Abstract

Ring-structured Bragg fibres that support a single TE-polarisation mode are investigated. The fibre designs consist of a hollow core and rings of holes concentric with the core, which form the low-index layers of the Bragg reflector in the cladding. The effects of varying the air fraction in each ring of holes on the transmission properties of the fibres are analysed and an approximate model based on homogenisation is explored. Surface modes and transitions thereof are also discussed.

©2004 Optical Society of America

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References

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  1. P. Russell, “Photonic crystal fibers,” Science 299, 358–62 (2003).
    [Crossref] [PubMed]
  2. T. A. Birks, J.C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–3 (1997).
    [Crossref] [PubMed]
  3. R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
    [Crossref] [PubMed]
  4. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–79 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748.
    [Crossref] [PubMed]
  5. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–3 (2002)
    [Crossref] [PubMed]
  6. A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, M. C. J. large, J. Zagari, N. A. P. Nicorovici, R. C. McPhedran, and C. M. de Sterke, “Ring structures in microstructured polymer optical fibres,” Opt. Express 9, 813–20 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-813.
    [Crossref] [PubMed]
  7. A. Argyros, N.A. Issa, I. Bassett, and M. A. van Eijkelenborg, “Microstructured optical fiber for single-polarization air guidance,” Opt. Lett. 29, 20–2 (2004).
    [Crossref] [PubMed]
  8. G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.
  9. I. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342–6 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1342.
    [Crossref] [PubMed]
  10. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–7 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411.
    [Crossref] [PubMed]
  11. J. Noda, K. Okamoto, and Y. Sasaki, “Polarization maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–89 (1986).
    [Crossref]
  12. A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers,” App. Phys. Lett. 78, 3184–6 (2001).
    [Crossref]
  13. N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Technol. 21, 1005–12 (2003).
    [Crossref]
  14. K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer-Verlag, Berlin, 2001).
  15. H. A. McLeod, Thin-film optical filters, Chapter 5 (Adam Hilger Ltd, London, 1969)
  16. Y. Xu, A. Yariv, J. G. Fleming, and S.-Y. Lin, “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039–49 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.
    [Crossref] [PubMed]
  17. S. E. Golowich and M. I. Weinstein, “Homogenisation expansion for resonances of microstructured photonic waveguides,” J. Opt. Soc. Am. B 20, 633–47 (2003).
    [Crossref]
  18. V. Rastogi and K. S. Chiang, “Holey optical fiber with circularly distributed holes analyzed by the radial effective-index method,” Opt. Lett. 28, 2449–51 (2003).
    [Crossref] [PubMed]
  19. The example used for comparison in Ref. [14] was re-modelled using the same method as for the rest of the calculations presented here. This method is more accurate than that used for the original comparison, and yielded the agreement quoted here, which differs from the original in [14].
  20. G.W. Milton, The theory of composites, (Cambridge University Press, London, 2001).
  21. A.W. Snyder and J.D. Love, Optical Waveguide Theory, Chapter 30 (Chapman and Hall, New York, 1983).
  22. 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.
    [Crossref] [PubMed]

2004 (2)

2003 (5)

2002 (3)

2001 (3)

1999 (1)

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

1997 (1)

1986 (1)

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–89 (1986).
[Crossref]

Allen, D. C.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

Argyros, A.

Bassett, I.

Bassett, I. M.

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–3 (2002)
[Crossref] [PubMed]

Birks, T. A.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

T. A. Birks, J.C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–3 (1997).
[Crossref] [PubMed]

Bjarklev, A.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Broeng, J.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Chiang, K. S.

Cregan, R. F.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

de Sterke, C. M.

Deyerl, H. J.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Engeness, T. D.

Ferrando, A.

A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers,” App. Phys. Lett. 78, 3184–6 (2001).
[Crossref]

Fink, Y.

Fleming, J. G.

Golowich, S. E.

Hansen, T. P.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–3 (2002)
[Crossref] [PubMed]

Huang, Y.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Ibanescu, M.

Issa, N.A.

Jacobs, S. A.

Jensen, J. B.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Joannopoulos, J. D.

Johnson, S. G.

Knight, J. C.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

Knight, J.C.

Koshiba, M .

large, M. C. J.

Larsen, B. H.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Lee, R.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Lin, S.-Y.

Love, J.D.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, Chapter 30 (Chapman and Hall, New York, 1983).

Managan, B. J.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

McLeod, H. A.

H. A. McLeod, Thin-film optical filters, Chapter 5 (Adam Hilger Ltd, London, 1969)

McPhedran, R. C.

Milton, G.W.

G.W. Milton, The theory of composites, (Cambridge University Press, London, 2001).

Miret, J. J.

A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers,” App. Phys. Lett. 78, 3184–6 (2001).
[Crossref]

Mortensen, N. A.

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.
[Crossref] [PubMed]

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Nicorovici, N. A. P.

Noda, J.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–89 (1986).
[Crossref]

Okamoto, K.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–89 (1986).
[Crossref]

Poladian, L.

Rastogi, V.

Roberts, P. J.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299, 358–62 (2003).
[Crossref] [PubMed]

Russell, P. St. J.

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

T. A. Birks, J.C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–3 (1997).
[Crossref] [PubMed]

Saitoh, K.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer-Verlag, Berlin, 2001).

Sasaki, Y.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–89 (1986).
[Crossref]

Simonsen, H.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Skorobogatiy, M.

Snyder, A.W.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, Chapter 30 (Chapman and Hall, New York, 1983).

Soljacic, M.

Sorensen, T.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–3 (2002)
[Crossref] [PubMed]

Terrel, M.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

van Eijkelenborg, M. A.

Vienne, G.

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Weinstein, M. I.

Weisberg, O.

Xu, Y.

Y. Xu, A. Yariv, J. G. Fleming, and S.-Y. Lin, “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039–49 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.
[Crossref] [PubMed]

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Yariv, A.

Y. Xu, A. Yariv, J. G. Fleming, and S.-Y. Lin, “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039–49 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1039.
[Crossref] [PubMed]

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

Zagari, J.

App. Phys. Lett. (1)

A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystals fibers,” App. Phys. Lett. 78, 3184–6 (2001).
[Crossref]

J. Lightwave Technol. (2)

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–89 (1986).
[Crossref]

N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Technol. 21, 1005–12 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–3 (2002)
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (3)

Science (2)

P. Russell, “Photonic crystal fibers,” Science 299, 358–62 (2003).
[Crossref] [PubMed]

R. F. Cregan, B. J. Managan, 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, 1537–9 (1999).
[Crossref] [PubMed]

Other (6)

G. Vienne, Y. Xu, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sorensen, M. Terrel, Y. Huang, R. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “First demonstration of air-silica Bragg fiber,” Optical Fiber Communication Conference & Exposition Postdeadline Papers (Institute of Electrical and Electronics Engineers, New York, 2004), PDP25.

K. Sakoda, Optical Properties of Photonic Crystals, Chapter 2 (Springer-Verlag, Berlin, 2001).

H. A. McLeod, Thin-film optical filters, Chapter 5 (Adam Hilger Ltd, London, 1969)

The example used for comparison in Ref. [14] was re-modelled using the same method as for the rest of the calculations presented here. This method is more accurate than that used for the original comparison, and yielded the agreement quoted here, which differs from the original in [14].

G.W. Milton, The theory of composites, (Cambridge University Press, London, 2001).

A.W. Snyder and J.D. Love, Optical Waveguide Theory, Chapter 30 (Chapman and Hall, New York, 1983).

Supplementary Material (1)

» Media 1: MOV (1653 KB)     

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the ring-structured Bragg fibre (a) with some of the parameters indicated in (b). (c) The 2D lattice of holes that forms at large distances from the core. The wavevector is indicated by k, its longitudinal component by β and its transverse component by κ.

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Fig. 2.
Fig. 2. (a) The loss of the TE01 mode (at the lowest loss wavelength) as a function of the number of rings for the different values of the air fraction. (b) The loss of the TE01 mode as a function of wavelength for the different values of f with 9 rings of holes, showing the loss and the wavelength of lowest loss increasing with decreasing air fraction. (c) The loss of the HE11 and TM01 modes for f=0.65 (Λi=0.403 µm) with 9 rings of holes. (d) The loss of the HE11 and TM01 modes for f=0.43 (Λi=0.607 µm) with 9 rings of holes. The loss of the hollow waveguide HE11 and TM01 modes is also shown in (c) and (d) for comparison. The mechanism for isolating the TE01 mode is the large difference in the loss of the modes [9]. The HE11 mode has a loss of 103 to 104 times larger than that of the TE01 mode, depending on the value of Λi.

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Fig. 3.
Fig. 3. Band diagrams for the highest and lowest air fraction of each ring of holes used for the TE and TM polarisations with the band gap regions denoted by white. The dispersion lines of the TE01 modes are shown on the TE diagrams, and of the HE11 mode on the TM diagrams (mostly indistinguishable from the light line on these graphs). The point where the TM band gap closes falls very close to the light line. This was achieved by incorporating the Brewster angle in these fibre designs [9].

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Fig. 4.
Fig. 4. The average refractive index profile n(r) obtained for one ring of holes with f=0.65 (Λi=0.403 µm) using the various methods discussed in the text.

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Fig. 5.
Fig. 5. Loss at the lowest loss wavelength for the ring-structured fibres and the various homogenisation approaches used. Only the curve for x=1 is plotted for Eq. (5) as it gave the best approximation compared to other values of x as described in the text.

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Fig. 6.
Fig. 6. (a) Schematic of Re{n eff} as a function of wavelength indicating the effect of the avoided crossings and the surface mode “s”, core mode “c” and transition stage “t” of each mode. The Re{n eff} (b) and loss (c) as a function of wavelength for f=0.43 (Λi=0.607 µm) and 9 rings of holes for modes of TE and HE polarisation from the middle of the primary band gap to longer wavelengths. The number of lobes in the intensity profile Sz of each mode is indicated. The intensity profile for the HE mode with 8 lobes is shown in Fig. 7 (for the wavelengths indicated) and in Fig. 8.

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Fig. 7.
Fig. 7. The z-component of the Poynting vector Sz across the fibre for a HE polarisation mode in the surface mode stage (a), in the core mode stage (b), and in the transition stage (c and d).

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Fig. 8.
Fig. 8. The z-component of the Poynting vector Sz across the fibre for a HE polarisation mode as the wavelength is increased from 1.18 to 1.57 µm, showing how the mode changes from a surface mode to a core mode and the transitions thereafter. Red indicates large intensities and violet low intensities. (animation -1.61 MB)

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Equations (6)

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

κ d ¯ = k 2 ( n 2 − Re { n eff } 2 ) d ¯ = π ⁄ 2 ,
n ( r ) = 1 2 π ∫ 0 2 π n ( r , ϕ ) d ϕ ,
n av = π d 4 Λ i + ( 1 − π d 4 Λ i ) n 1 = f + ( 1 − f ) n 1 ,
n av = [ π d 4 Λ i + ( 1 − π d 4 Λ i ) n 1 x ] 1 ⁄ x = [ f + ( 1 − f ) n 1 x ] 1 ⁄ x , − 2 ≤ x ≤ 2 ,
n ( r ) = [ 1 2 π ∫ 0 2 π n x ( r , ϕ ) d ϕ ] 1 ⁄ x , − 2 ≤ x ≤ 2 .
n ( r ) = e 1 2 π ∫ 0 2 π ln [ n ( r , ϕ ) ] d ϕ ,

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