Abstract

We study in general the property of dielectric optical waveguides that are translationally invariant along the field propagation direction. The core and the cladding can be made of either homogeneous or two-dimensional (2D) composite material (or metamaterial). When the metamaterial involved exhibits 2D periodicity, the waveguides are so-called photonic crystal fibers (PCFs). Guidance varieties in such waveguides are studied by examining separately the optical properties of the bulk core and cladding materials as well as by explicit numerical calculation of the overall waveguide structure. Our analysis reveals several neglected modal characteristics of such waveguides. For example, we show that different modes can copropagate in some PCFs with different guidance mechanisms, i.e., index guidance and photonic bandgap guidance. Such a feature can offer additional flexibility in light-wave manipulation. We also show that there exists a cutoff wavelength for the fundamental index-guided mode in some PCFs, which breaks the convention in traditional step-index fibers where such a cutoff does not exist.

© 2006 Optical Society of America

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References

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  1. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  2. P. S. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
    [CrossRef] [PubMed]
  3. J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, "Analysis of air-guiding photonic bandgap fibers," Opt. Lett. 25, 96-98 (2000).
    [CrossRef]
  4. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  5. T. A. Birks, J. C. Knight, and P. S. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  6. A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994).
    [CrossRef]
  7. M. Yan, P. Shum, and X. Yu, "Heterostructured photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 1438-1440 (2005).
    [CrossRef]
  8. T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
    [CrossRef]
  9. T. Okoshi, Optical Fibers, 1st ed. (Academic, 1982).
  10. D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.
  11. S. E. Barkou, J. Broeng, and A. Bjarklev, "Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect," Opt. Lett. 24, 46-48 (1999).
    [CrossRef]
  12. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1746-1748 (1998).
    [CrossRef]
  13. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
    [CrossRef] [PubMed]
  14. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657-659 (2003).
    [CrossRef] [PubMed]
  15. N. A. Issa and L. Poladian, "Vector wave expansion method for leaky modes of microstructured optical fibers," J. Lightwave Technol. 21, 1005-1012 (2003).
    [CrossRef]
  16. P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides--I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
    [CrossRef]
  17. S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, "Loss and dispersion analysis of microstructured fibers by finite-difference method," Opt. Express 12, 3341-3352 (2004).
    [CrossRef] [PubMed]
  18. M. Yan, P. Shum, and J. Hu, "Design of air-guiding honeycomb photonic bandgap fiber," Opt. Lett. 30, 465-467 (2005).
    [CrossRef] [PubMed]
  19. M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
    [CrossRef]
  20. B. J. Mangan, J. Arriaga, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Fundamental-mode cutoff in a photonic crystal fiber with a depressed-index core," Opt. Lett. 26, 1469-1471 (2001).
    [CrossRef]

2005 (2)

M. Yan, P. Shum, and X. Yu, "Heterostructured photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 1438-1440 (2005).
[CrossRef]

M. Yan, P. Shum, and J. Hu, "Design of air-guiding honeycomb photonic bandgap fiber," Opt. Lett. 30, 465-467 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, "Loss and dispersion analysis of microstructured fibers by finite-difference method," Opt. Express 12, 3341-3352 (2004).
[CrossRef] [PubMed]

2003 (3)

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

N. A. Issa and L. Poladian, "Vector wave expansion method for leaky modes of microstructured optical fibers," J. Lightwave Technol. 21, 1005-1012 (2003).
[CrossRef]

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

2001 (2)

2000 (1)

1999 (2)

S. E. Barkou, J. Broeng, and A. Bjarklev, "Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect," Opt. Lett. 24, 46-48 (1999).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

1998 (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1746-1748 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

1994 (1)

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994).
[CrossRef]

1975 (1)

P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides--I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

Albin, S.

Allan, D. C.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, 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. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

Arriaga, J.

Atkin, D. M.

J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Barkou, S. E.

Birks, T. A.

B. J. Mangan, J. Arriaga, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Fundamental-mode cutoff in a photonic crystal fiber with a depressed-index core," Opt. Lett. 26, 1469-1471 (2001).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1746-1748 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. S. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (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. Allan, and K. W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657-659 (2003).
[CrossRef] [PubMed]

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

Broeng, J.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Fajardo, J. C.

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

Gallagher, M. T.

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

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

Guo, S.

Hu, J.

Issa, N. A.

Joannopoulos, J. D.

Johnson, S. G.

Knight, J. C.

Koch, K. W.

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

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

Lu, C.

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

Mangan, B. J.

B. J. Mangan, J. Arriaga, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Fundamental-mode cutoff in a photonic crystal fiber with a depressed-index core," Opt. Lett. 26, 1469-1471 (2001).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Maradudin, A. A.

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994).
[CrossRef]

McGurn, A. R.

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994).
[CrossRef]

McIsaac, P.

P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides--I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

Müller, D.

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

Okoshi, T.

T. Okoshi, Optical Fibers, 1st ed. (Academic, 1982).

Poladian, L.

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Rogowski, R. S.

Russell, P. S. J.

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

B. J. Mangan, J. Arriaga, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Fundamental-mode cutoff in a photonic crystal fiber with a depressed-index core," Opt. Lett. 26, 1469-1471 (2001).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1746-1748 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, and P. S. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
[CrossRef] [PubMed]

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Shepherd, T. J.

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

Shum, P.

M. Yan, P. Shum, and J. Hu, "Design of air-guiding honeycomb photonic bandgap fiber," Opt. Lett. 30, 465-467 (2005).
[CrossRef] [PubMed]

M. Yan, P. Shum, and X. Yu, "Heterostructured photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 1438-1440 (2005).
[CrossRef]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

Smith, C. M.

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

Søndergaard, T.

Tai, H.

Venkataraman, N.

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

West, J. A.

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

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

Wu, F.

Yan, M.

M. Yan, P. Shum, and X. Yu, "Heterostructured photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 1438-1440 (2005).
[CrossRef]

M. Yan, P. Shum, and J. Hu, "Design of air-guiding honeycomb photonic bandgap fiber," Opt. Lett. 30, 465-467 (2005).
[CrossRef] [PubMed]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

Yu, X.

M. Yan, P. Shum, and X. Yu, "Heterostructured photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 1438-1440 (2005).
[CrossRef]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

Zhu, Y.

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

Electron. Lett. (1)

T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, "Full 2-D photonic bandgaps in silica/air structures," Electron. Lett. 31, 1941-1943 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

M. Yan, P. Shum, and X. Yu, "Heterostructured photonic crystal fiber," IEEE Photon. Technol. Lett. 17, 1438-1440 (2005).
[CrossRef]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, "Honeycomb photonic bandgap fiber with a modified core design," IEEE Photon. Technol. Lett. 16, 2051-2053 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides--I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in a two dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994).
[CrossRef]

Nature (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, 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. (6)

Science (3)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1746-1748 (1998).
[CrossRef]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

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

Other (2)

T. Okoshi, Optical Fibers, 1st ed. (Academic, 1982).

D. C. Allan, J. A. West, J. C. Fajardo, M. T. Gallagher, K. W. Koch, and N. F. Borrelli, "Photonic crystal fibers: effective-index and bandgap guidance," in Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001), Vol. 4, pp. 305-320.

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Figures (12)

Fig. 1
Fig. 1

Square-lattice 2D composite. Notice that the cylinders can have a higher or lower index than the background material index.

Fig. 2
Fig. 2

β - k spectral diagram for a hole-in-silica PC with d Λ = 0.8 . The DOM at each ( β , k ) point is represented by its degree in redness. For this calculation, transverse wave vector k t is uniformly sampled on 3240 points over the irreducible Brillouin zone in reciprocal space, and 32 × 32 plane waves are used for each polarization. The inset shows the PC. Black regions are air.

Fig. 3
Fig. 3

Spectrum diagrams for various optical fibers, whose core (material A) and cladding (material B) can be either homogeneous or composite. The thick dashed line is the radiation line of the core material ( β = k n A 0 ) . The thick solid line is the radiation line of the cladding material ( β = k n B 0 ) . When the material is homogeneous, n x 0 = n x , where x = A , B . Bandgap regions are shown as patches with thin boundary lines, which may belong to the core (denoted by thin dashed boundary line) or the cladding (denoted by thin solid boundary line). Spectral regions where modes are allowed by the core material but disallowed by the cladding material are shaded with slanted lines. (a) A conventional SIF; (b) a PCF with a homogeneous core whose index is larger than the indices of two element materials forming the PC cladding; (c) a PCF with a homogeneous core whose index is in between the indices of two element materials forming the PC cladding; (d) a PCF with a homogeneous core whose index is lower than the indices of two element materials forming the PC cladding; (e) a PCF with a homogeneous cladding whose index is in between the two element materials forming the PC core; (f) a PCF whose core and cladding are both PCs.

Fig. 4
Fig. 4

Spectrum diagram of a PCF with a hole-in-silica PC cladding ( d Λ = 0.8 ) and a pure silica core. The light-gray region is where propagation is allowed in the cladding PC. The dark-gray region is where propagation in the core is disallowed. Only the cladding bandgaps that appeared among the first 12 bands are recorded. At λ = 1.55 μ m (with Λ = 3.1 μ m ), confined modes are marked as short dashes.

Fig. 5
Fig. 5

Distributions of S z and E t for a PBG-guided mode (of C 3 symmetry) with the least radiation losses. n eff = 1.2289 (loss of 2.1 dB m ).

Fig. 6
Fig. 6

Power spectra under different launch conditions for a solid-core PCF with a hole-in-silica cladding characterized by d Λ = 0.8 . The red (nearest) line is for a centered Gaussian beam launch whose width is 0.25 Λ . The green (middle) line is for an offset Gaussian beam launch with the same width. The offset is along the x direction, by Λ 2 . The blue (furthest) line is for an offset launch by the same amount as in the previous case, but with the beam width at 1.5 Λ . The propagation length is 2048 μ m for all three calculations.

Fig. 7
Fig. 7

Spectrum diagram of a PCF with a hole-in-silica cladding ( d Λ = 0.5 ) and a pure silica core. Only cladding bandgaps appearing among the first 12 bands are recorded. Confined modes at λ = 1.55 μ m ( Λ = 2.3 μ m ) are marked as short dashes. Bandgap regions found for a cladding with d Λ = 0.4 and d Λ = 0.3 are shown as green and red patches, respectively. Inset, Poynting vector of the bandgap-guided mode (of C 1 symmetry).

Fig. 8
Fig. 8

Variation of leakage loss as the number of air-hole rings in the cladding is increased. The radial–azimuthal resolutions are at 400 × 60 , 500 × 90 , 600 × 120 , and 700 × 150 , and domain radii are at 20, 25, 30, and 35 μ m for the number of cladding rings at 8, 10, 12, and 14, respectively.

Fig. 9
Fig. 9

Spectral diagram for a fiber made of two PCs. Both PCs are of triangular lattice, as examined according to the silica pillar locations. The light-gray region is where propagation is allowed in a cladding PC. The dark-gray region is where propagation is prohibited in a core PC. Only the cladding bandgaps appearing among the first 16 bands are recorded. Inset shows the cladding PC, with the air holes shaded in black. Confined modes at λ = 1.55 μ m ( Λ = 2.5 μ m ) are marked by short dashes. The PBG-guided modes, from top to bottom, are, respectively, TE 01 -like, TM 01 -like, and HE 21 -like.

Fig. 10
Fig. 10

(a) S z 4 and E t fields for the HE 11 -like mode. (b) S z 3 and E t fields for the HE 21 -like mode.

Fig. 11
Fig. 11

Trace of water-guided mode (open circles) as the wavelength varies. The gray region is where propagation in the PC cladding is allowed.

Fig. 12
Fig. 12

Transition of the water-guided mode as the wavelength varies. (a) λ = 0.70 μ m , (b) λ = 0.65 μ m , (c) λ = 0.60 μ m .

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