Abstract

Cladding structures of photonic bandgap fibers often have airholes of noncircular shape, and, typically, close-to-hexagonal airholes with curved corners are observed. We study photonic bandgaps in such structures by aid of a two-parameter representation of the size and curvature. For the fundamental bandgap we find that the bandgap edges (the intersections with the air line) shift toward shorter wavelengths when the air-filling fraction f is increased. The bandgap also broadens, and the relative bandwidth increases exponentially with f2. Compared with recent experiments [Nature 424, 657 (2003)] we find very good agreement.

© 2004 Optical Society of America

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  1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
    [CrossRef] [PubMed]
  2. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
    [CrossRef] [PubMed]
  3. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
    [CrossRef] [PubMed]
  4. P. Russell, Science 299, 358 (2003).
    [CrossRef] [PubMed]
  5. J. C. Knight, Nature 424, 847 (2003).
    [CrossRef] [PubMed]
  6. J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, Opt. Lett. 25, 96 (2000).
    [CrossRef]
  7. T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
    [CrossRef]
  8. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, Opt. Lett. 21, 1547 (1996).
    [CrossRef] [PubMed]
  9. G. Bouwmans, F. Luan, J. C. Knight, P. St. J. Russell, L. Farr, B. J. Mangan, and H. Sabert, Opt. Express 11, 1613 (2003), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  10. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
    [CrossRef] [PubMed]
  11. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  12. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org.
    [CrossRef] [PubMed]

2003

P. Russell, Science 299, 358 (2003).
[CrossRef] [PubMed]

J. C. Knight, Nature 424, 847 (2003).
[CrossRef] [PubMed]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

G. Bouwmans, F. Luan, J. C. Knight, P. St. J. Russell, L. Farr, B. J. Mangan, and H. Sabert, Opt. Express 11, 1613 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

2002

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
[CrossRef] [PubMed]

2001

2000

1999

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

1998

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
[CrossRef] [PubMed]

1996

1995

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

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, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Allen, D. C.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

Antonopoulos, G.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
[CrossRef] [PubMed]

Atkin, D. M.

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, Opt. Lett. 21, 1547 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

Barkou, S. E.

Benabid, F.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
[CrossRef] [PubMed]

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, Opt. Lett. 21, 1547 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

Bjarklev, A.

Borrelli, N. F.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

Bouwmans, G.

Broeng, J.

J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, Opt. Lett. 25, 96 (2000).
[CrossRef]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
[CrossRef] [PubMed]

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, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Farr, L.

Gallagher, M. T.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org.
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Johnson, S. G.

Knight, J. C.

J. C. Knight, Nature 424, 847 (2003).
[CrossRef] [PubMed]

G. Bouwmans, F. Luan, J. C. Knight, P. St. J. Russell, L. Farr, B. J. Mangan, and H. Sabert, Opt. Express 11, 1613 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
[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, Science 285, 1537 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, Opt. Lett. 21, 1547 (1996).
[CrossRef] [PubMed]

Koch, K. W.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

Luan, F.

Mangan, B. J.

G. Bouwmans, F. Luan, J. C. Knight, P. St. J. Russell, L. Farr, B. J. Mangan, and H. Sabert, Opt. Express 11, 1613 (2003), http://www.opticsexpress.org.
[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, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Müller, D.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

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, Science 285, 1537 (1999).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

Russell, P.

P. Russell, Science 299, 358 (2003).
[CrossRef] [PubMed]

Russell, P. St. J.

G. Bouwmans, F. Luan, J. C. Knight, P. St. J. Russell, L. Farr, B. J. Mangan, and H. Sabert, Opt. Express 11, 1613 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
[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, Science 285, 1537 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, Opt. Lett. 21, 1547 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

Sabert, H.

Shephard, T. J.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

Smith, C. M.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

Søndergaard, T.

Venkataraman, N.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

West, J. A.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Electron. Lett.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shephard, Electron. Lett. 31, 1941 (1995).
[CrossRef]

Nature

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, Nature 424, 657 (2003).
[CrossRef] [PubMed]

J. C. Knight, Nature 424, 847 (2003).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Science

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998).
[CrossRef] [PubMed]

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, Science 298, 399 (2002).
[CrossRef] [PubMed]

P. Russell, Science 299, 358 (2003).
[CrossRef] [PubMed]

Other

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

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Figures (5)

Fig. 1
Fig. 1

(a) SEM of the cladding structure of a PBG fiber with a pitch of Λ2.7 µm. (b) Two-parameter representation of the airhole shape.

Fig. 2
Fig. 2

Plot of β/k versus k along principal directions in the Brillouin zone for βΛ=14.45 and a structure with D/Λ=0.96 and Dc/Λ=0.55 (see inset of Fig. 3). The filled curve shows the corresponding density of states (the projection of the data onto the y axis).

Fig. 3
Fig. 3

Plot of β/k versus λ/Λ for a structure with D/Λ=0.96 and Dc/Λ=0.55 (see inset). The data points result from a sampling of the Brillouin zone along the usual path from the Γ point to the M point and through the K point back to the Γ point. The solid curves indicate the bandgap boundaries, and the dashed line β/k=1 shows the air line that passes through several bandgaps.

Fig. 4
Fig. 4

Plot of the bandgap edges λu and λl of the fundamental bandgap as a function of air-filling fraction f. The dashed curves are guides for the eyes. Recent experimental data10 are also included.

Fig. 5
Fig. 5

Relative bandwidth versus f2. The dashed line indicates the close-to-exponential dependence [see relation (4)]. Recent experimental data10 are also included.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

f=AhexD-AhexDc+AcircDcAhexΛ=DΛ21-1-π23DcD2,
×1r×Hωr=ω2c2Hωr.
Hωr=hωrexpik·r+iβz.
Δλ/λcB×expA×f2,

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