Abstract

Scaling laws for photonic bandgaps in photonic crystal fibres are described. Although only strictly valid for small refractive index contrast, they successfully identify corresponding features in structures with large index contrast. Furthermore, deviations from the scaling laws distinguish features that are vector phenomena unique to electromagnetic waves from those that would be expected for generic scalar waves.

© 2004 Optical Society of America

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References

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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. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
    [CrossRef] [PubMed]
  2. T. A. Birks, P. J. Roberts, P. St.J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
    [CrossRef]
  3. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
    [CrossRef]
  4. F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
    [CrossRef]
  5. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
    [CrossRef] [PubMed]
  6. G. Bouwmans, F. Luan, J. C. Knight, P. St.J. Russell, L. Farr, B. J. Mangan, and H. Sabert, “Properties of a hollow-core photonic bandgap fiber at 850 nm wavelength,” Opt. Express 11, 1613–1620 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-14-1613.
    [CrossRef] [PubMed]
  7. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.
  8. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002).
    [CrossRef]
  9. P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. E 46, 4946–4972 (1992).
  10. J. Riishede, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Proc. Conference on Lasers and Electro-Optics (2003), paper CTuC5.
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  12. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995), pp. 19–20.
  13. 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]
  14. 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]
  15. P. J. Roberts, T. A. Birks, P. St.J. Russell, T. J. Shepherd, and D. M. Atkin, “Two-dimensional photonic band-gap structures as quasi-metals,” Opt. Lett. 21, 507–509 (1996).
    [CrossRef] [PubMed]
  16. J. Riishede, N. A. Mortensen, and J. Laegsgaard, “A ‘poor man’s approach’ to modelling micro-structured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003).
    [CrossRef]

2003 (4)

2002 (1)

2000 (1)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[CrossRef]

1999 (2)

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

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

1997 (1)

1996 (1)

1995 (1)

T. A. Birks, P. J. Roberts, P. St.J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

1992 (1)

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. E 46, 4946–4972 (1992).

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (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. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

Atkin, D. M.

P. J. Roberts, T. A. Birks, P. St.J. Russell, T. J. Shepherd, and D. M. Atkin, “Two-dimensional photonic band-gap structures as quasi-metals,” Opt. Lett. 21, 507–509 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St.J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Bird, D. M.

Birks, T. A.

Bise, R. T.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

Bjarklev, A.

J. Riishede, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Proc. Conference on Lasers and Electro-Optics (2003), paper CTuC5.

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Bouwmans, G.

Brechet, F.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[CrossRef]

Broeng, J.

J. Riishede, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Proc. Conference on Lasers and Electro-Optics (2003), paper CTuC5.

Chen, C.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

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. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[CrossRef] [PubMed]

de Sterke, C. M.

Eggleton, B. J.

T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002).
[CrossRef]

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

Fan, S.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

Farr, L.

Fink, Y.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Hedley, T. D.

Joannopoulos, J. D.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995), pp. 19–20.

Kerbage, C.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

Knight, J. C.

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Kranz, K. S.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

Laegsgaard, J.

J. Riishede, N. A. Mortensen, and J. Laegsgaard, “A ‘poor man’s approach’ to modelling micro-structured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003).
[CrossRef]

Litchinitser, N. M.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Luan, F.

Mangan, B. J.

Marcou, J.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[CrossRef]

McPhedran, R. C.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995), pp. 19–20.

Mortensen, N. A.

J. Riishede, N. A. Mortensen, and J. Laegsgaard, “A ‘poor man’s approach’ to modelling micro-structured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003).
[CrossRef]

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Pagnoux, D.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[CrossRef]

Piché, M.

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. E 46, 4946–4972 (1992).

Pottage, J. M.

Riishede, J.

J. Riishede, N. A. Mortensen, and J. Laegsgaard, “A ‘poor man’s approach’ to modelling micro-structured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003).
[CrossRef]

J. Riishede, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Proc. Conference on Lasers and Electro-Optics (2003), paper CTuC5.

Ripin, D. J.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

Roberts, P. J.

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]

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

P. J. Roberts, T. A. Birks, P. St.J. Russell, T. J. Shepherd, and D. M. Atkin, “Two-dimensional photonic band-gap structures as quasi-metals,” Opt. Lett. 21, 507–509 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St.J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Roy, P.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[CrossRef]

Russell, P. St.J.

Sabert, H.

Shepherd, T. J.

P. J. Roberts, T. A. Birks, P. St.J. Russell, T. J. Shepherd, and D. M. Atkin, “Two-dimensional photonic band-gap structures as quasi-metals,” Opt. Lett. 21, 507–509 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. St.J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Thomas, E. L.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

Trevor, D. J.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Villeneuve, P. R.

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. E 46, 4946–4972 (1992).

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, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

White, T. P.

Windeler, R. S.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995), pp. 19–20.

Electron. Lett. (2)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[CrossRef]

T. A. Birks, P. J. Roberts, P. St.J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

IEEE J. Lightwave Technol. (1)

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” IEEE J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

J. Riishede, N. A. Mortensen, and J. Laegsgaard, “A ‘poor man’s approach’ to modelling micro-structured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003).
[CrossRef]

Nature (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allen, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. E (1)

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. E 46, 4946–4972 (1992).

Science (1)

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

Other (4)

J. Riishede, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Proc. Conference on Lasers and Electro-Optics (2003), paper CTuC5.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995), pp. 19–20.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” Proc. Optical Fiber Communication Conference (2002) pp. 466–468.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram illustrating our new way of mapping the states of a photonic crystal, on normalised axes v 2 (or v) against w 2. Propagation is forbidden in the regions coloured red.

Fig. 2.
Fig. 2.

Plots of DOS on normalised axes for d/Λ=(a) 0.96 and (b) 0.80, and for n=(i) 1.02 and (ii) 1.45 (the “scalar” and “vector” cases respectively). Regions with no propagating states (DOS≡0) are coloured red for emphasis. These correspond to cutoff (bottom right, β>kneff ), evanescent states (bottom left, β2<0) and bandgaps (elsewhere). The arrows on the top edge locate the resonances of the circular low-index regions, whose loci would be vertical lines on these plots.

Fig. 3.
Fig. 3.

Plots of DOS on the low-index line as index contrast n is varied, for d/Λ=(a) 0.96 and (b) 0.80. The colour map for DOS is the same as that in Fig. 2. The shrinking bandgap around v=12 in (a) is the fundamental bandgap used in current air-guiding silica PCFs. The growing bandgap around v=12 in (b) is the robust type II bandgap of [13].

Equations (7)

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

k t 1 k t 2 = k 2 n 1 2 β 2 k 2 n 2 2 β 2
( t 2 + k 2 n 0 2 β 2 ) h = ( t × h ) × ( t ln n 0 2 ) ,
f ( X , Y ) = { 0 low index regions 1 high index regions ,
T 2 Ψ + ( v 2 f w 2 ) Ψ = 0 ,
v 2 = Λ 2 k 2 ( n 1 2 n 2 2 ) .
w 2 = Λ 2 ( β 2 k 2 n 2 2 )
w 2 = ( 2 j l m d Λ ) 2 ,

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