Abstract

We describe a finite-difference numerical method that allows us to simulate the modes of air-core photonic-bandgap fibers (PBF) of any geometry in minutes on a standard PC. The modes’ effective indices and fields are found by solving a vectorial transverse magnetic-field equation in a matrix form, which can be done quickly because this matrix is sparse and because we reduce its bandwidth by rearranging its elements. The Stanford Photonic-Bandgap Fiber code, which is based on this method, takes about 4 minutes to model 20 modes of a typical PBF on a PC. Other advantage; include easy coding, faithful modeling of the abrupt discontinuities in the index profile, high accuracy, and applicability to waveguides of arbitrarily complex profile.

© 2006 Optical Society of America

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  1. X. Yong, R. K. Lee, and A. Yariv, "Asymptotic analysis of Bragg fibers," Opt. Lett. 25, 1756-815 (2000).
    [CrossRef]
  2. P.V. Kaiser and H. W. Astle, "Low-loss single material fibers made from pure fused silica," Bell Syst. Tech. J. 53, 1021-1039 (1974).
  3. J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
    [CrossRef]
  4. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, "All-silica single mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  5. J. C. Knight, T. A. Birks, R.F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
    [CrossRef]
  6. R. S. Windeler, J. L. Wagener, and D. J. Giovanni, "Silica-air microstructured fibers: Properties and applications," Optical Fiber Communications conference, San Diego (1999).
  7. B. Kuhlmey, G. Renversez, and D. Maystre, "Chromatic dispersion and losses of microstructured optical fibers," Appl. Opt. 42, 634-639 (2003).
    [CrossRef] [PubMed]
  8. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843.
    [CrossRef] [PubMed]
  9. G. Renversez, B. Kuhlmey, and R. McPhedran, "Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses," Opt. Lett. 28, 989-991 (2003).
    [CrossRef] [PubMed]
  10. D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, "Surface modes and loss in air-core photonic band-gap fibers," in Photonic Crystal Devices, A. Adibi, A. Scherer, and S.Y. Lin, eds., Proc. SPIE 5000, 161-174, (2003).
    [CrossRef]
  11. 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]
  12. J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485.
    [CrossRef] [PubMed]
  13. M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, "Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers," Opt. Express 12, 1864-1872, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864.
    [CrossRef] [PubMed]
  14. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Electron. 40, 551-556 (2004).
    [CrossRef]
  15. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt. Lett. 26, 1660-1662 (2001).
    [CrossRef]
  16. D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, "Leakage properties of photonic crystal fibers," Opt. Express 10, 1285-1290 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314.
  17. 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]
  18. S. G. Johnson, and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in plane wave basis," Opt. Express 8, 173-190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
    [CrossRef] [PubMed]
  19. MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/.
  20. T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
    [CrossRef]
  21. L. Poladian, N. A. Issa, and T. M. Monro, "Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry," Opt. Express 10, 449-454 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449.
    [PubMed]
  22. Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853.
    [PubMed]
  23. C. Yu and H. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
    [CrossRef]
  24. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
    [CrossRef] [PubMed]
  25. K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum. Electron. 38, 927-933 (2002).
    [CrossRef]
  26. M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-30 (2001).
    [CrossRef]
  27. J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. J. Roberts, and P. St. J. Russell, "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]
  28. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983), Chap. 29.
  29. N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
    [CrossRef]
  30. A. George and J. Liu, Computer Solution of Large Sparse Positive Definite Systems, (Prentice-Hall, 1981).
  31. R. A. Horn and C. R. Johnson, Matrix Analysis, (Cambridge University Press, 1990), Chap. 4.

2004

2003

2002

2001

M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-30 (2001).
[CrossRef]

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

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

T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt. Lett. 26, 1660-1662 (2001).
[CrossRef]

2000

1999

J. C. Knight, T. A. Birks, R.F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

1997

N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

1996

1974

P.V. Kaiser and H. W. Astle, "Low-loss single material fibers made from pure fused silica," Bell Syst. Tech. J. 53, 1021-1039 (1974).

Allan, D. C.

Astle, H. W.

P.V. Kaiser and H. W. Astle, "Low-loss single material fibers made from pure fused silica," Bell Syst. Tech. J. 53, 1021-1039 (1974).

Atkin, D. M.

Barkou, S. E.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Bird, D. M.

Birks, T. A.

Bjarklev, A.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Borelli, N. F.

Botten, L. C.

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

Botton, L. C.

Broeng, J.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Brown, T. G.

Chang, H.

C. Yu and H. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Cregan, R.F.

J. C. Knight, T. A. Birks, R.F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Cucinotta, A.

de Sterke, C. M.

Digonnet, M. J. F.

Fan, S. H.

Ferrarini, D.

Guobin, R.

Hasegawa, T.

Hedley, T. D.

Houshmand, B.

N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Issa, N. A.

Itoh, T.

N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kaiser, P.V.

P.V. Kaiser and H. W. Astle, "Low-loss single material fibers made from pure fused silica," Bell Syst. Tech. J. 53, 1021-1039 (1974).

Kaneda, N.

N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Kim, H. K.

Kino, G. S.

Knight, J. C.

Koch, K. W.

Koshiba, M.

Kuhlmey, B.

Lee, R. K.

Martijn de Sterke, C.

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

Maystre, D.

McPhedran, R.

McPhedran, R. C.

T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt. Lett. 26, 1660-1662 (2001).
[CrossRef]

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

Mogilevstev, D.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Monro, T. M.

Mortensen, N. A.

Poladian, L.

Pottage, J. M.

Qiu, M.

M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-30 (2001).
[CrossRef]

Renversez, G.

Roberts, P. J.

Russell, P. St. J.

Saitoh, K.

Sandro, J. P.

J. C. Knight, T. A. Birks, R.F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Sasaoka, E.

Selleri, S.

Shin, J.

Shuisheng, J.

Shuqin, L.

Smith, C. M.

Smith, G. H.

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

Steel, M. J.

Vincetti, L.

West, J. A.

White, T. P.

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt. Lett. 26, 1660-1662 (2001).
[CrossRef]

Yariv, A.

Yong, X.

Yu, C.

C. Yu and H. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Zhi, W.

Zhu, Z.

Zoboli, M.

Appl. Opt.

Bell Syst. Tech. J.

P.V. Kaiser and H. W. Astle, "Low-loss single material fibers made from pure fused silica," Bell Syst. Tech. J. 53, 1021-1039 (1974).

IEEE J. Quantum Electron.

H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Electron. 40, 551-556 (2004).
[CrossRef]

IEEE J. Quantum. Electron.

K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum. Electron. 38, 927-933 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

N. Kaneda, B. Houshmand, and T. Itoh, "FDTD analysis of dielectric resonators with curved surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997).
[CrossRef]

Microwave Opt. Technol. Lett.

M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-30 (2001).
[CrossRef]

Opt. Express

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 8, 721-732 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-721.
[CrossRef]

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

L. Poladian, N. A. Issa, and T. M. Monro, "Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry," Opt. Express 10, 449-454 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-10-449.
[PubMed]

Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853.
[PubMed]

D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, "Leakage properties of photonic crystal fibers," Opt. Express 10, 1285-1290 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314.

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843.
[CrossRef] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
[CrossRef] [PubMed]

J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. J. Roberts, and P. St. J. Russell, "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]

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]

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]

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

M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, "Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers," Opt. Express 12, 1864-1872, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864.
[CrossRef] [PubMed]

Opt. Fiber Technol.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Opt. Lett.

Opt. Mater.

J. C. Knight, T. A. Birks, R.F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Opt. Quantum Electron.

C. Yu and H. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983), Chap. 29.

R. S. Windeler, J. L. Wagener, and D. J. Giovanni, "Silica-air microstructured fibers: Properties and applications," Optical Fiber Communications conference, San Diego (1999).

D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, "Surface modes and loss in air-core photonic band-gap fibers," in Photonic Crystal Devices, A. Adibi, A. Scherer, and S.Y. Lin, eds., Proc. SPIE 5000, 161-174, (2003).
[CrossRef]

MIT Photonic Bandgap software website, http://ab-initio.mit.edu/mpb/.

A. George and J. Liu, Computer Solution of Large Sparse Positive Definite Systems, (Prentice-Hall, 1981).

R. A. Horn and C. R. Johnson, Matrix Analysis, (Cambridge University Press, 1990), Chap. 4.

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Figures (11)

Fig. 1.
Fig. 1.

Example of a matrix M representing the mode equation for a particular PBF using a rectangular boundary.

Fig. 2.
Fig. 2.

Matrix N obtained by rearranging matrix M of Fig. 1 to minimize the matrix bandwidth.

Fig. 3.
Fig. 3.

Summary of the SPBF code algorithm (see text for details).

Fig. 4.
Fig. 4.

Sampling of the refractive index profile of an air-core PBF with an air-filling ratio ρ = 0.47Λ and a core radius R = 0 8Λ used to simulate this fiber using periodic boundary conditions.

Fig. 5.
Fig. 5.

Mode intensity profile of the fundamental core mode of the PBF of Fig. 4 at λ = 0.60Λ. The contour lines are magnified in the right-side figure for easier viewing.

Fig. 6.
Fig. 6.

Cut at x = 0 of the fundamental core mode intensity of Fig. 5.

Fig. 7.
Fig. 7.

Band profile and dispersion of the two fundamental modes of the PBF of Fig. 4, except for a core radius R = 1.00Λ.

Fig. 8.
Fig. 8.

Convergence of the fundamental mode effective index vs. grid size for the PBF in Fig. 4; the reference value used is the result obtained by SPBF with 600 grid points per side.

Fig. 9.
Fig. 9.

Convergence of the fundamental mode effective index vs. grid size for a silica fiber in air (red circles) and for the holey microstructured optical fiber (MOF) of Ref. 22 (blue dots); the analytical solution is used as a reference for the silica fiber, and the multipole result as a reference for the MOF.

Fig. 10.
Fig. 10.

Band profile and dispersion of the two fundamental modes and surface modes of the same PBF as shown in Fig. 4 except for a core radius R = 1.15Λ.

Fig. 11
Fig. 11

Intensity profile of an exemplary surface mode of the fiber modeled in Fig. 8 (ρ = 0.47Λ and R = 1.15Λ) at a wavelength λ = 0 607Λ The contour lines are magnified in the right-side figure for easier viewing.

Equations (8)

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

× { 1 ε ( r ) × H ( r ) } = ω 2 c 2 H ( r ) ,
{ × E = μ 0 H t × H = ε 0 ε ( r ) E t ,
{ T 2 + k 0 2 ( n 2 ( x , y ) n eff 2 ) } h T = ( T × h T ) × T ( In n 2 ( x , y ) ) ,
{ h z = i k 0 n eff T h T e T = ( μ 0 ε 0 ) ½ 1 k 0 n 2 u z × { k 0 n eff u z 1 k 0 n eff T ( T h T ) } , e z = i ( μ 0 ε 0 ) ½ 1 k 0 n 2 u z ( T × h T )
h T = k 0 2 n eff 2 h T ,
M h = k 0 2 n eff 2 h ,
B M = max ( i , j ) ; M ( i , j ) 0 i j ,
A = ( N k 0 2 n 0 2 I ) 1 ,

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