Abstract

We calculate the minimum leakage loss and maximum relative bandgap as a function of the air-filling factor in a photonic bandgap fiber for a refractive index between 1.4 and 2.8. We analyze the mode properties for two maxima of the relative bandgap when we vary the air-filling factor and the refractive index. The maximum relative bandgap accurately predicts the optimal operating air-filling factor corresponding to lowest leakage loss.

© 2007 Optical Society of America

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  1. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science,  282, 1476-1478 (1998).
    [CrossRef] [PubMed]
  2. J. Broeng, S. E. Barkou, T. Søndergaard, A. Bjarklev, "Analysis of air-guiding photonic bandgap fibers," Opt. Lett. 25, 96-98 (2000).
    [CrossRef]
  3. K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100.
    [CrossRef] [PubMed]
  4. B. Kuhlmey, R. McPhedran, C. de Sterke, P. Robinson, G. Renversez, and D. Maystre, "Microstructured optical fibers: where’s the edge?," Opt. Express 10, 1285-1290 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-22-1285.
    [PubMed]
  5. J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. St.J. Russell and P. J. Roberts, "Robust photonic band gaps for hollow core guidance in PCF made from high index glass," Opt. Express 11, 2854-2861 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854.
    [CrossRef] [PubMed]
  6. T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
    [CrossRef]
  7. L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "As-S and As-Se based photonic band gap fiber for IR laser transmission," Opt. Express 11, 3455-3460 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-25-3455.
    [CrossRef] [PubMed]
  8. S. G. Johnson, and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in planewave basis," Opt. Express 8, 173-190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
    [CrossRef] [PubMed]
  9. X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, "Nonsilica glasses for holey fibers," J. Lightwave Technol. 23, 2046-2054 (2005).
    [CrossRef]
  10. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, (Princeton University Press, 1995).
  11. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  12. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, R. C. McPhedran, "Multipole method for microstructured optical fibers. II. Implementation and results," J. Opt. Soc. Am. B 19, 2331-2340 (2002).
    [CrossRef]
  13. H. K. Kim, M. J. F. Digonnet, G. S. Kino, J. Shin, and S. Fan, "Simulations of the effect of the core ring on surface and air-core modes in photonic bandgap fibers," Opt. Express 12, 3436-3442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3436.
    [CrossRef] [PubMed]
  14. G. J. Pearce, J. M. Pottage, D. M. Bird, P. J. Roberts, J. C. Knight, and P. S. Russell, "Hollow-core PCF for guidance in the mid to far infra-red," Opt. Express 13, 6937-6946 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6937.
    [CrossRef] [PubMed]
  15. J. West, C. Smith, N. Borrelli, D. Allan, and K. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1485.
    [CrossRef] [PubMed]

2005

2004

2003

2002

2001

2000

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

J. Broeng, S. E. Barkou, T. Søndergaard, A. Bjarklev, "Analysis of air-guiding photonic bandgap fibers," Opt. Lett. 25, 96-98 (2000).
[CrossRef]

1998

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science,  282, 1476-1478 (1998).
[CrossRef] [PubMed]

Aggarwal, I. D.

Allan, D.

Barkou, S. E.

Bird, D. M.

Birks, T. A.

Bjarklev, A.

Borrelli, N.

Botten, L. C.

Broderick, N. G. R.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

Broeng, J.

J. Broeng, S. E. Barkou, T. Søndergaard, A. Bjarklev, "Analysis of air-guiding photonic bandgap fibers," Opt. Lett. 25, 96-98 (2000).
[CrossRef]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science,  282, 1476-1478 (1998).
[CrossRef] [PubMed]

de Sterke, C.

de Sterke, C. M.

Digonnet, M. J. F.

Fan, S.

Feng, X.

Hedley, T. D.

Hewak, D. W.

X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, "Nonsilica glasses for holey fibers," J. Lightwave Technol. 23, 2046-2054 (2005).
[CrossRef]

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kim, H. K.

Kino, G. S.

Knight, J. C.

Koch, K.

Koshiba, M.

Kuhlmey, B.

Kuhlmey, B. T.

Mairaj, A. K.

Maystre, D.

McPhedran, R.

McPhedran, R. C.

Monro, T. M.

X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, "Nonsilica glasses for holey fibers," J. Lightwave Technol. 23, 2046-2054 (2005).
[CrossRef]

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

Pearce, G. J.

Pottage, J. M.

Renversez, G.

Richardson, D. J.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

Roberts, P. J.

Robinson, P.

Russell, P. S.

Russell, P. St. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science,  282, 1476-1478 (1998).
[CrossRef] [PubMed]

Russell, P. St.J.

Saitoh, K.

Sanghera, J. S.

Shaw, L. B.

Shin, J.

Smith, C.

Søndergaard, T.

West, J.

West, Y. D.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

White, T. P.

Electron. Lett.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, "Chalcogenide holey fibres," Electron. Lett. 36, 1998-2000 (2000).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

H. K. Kim, M. J. F. Digonnet, G. S. Kino, J. Shin, and S. Fan, "Simulations of the effect of the core ring on surface and air-core modes in photonic bandgap fibers," Opt. Express 12, 3436-3442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3436.
[CrossRef] [PubMed]

G. J. Pearce, J. M. Pottage, D. M. Bird, P. J. Roberts, J. C. Knight, and P. S. Russell, "Hollow-core PCF for guidance in the mid to far infra-red," Opt. Express 13, 6937-6946 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6937.
[CrossRef] [PubMed]

J. West, C. Smith, N. Borrelli, D. Allan, and K. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1485.
[CrossRef] [PubMed]

L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "As-S and As-Se based photonic band gap fiber for IR laser transmission," Opt. Express 11, 3455-3460 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-25-3455.
[CrossRef] [PubMed]

S. G. Johnson, and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in planewave basis," Opt. Express 8, 173-190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
[CrossRef] [PubMed]

K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100.
[CrossRef] [PubMed]

B. Kuhlmey, R. McPhedran, C. de Sterke, P. Robinson, G. Renversez, and D. Maystre, "Microstructured optical fibers: where’s the edge?," Opt. Express 10, 1285-1290 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-22-1285.
[PubMed]

J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. St.J. Russell and P. J. Roberts, "Robust photonic band gaps for hollow core guidance in PCF made from high index glass," Opt. Express 11, 2854-2861 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854.
[CrossRef] [PubMed]

Opt. Lett.

Science

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science,  282, 1476-1478 (1998).
[CrossRef] [PubMed]

Other

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, (Princeton University Press, 1995).

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Figures (13)

Fig. 1.
Fig. 1.

Bandgap diagrams for (a) a refractive index of 1.45 with an air-filling factor of 0.9 and (b) a refractive index of 2.4 with an air-filling factor of 0.8, respectively. The refractive indices of 1.45 and 2.4 correspond to silica and chalcogenide glass, respectively.

Fig. 2.
Fig. 2.

The contour plot for the maximum relative bandgap as a function of the air-filling factor and the refractive index. The corresponding air-filling fraction is also shown. The black x-marks represent the air-filling factors corresponding to the maximum relative bandgap at each refractive index. Two black dashed curves connect x-marks in the two relative bandgap maxima in the contour plot. There exists a discontinuity of the maximum relative bandgap around a refractive index of 1.8.

Fig. 3.
Fig. 3.

Band diagram when we use a refractive index of 1.45 and an air-filling factor of 0.92. Yellow indicates the largest relative bandgap.

Fig. 4.
Fig. 4.

Magnitude of Poynting flux, normalized to its peak, in the (a) 4th and (b) 5th band. The red regions correspond to a large Poynting flux. Most of the flux in the 4th band is located in the spots, which are the large glass regions between three holes in the triangular structure. However, most of the flux in the 5th band is located in the veins, which are the long thin connections between two spots.

Fig. 5.
Fig. 5.

Fill factor as a function of the air-filling factor. The corresponding air-filling fraction is also shown. Red dashed and blue solid curves represent the fill factor for the 4th and 5th bands, respectively.

Fig. 6.
Fig. 6.

Band diagram when we use a refractive index of 2.4 and an air-filling factor of 0.75. Yellow indicates the largest relative bandgap.

Fig. 7.
Fig. 7.

(a) Red solid and dashed curves represent the lowest frequency in the 9th band and highest frequency in the 8th band as functions of the air-filling factor. (b) The red dashed curve is the same as in (a). Three solid blue curves represent the normalized frequency in the Γ, M, and M-K points as a function of air-filling factor. The corresponding air-filling fraction is also shown.

Fig. 8.
Fig. 8.

The leakage loss vs. normalized frequency for air-core PBGFs with 5 air-hole rings, where the air-filling factor is taken as a parameter. The dashed curve represents the loss for the optimized normalized frequency with different air-filling factors (d/Λ). We recall that normalized frequency ωΛ/2πc equals the pitch-wavelength ratio Λ/λ. Loss is calculated with a wavelength of 4 μm and a refractive index of 2.4.

Fig. 9.
Fig. 9.

(a) The normalized leakage loss (loss × wavelength) as a function of refractive index. (b) The corresponding real part of the effective index.

Fig. 10.
Fig. 10.

Geometry and corresponding normalized mode intensity for a 5-layer air-guided PBGF corresponding to (a) the mode found by our optimization procedure without any constraints and (b) the mode found by our optimization procedure with the constraint that more than 75% of the mode power is located in the central air core. Normalized mode intensity smaller than 0.1 is not shown.

Fig. 11.
Fig. 11.

(a) The solid curves represent the air-filling factor corresponding to minimum loss. The dashed curves with x-marks represent the air-filling factor corresponding to the maximum relative bandgap. (b) Relative bandgap when the air-filling factor corresponds to the dashed curve of (a).

Fig. 12.
Fig. 12.

The optimized normalized frequency corresponding to curves in Fig. 9 as a function of refractive index.

Fig. 13.
Fig. 13.

(a) Geometry and corresponding normalized mode intensity for a 5-layer air-guided PBGF corresponding to the minima shown in the dashed curves in Fig. 9. Normalized mode intensity smaller than 0.1 is not shown. (b) The loss as a function of the number of air hole rings on a logarithmic scale with the same fiber parameter.

Equations (1)

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f = glass [ D ( x , y ) 2 ε ( x , y ) ] dxdy all space [ D ( x , y ) 2 ε ( x , y ) ] dxdy .

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