Abstract

Photonic crystal fibers often consist of rotationally symmetric inclusions in an otherwise uniform background medium. The band diagrams and modes of such structures can be efficiently calculated using geometry-specific methods that exploit this rotational symmetry. Until now, these have only been applied to fibers in which the inclusions are circular and have a uniform refractive index. Here, we generalize this to arbitrary rotationally symmetric inclusions using a transfer matrix approach, and we implement this approach in an approximate scalar method, which is valid for low-index contrasts and in the rigorous Rayleigh multipole method. We apply the methods to structures incorporating inclusions with graded refractive indices and to structures incorporating metal rings.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2008 (2)

2007 (4)

2006 (5)

2004 (5)

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers. I: Theory,” Opt. Express 12, 3791-3805 (2004).
[CrossRef] [PubMed]

S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919-1928 (2004).
[CrossRef]

J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibers with high-index rods,” J. Opt. A 6, 798-804 (2004).
[CrossRef]

T. L. Wu and C. H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126-128 (2004).
[CrossRef]

2003 (3)

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

J. C. Knight, “Photonic crystal fibers,” Nature 424, 847-851 (2003).
[CrossRef] [PubMed]

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787-1792 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808-7816 (2000).
[CrossRef]

1997 (1)

1995 (1)

C. T. Chan, Q. L.Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

1994 (1)

S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, “Green's function and lattice sums for electromagnetic scattering by a square array of cylinders,” Phys. Rev. E 49, 4590-4602 (1994).
[CrossRef]

1990 (1)

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

1978 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), ninth Dover printing, tenth GPO printing ed.

Agrawal, G. P.

G. P. Agrawal and R. W. Boyd, Nonlinear Fiber Optics (Springer, 2001).

Amezcua-Correa, A.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Baril, N. F.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Bird, D.

Bird, D. D. M.

Bird, D. M.

Birks, T. A.

Botten, L. C.

S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919-1928 (2004).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322-2330 (2002).
[CrossRef]

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002).
[CrossRef]

T. P. White, R. C. McPhedran, L. C. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721-732 (2001).
[CrossRef] [PubMed]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808-7816 (2000).
[CrossRef]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Boyd, R. W.

G. P. Agrawal and R. W. Boyd, Nonlinear Fiber Optics (Springer, 2001).

Busch, K.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

Campbell, S.

Chan, C. T.

C. T. Chan, Q. L.Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Chao, C. H.

T. L. Wu and C. H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126-128 (2004).
[CrossRef]

Cheng, H.

Chin, S. K.

S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, “Green's function and lattice sums for electromagnetic scattering by a square array of cylinders,” Phys. Rev. E 49, 4590-4602 (1994).
[CrossRef]

Crutchfield, W.

de Sterke, C. M.

S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919-1928 (2004).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322-2330 (2002).
[CrossRef]

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002).
[CrossRef]

T. P. White, R. C. McPhedran, L. C. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721-732 (2001).
[CrossRef] [PubMed]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Doery, M.

Eggleton, B. J.

Finlayson, C. E.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Fu, L.

Fussell, D. P.

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

George, A.

George, A. K.

Greengard, L.

Grubits, K. A.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808-7816 (2000).
[CrossRef]

Guan, N.

Habu, S.

Hassani, A.

Hayes, J. R.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Himeno, K.

Ho, K. M.

C. T. Chan, Q. L.Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Hochman, A.

Hou, J.

Jackson, B. R.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Kabashin, A.

Kakarantzas, G.

Knight, J.

Knight, J. C.

Kuhlmey, B.

J. Hou, D. Bird, A. George, S. Maier, B. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibers,” Opt. Express 16, 5983-5990 (2008).
[CrossRef] [PubMed]

B. Kuhlmey, “Theoretical and numerical investigation of the physics of microstructured optical fibers,” Ph.D. thesis, University of Sydney and Université Aix-Marseille III (2006). http://setis.library.usyd.edu.au/adt/public html/adt-NU/public/adt-NU20040715.171105/.

Kuhlmey, B. T.

Lacroix, S.

Laegsgaard, J.

J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibers with high-index rods,” J. Opt. A 6, 798-804 (2004).
[CrossRef]

Langtry, T. N.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Leung, K. M.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Leviatan, Y.

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Academic, 1983).

Luan, F.

Maier, S.

Mangan, B. J.

P. S. J. Russell, T. A. Birks, J. C. Knight, and B. J. Mangan, “Photonic crystal fibers,” US Patent 6,990,282 (2006).

Margine, E. R.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Marom, E.

Maystre, D.

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

McPhedran, R. C.

B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express 14, 10851-10864 (2006).
[CrossRef] [PubMed]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919-1928 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322-2330 (2002).
[CrossRef]

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002).
[CrossRef]

T. P. White, R. C. McPhedran, L. C. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721-732 (2001).
[CrossRef] [PubMed]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808-7816 (2000).
[CrossRef]

S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, “Green's function and lattice sums for electromagnetic scattering by a square array of cylinders,” Phys. Rev. E 49, 4590-4602 (1994).
[CrossRef]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808-7816 (2000).
[CrossRef]

S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, “Green's function and lattice sums for electromagnetic scattering by a square array of cylinders,” Phys. Rev. E 49, 4590-4602 (1994).
[CrossRef]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Pathmanandavel, K.

Pearce, G. J.

Pone, E.

Poulton, C. G.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys Rev B 77, 033417 (2007).
[CrossRef]

C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. S. J. Russell, “Numerical study of guided modes in arrays of metallic nanowires,” Opt. Lett. 32, 1647-1649 (2007).
[CrossRef] [PubMed]

Renversez, G.

Robinson, P. A.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

Russell, P. S. J.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys Rev B 77, 033417 (2007).
[CrossRef]

C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. S. J. Russell, “Numerical study of guided modes in arrays of metallic nanowires,” Opt. Lett. 32, 1647-1649 (2007).
[CrossRef] [PubMed]

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]

P. S. J. Russell, T. A. Birks, J. C. Knight, and B. J. Mangan, “Photonic crystal fibers,” US Patent 6,990,282 (2006).

Sazio, P. J. A.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Scheidemantel, T. J.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Schmidt, M. A.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys Rev B 77, 033417 (2007).
[CrossRef]

C. G. Poulton, M. A. Schmidt, G. J. Pearce, G. Kakarantzas, and P. S. J. Russell, “Numerical study of guided modes in arrays of metallic nanowires,” Opt. Lett. 32, 1647-1649 (2007).
[CrossRef] [PubMed]

Sempere, L. N. P.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys Rev B 77, 033417 (2007).
[CrossRef]

Skorobogatiy, M.

Smith, G.

Smith, G. H.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Academic, 1983).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), ninth Dover printing, tenth GPO printing ed.

Stone, J. M.

Takenaga, K.

Tyagi, H. K.

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys Rev B 77, 033417 (2007).
[CrossRef]

Wada, A.

Wang, A.

White, T. P.

Won, D. J.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Wu, T. L.

T. L. Wu and C. H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126-128 (2004).
[CrossRef]

Yariv, A.

Yeh, P.

Yeom, D. I.

Yu, Q. L.

C. T. Chan, Q. L.Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Zhang, F.

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett. (1)

T. L. Wu and C. H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126-128 (2004).
[CrossRef]

J. Lightwave Technol. (1)

J. Math. Phys. (1)

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808-7816 (2000).
[CrossRef]

J. Opt. A (1)

J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibers with high-index rods,” J. Opt. A 6, 798-804 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (3)

Nature (1)

J. C. Knight, “Photonic crystal fibers,” Nature 424, 847-851 (2003).
[CrossRef] [PubMed]

Opt. Express (10)

T. P. White, R. C. McPhedran, L. C. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721-732 (2001).
[CrossRef] [PubMed]

T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibers,” Opt. Express 14, 5688-5698 (2006).
[CrossRef] [PubMed]

J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14, 6291-6296 (2006).
[CrossRef] [PubMed]

T. A. Birks, G. J. Pearce, and D. D. M. Bird, “Approximate band structure calculation for photonic bandgap fibers,” Opt. Express 14, 9483-9490 (2006).
[CrossRef] [PubMed]

B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express 14, 10851-10864 (2006).
[CrossRef] [PubMed]

H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers. I: Theory,” Opt. Express 12, 3791-3805 (2004).
[CrossRef] [PubMed]

E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics, and sensing,” Opt. Express 15, 10231-10246 (2007).
[CrossRef] [PubMed]

A. Hochman and Y. Leviatan, “Efficient and spurious-free integral-equation-based optical waveguide mode solver,” Opt. Express 15, 14431-14453 (2007).
[CrossRef] [PubMed]

J. Hou, D. Bird, A. George, S. Maier, B. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibers,” Opt. Express 16, 5983-5990 (2008).
[CrossRef] [PubMed]

B. T. Kuhlmey, F. Luan, L. Fu, D. I. Yeom, B. J. Eggleton, A. Wang, and J. Knight, “Experimental reconstruction of bands in solid core photonic bandgap fibers using acoustic gratings,” Opt. Express 16, 13845-13856 (2008).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys Rev B (1)

M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys Rev B 77, 033417 (2007).
[CrossRef]

Phys. Rev. B (1)

C. T. Chan, Q. L.Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Phys. Rev. E (2)

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, “Green's function and lattice sums for electromagnetic scattering by a square array of cylinders,” Phys. Rev. E 49, 4590-4602 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

PIER (1)

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” PIER 41, 21-60 (2003), doi:10.2528/PIER02010802.
[CrossRef]

Science (1)

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, and E. R. Margine, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583-1586 (2006).
[CrossRef] [PubMed]

Other (6)

G. P. Agrawal and R. W. Boyd, Nonlinear Fiber Optics (Springer, 2001).

L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, From Multipole Methods to Photonic Crystal Device Modelling (CRC Press, 2005).

P. S. J. Russell, T. A. Birks, J. C. Knight, and B. J. Mangan, “Photonic crystal fibers,” US Patent 6,990,282 (2006).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Academic, 1983).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964), ninth Dover printing, tenth GPO printing ed.

B. Kuhlmey, “Theoretical and numerical investigation of the physics of microstructured optical fibers,” Ph.D. thesis, University of Sydney and Université Aix-Marseille III (2006). http://setis.library.usyd.edu.au/adt/public html/adt-NU/public/adt-NU20040715.171105/.

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Figures (6)

Fig. 1
Fig. 1

(a) Cross section of an example fiber whose cladding consists of an array of layered inclusions—in this case, dielectric rings representing materials with homogeneous refractive indices. (b) Many of the spectral properties of the fiber follow from those of the associated infinite periodic lattice.

Fig. 2
Fig. 2

(a) Infinite hexagonal lattice geometry, with coordinate system indicated. (b) Unit cell of hexagonal lattice of layered inclusions of pitch Λ representing different component materials. The index of the p th annulus is n p , with the interface between the p th and ( p + 1 ) th layers at ρ p .

Fig. 3
Fig. 3

Graded index profile of each cylinder in the hexagonal infinite lattice (black curve) with an example discretization into four layers.

Fig. 4
Fig. 4

Band edges for an infinite lattice of graded inclusions, generated by the scalar method of Section 4, for different numbers of approximating discretized layers N. Here n bg is the wavelength dependent background index of silica, obtained using a Sellmeier expansion [31].

Fig. 5
Fig. 5

Comparison of approximate band-edge finding algorithm (curves) with fully vectorial multipole method (black solid regions). Also shown are the results of a commercial plane-wave package (lines). White dots are the cladding modes of a finite fiber consisting of three layers of graded inclusions around a central missing inclusion. Gray dashed curves are core modes for this finite fiber.

Fig. 6
Fig. 6

Propagation diagrams for infinite hexagonal lattices, pitch Λ = 1 μ m , of metallic rings of varying thicknesses. Dots indicate the bound modes of single metallic rings in a silica background, which exist for n eff > n bg . Metal permittivity is ε m = 125.4 and background permittivity is 2.085.

Equations (42)

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E ( r , θ , z , t ) = E ( r , θ ) exp [ i ( β z ω t ) ] + c.c . ,
H ( r , θ , z , t ) = H ( r , θ ) exp [ i ( β z ω t ) ] + c.c. ,
V p ( r , θ ) = m = ( A m V , p J m ( k p r ) + B m V , p H m ( 1 ) ( k p r ) ) exp ( i m θ ) ,
k p = k 2 n p 2 β 2 .
v p ± 1 m = T p , p ± 1 m v p m .
v 1 m = ( T 2 , 1 m T 3 , 2 m T N + 1 , N m ) v N + 1 m .
= [ H m ( 1 ) ( k N + 1 r UC ) J m ( k N + 1 r UC ) , 1 ] T , for d Ψ d s = 0 ,
= [ H m ( 1 ) ( k N + 1 r UC ) J m ( k N + 1 r UC ) , 1 ] T , for Ψ = 0 ,
[ A 1 B 1 ] = T 2 , 1 2 × 2 T 3 , 2 2 × 2 T N + 1 , N 2 × 2 [ A N + 1 B N + 1 ] .
[ B m E , N + 1 B m H , N + 1 ] = S m [ A m E , N + 1 A m H , N + 1 ] .
[ A 1 E 0 A 1 H 0 ] = T 2 , 1 4 × 4 T 3 , 2 4 × 4 T N + 1 , N 4 × 4 [ A N + 1 E B N + 1 E A N + 1 H B N + 1 H ] T 4 × 4 [ A N + 1 E B N + 1 E A N + 1 H B N + 1 H ] .
[ I S K ] B P ( λ , k 0 , n eff ) B = 0 .
n ( r ) = { n silica ( 1 + Δ n GI ( 1 ( r r 0 ) α ) ) , if r < 0 n silica , if r r 0 } ,
r ¯ i = 1 Δ n n i n i + 1 r 2 ( n ) d n .
ψ p = [ A l p J l ( k p r ) + B l p H l ( k p r ) ] exp ( i l θ ) .
T i , i + 1 2 × 2 = 1 k i + 1 [ J l ( y ) H l ( 1 ) ( y ) J l ( y ) H l ( 1 ) ( y ) ] ( k i + 1 H l ( 1 ) ( y ) J l ( x ) k i H l ( 1 ) ( y ) J l ( x ) k i + 1 H l ( 1 ) ( y ) H l ( 1 ) ( x ) k i H l ( 1 ) ( y ) H l ( 1 ) ( x ) k i J l ( y ) J l ( x ) k i + 1 J l ( x ) J l ( y ) k i J l ( y ) H l ( 1 ) ( x ) k i + 1 H l ( 1 ) J l ( y ) ) ,
E θ = i β k 2 n p 2 β 2 ( r θ E z ω β r H z ) ,
H θ = i β k 2 n p 2 β 2 ( r θ H z + ω β r E z ) .
A p E J m ( k , i ρ ) + B p E H m ( k , i ρ ) = A i + 1 E J m ( k , i + 1 ρ ) + B i + 1 E H m ( k , i + 1 ρ ) .
1 ( k p ) 2 ( i m ρ [ A p E J m ( k p ρ ) + B p E H m ( k p ρ ) ] ω k p β [ A p H J m ( k p ρ ) + B p H H m ( k p ρ ) ] ) ,
= 1 ( k i + 1 ) 2 ( i m ρ [ A 2 E J m ( k i + 1 ρ ) + B 2 E H m ( k i + 1 ρ ) ] ω k i + 1 β [ A 2 H J m ( k i + 1 ρ ) + B 2 H H m ( k i + 1 ρ ) ] ) .
M ( i , ρ ) [ A p E B p E A p H B p H ] = M ( i + 1 , ρ ) [ A i + 1 E B i + 1 E A i + 1 H B i + 1 H ] .
T i , i + 1 4 × 4 = π y 2 ( t 11 t 12 t 13 t 14 t 21 t 22 t 23 t 24 t 31 t 32 t 33 t 34 t 41 t 42 t 43 t 44 ) ,
t 11 = J l ( x ) H l ( 1 ) ( y ) ( k i + 1 ε p k p ε i + 1 ) J l ( x ) H l ( 1 ) ( y ) ,
t 12 = H l ( 1 ) ( x ) H l ( 1 ) ( y ) ( k i + 1 ε p k p ε i + 1 ) H l ( 1 ) ( x ) H ( 1 ) ( y ) ,
t 13 = ( i β l ω ε i + 1 ) ( 1 y k i + 1 ( x k p ) ) J l ( x ) H l ( 1 ) ( y ) ,
t 14 = ( i β l ω ε i + 1 ) ( 1 y k i + 1 ( x k p ) ) H l ( 1 ) ( x ) H l ( 1 ) ( y ) ,
t 21 = ( k i + 1 ε p k p ε i + 1 ) J l ( x ) J l ( y ) J l ( x ) J l ( y ) ,
t 22 = H l ( 1 ) ( x ) H l ( 1 ) ( y ) ( k i + 1 ε p k p ε i + 1 ) H l ( 1 ) ( x ) H ( 1 ) ( y ) ,
t 23 = ( i β l ω ε i + 1 ) ( 1 y k i + 1 ( x k p ) ) J l ( x ) H l ( 1 ) ( y ) ,
t 24 = ( i β l ω ε i + 1 ) ( 1 y k i + 1 ( x k p ) ) H l ( 1 ) ( x ) H l ( 1 ) ( y ) ,
t 31 = ( i β l ω ) ( k i + 1 ( x k p ) 1 y ) J l ( x ) H l ( 1 ) ( y ) ,
t 32 = ( i β l ω ) ( k i + 1 ( x k p ) 1 y ) H l ( 1 ) ( x ) H l ( 1 ) ( y ) ,
t 33 = J l ( x ) H l ( 1 ) ( y ) ( k i + 1 k p ) J l ( x ) H l ( 1 ) ( y ) ,
t 34 = H ( 1 ) ( x ) H l ( 1 ) ( y ) ( k i + 1 k p ) H ( 1 ) ( x ) H l ( 1 ) ( y ) ,
t 41 = ( i β l ω ) ( 1 y k i + 1 ( x k p ) ) J l ( x ) J l ( y ) ,
t 42 = ( i β l ω ) ( 1 y k i + 1 ( x k p ) ) H l ( 1 ) ( x ) J l ( y ) ,
t 43 = ( k i + 1 k p ) J l ( x ) J l ( y ) J l ( x ) J l ( y ) ,
t 44 = ( k i + 1 k p ) H l ( 1 ) ( x ) J l ( y ) H l ( 1 ) ( x ) J l ( y ) .
T 21 A N + 1 E + T 22 B N + 1 E + T 23 A N + 1 H + T 24 B N + 1 H = 0 ,
T 41 A N + 1 E + T 42 B N + 1 E + T 43 A N + 1 H + T 44 B N + 1 H = 0 .
[ B E B H ] = 1 Δ [ T 24 T 41 T 44 T 21 T 24 T 43 T 44 T 23 T 42 T 21 T 22 T 41 T 42 T 23 T 22 T 43 ] [ A E A H ] S m [ A E A H ] ,

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