Opt. Express 11, 2832-2837 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2832.
[CrossRef] [PubMed]

J. D. Shephard, W. N. MacPherson, R. R. J. Maier, J. D. C. Jones, M. Mohebbi, A. K. George, P. J. Roberts, and J. C. Knight, "Single-mode mid-IR guidance in a hollow-core photonic crystal fiber," Opt. Express 13, 7139-7144 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-7139.
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N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. Martijn de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1243.
[CrossRef] [PubMed]

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236-244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236.
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P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround," Opt. Express 13,8277-8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277.
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P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14,7329-7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329.
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J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12,1485-1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485.
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K. Saitoh, N. A. Mortensen, and M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes," Opt. Express 12,394-400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394.
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R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti1, D. J. Richardson, "Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers," Opt. Express 14, 7974-7985 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-17-7974.
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Opt. Lett.

Science

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, "Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber," Science 298, 399-402 (2002).
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P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
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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 band gap guidance of light in air," Science 285, 1537-1539 (1999).
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P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St. J. Russell, B. J. Mangan, H. Sabert, D. P. Willliams, and L. Farr, "Achieving low loss and low nonlinearity in hollow-core photonic crystal fibers," in Proc. CLEO2005 (Baltimore, 2005), paper CWA7.

S. Fevrier, P. Viale, M. Lelek, F. Louradour, J. L. Auguste, P. Roy, J. M. Blondy, "Singlemode low-index liquid core holey fibre," in Proc. ECOC2005 (Glasgow, 2005), paper Tu1.4.3.

R. Amezcua-Correa, M. N. Petrovich, N. G. Broderick, D. J. Richardson, T. Delmonte, M. A. Watson, and E. J. O’Driscoll, "Broadband infrared transmission in a hollow-core photonic bandgap fibre free of surface modes," in Proc. ECOC2006 (Cannes, 2006), paper We4.4.4.

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Figures (7)

Fig. 1.
Fig. 1.

Cross section of the analyzed HC-PBGFs: (a) 7-cell core, (b) 19-cell core. (c) Structural parameters: d/Λ=0.98, dc /Λ=0.44, dp /Λ=0.22, tr /(Λ-d)=1, and Λ=4.7µm.

Fig. 2.
Fig. 2.

Normalized interface field intensity of the fundamental air-guided mode vs. wavelength for different values of the normalized ring thickness T. For the (a) 7-cell core fiber and (b) 19-cell core fiber. Note that the color scale is the same for both maps

Fig. 3.
Fig. 3.

Calculated useful bandwidth vs. normalized ring thickness, (a) for the 7-cell core, and (b) 19-cell core fibers.

Fig. 4.
Fig. 4.

Maximum of the power fraction in the core (solid) and minimum of the normalized field intensity F (dashed) vs. normalized ring thickness, (a) for the 7-cell core, and (b) 19-cell core fibers.

Fig. 5.
Fig. 5.

(a) Group velocity dispersion, and (b) normalized interface field intensity of the fundamental mode for a 7-cell core fiber with T=0.45, 0.5, 0.55, 0.6, and 0.65. The black circles in (b) indicate the zero-GVD wavelength for each design. (c) Calculated mode indices vs. wavelength for T=0.45 (red) and 0.65 (black). Solid lines correspond to the fundamental air-guided mode, while dashed lines correspond to surface modes.

Fig. 6.
Fig. 6.

(a) Schematic cross section of fiber with core surround of modified refractive index. (b) Stack of capillaries that can be used to fabricate the proposed fiber. Red represents glass of different refractive index, black is for silica and white air.

Fig. 7.
Fig. 7.

(a) Group velocity dispersion, and (b) normalized interface field intensity of the fundamental mode for a 7-cell core fiber with T=0.5, for different increments ring’s refractive index. The black circles in (b) indicate the zero-GVD wavelength for each design.

Equations (2)

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t r = ( 2 j + 1 ) λ 4 n s 2 1 ,
F = ( ε 0 μ 0 ) 1 / 2 holeperimeters d l E 2 cross sec tion d A E × H * · z ̂ ,

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