Abstract

For finite two-dimensional photonic crystals given as periodic arrays of circular cylinders in a square or triangular lattice, we develop an efficient method to compute the transmission and reflection spectra for oblique incident plane waves. The method relies on vector cylindrical wave expansions to approximate the Dirichlet-to-Neumann (DtN) map for each distinct unit cell and uses the DtN maps to derive an efficient method that works on the edges of the unit cells only. The DtN operator maps the two longitudinal field components to their derivatives on the boundary of the unit cell.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  2. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).
  3. S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures,” SIAM J. Appl. Math. 60, 1686-1706 (2000).
    [CrossRef]
  4. D. Pissoort, E. Michielssen, F. Olyslager, and D. De Zutter, “Fast analysis of 2-D electromagnetic crystal devices using a periodic Green function approach,” J. Lightwave Technol. 23, 2294-2308 (2005).
    [CrossRef]
  5. E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 17, 320-327 (2000).
    [CrossRef] [PubMed]
  6. G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
    [CrossRef]
  7. G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
    [CrossRef]
  8. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  9. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
    [CrossRef]
  10. J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 23, 3217-3222 (2006).
    [CrossRef] [PubMed]
  11. J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114-120 (2007).
    [CrossRef]
  12. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860-2867 (2007).
    [CrossRef]
  13. S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454-14466 (2007).
    [CrossRef] [PubMed]
  14. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
    [CrossRef]
  15. Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).
  16. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008).
    [CrossRef]
  17. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383-17399 (2008).
    [CrossRef] [PubMed]
  18. J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617-4629 (2008).
    [CrossRef]
  19. L. Li, “A modal analysis of lamellar diffraction gratings in cornical mountings,” J. Mod. Opt. 40, 553-573 (1993).
    [CrossRef]
  20. S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
    [CrossRef]
  21. B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
    [CrossRef] [PubMed]
  22. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 779-784 (1996).
    [CrossRef]
  23. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 1024-1035 (1996).
    [CrossRef]
  24. E. Popov and B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926-4932 (2000).
    [CrossRef]
  25. J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
    [CrossRef]
  26. G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 22, 1106-1114 (2005).
    [CrossRef] [PubMed]
  27. A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
    [CrossRef]
  28. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 34-43 (1997).
    [CrossRef]
  29. E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47-55 (1999).
    [CrossRef]
  30. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

2008 (4)

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617-4629 (2008).
[CrossRef]

B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
[CrossRef] [PubMed]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383-17399 (2008).
[CrossRef] [PubMed]

2007 (5)

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454-14466 (2007).
[CrossRef] [PubMed]

Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860-2867 (2007).
[CrossRef]

S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114-120 (2007).
[CrossRef]

2006 (3)

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 23, 3217-3222 (2006).
[CrossRef] [PubMed]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

2005 (2)

D. Pissoort, E. Michielssen, F. Olyslager, and D. De Zutter, “Fast analysis of 2-D electromagnetic crystal devices using a periodic Green function approach,” J. Lightwave Technol. 23, 2294-2308 (2005).
[CrossRef]

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 22, 1106-1114 (2005).
[CrossRef] [PubMed]

2004 (2)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

2003 (1)

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

2002 (2)

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[CrossRef]

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

2000 (3)

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 17, 320-327 (2000).
[CrossRef] [PubMed]

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures,” SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

E. Popov and B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926-4932 (2000).
[CrossRef]

1999 (1)

1997 (1)

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 34-43 (1997).
[CrossRef]

1996 (2)

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 779-784 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 1024-1035 (1996).
[CrossRef]

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in cornical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

1991 (1)

A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

Antoine, X.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617-4629 (2008).
[CrossRef]

Asatryan, A. A.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Bao, G.

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 22, 1106-1114 (2005).
[CrossRef] [PubMed]

Botten, L. C.

S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[CrossRef]

Bozhkov, B.

Campbell, S.

S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

Centeno, E.

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 17, 320-327 (2000).
[CrossRef] [PubMed]

Chen, Z. M.

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 22, 1106-1114 (2005).
[CrossRef] [PubMed]

De Sterke, C. M.

S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

De Zutter, D.

Elschner, J.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

Enoch, S.

B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
[CrossRef] [PubMed]

Felbacq, D.

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 17, 320-327 (2000).
[CrossRef] [PubMed]

Gralak, B.

B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
[CrossRef] [PubMed]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Haider, M. A.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures,” SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

Hinder, R.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

Hoose, J.

Hu, Z.

Huang, Y.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

Lalanne, P.

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 779-784 (1996).
[CrossRef]

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Li, L.

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 1024-1035 (1996).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in cornical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

Li, S.

Lu, Y. Y.

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383-17399 (2008).
[CrossRef] [PubMed]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008).
[CrossRef]

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617-4629 (2008).
[CrossRef]

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454-14466 (2007).
[CrossRef] [PubMed]

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114-120 (2007).
[CrossRef]

Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860-2867 (2007).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 23, 3217-3222 (2006).
[CrossRef] [PubMed]

Mait, J. N.

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 34-43 (1997).
[CrossRef]

Maystre, D.

McPhedran, R. C.

S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Michielssen, E.

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 34-43 (1997).
[CrossRef]

Morris, G. M.

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 779-784 (1996).
[CrossRef]

Nicorovici, N. A.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[CrossRef]

Olyslager, F.

Papanicolaou, V.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures,” SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

Pierre, R.

B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
[CrossRef] [PubMed]

Pissoort, D.

Pomp, A.

A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

Popov, E.

Prather, D. W.

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 34-43 (1997).
[CrossRef]

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

Schmidt, G.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

Smith, G. H.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Tayeb, G.

B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
[CrossRef] [PubMed]

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

Venakides, S.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures,” SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

White, T. P.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wu, H. J.

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 22, 1106-1114 (2005).
[CrossRef] [PubMed]

Wu, Y.

Yasumoto, K.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

Yuan, J.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617-4629 (2008).
[CrossRef]

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114-120 (2007).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 23, 3217-3222 (2006).
[CrossRef] [PubMed]

Adv. Comput. Math. (1)

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

Appl. Opt. (2)

Comm. Comp. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

IEEE Trans. Antennas Propag. (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

J. Comput. Math. (1)

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

J. Comput. Phys. (1)

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617-4629 (2008).
[CrossRef]

J. Lightwave Technol. (2)

J. Mod. Opt. (2)

A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in cornical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

J. Opt. Soc. Am. A Opt. Image Sci. Vis. (7)

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 22, 1106-1114 (2005).
[CrossRef] [PubMed]

B. Gralak, R. Pierre, G. Tayeb, and S. Enoch, “Solutions of Maxwell's equations in presence of lamellar gratings including infinitely conducting metal,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 3099-3110 (2008).
[CrossRef] [PubMed]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 779-784 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 1024-1035 (1996).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 23, 3217-3222 (2006).
[CrossRef] [PubMed]

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 17, 320-327 (2000).
[CrossRef] [PubMed]

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 14, 34-43 (1997).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

J. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice,” Opt. Commun. 273, 114-120 (2007).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (3)

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

SIAM J. Appl. Math. (1)

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures,” SIAM J. Appl. Math. 60, 1686-1706 (2000).
[CrossRef]

Waves Random Complex Media (1)

S. Campbell, L. C. Botten, C. M. De Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

Other (2)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Five arrays of circular cylinders in a square or triangular lattice. The domain S covering one period of the structure is divided into cells Ω j for j = 1 , 2 , , 5 .

Fig. 2
Fig. 2

Square, hexagon, and special unit cells, each containing one circular cylinder.

Fig. 3
Fig. 3

Unit cells that contain two half-cylinders each.

Fig. 4
Fig. 4

Logarithm of transmittance T versus normalized wavelength λ / L and conicity angle θ 0 for nine arrays of dielectric cylinders in a triangular lattice.

Fig. 5
Fig. 5

Transmittance T versus normalized wavelength λ / L and conicity angle θ 0 for nine arrays of air holes in a triangular lattice.

Fig. 6
Fig. 6

Transmission spectra of nine interpenetrating arrays of dielectric cylinders in free space for oblique incident waves in the E (top) or H (bottom) polarizations (example 3).

Fig. 7
Fig. 7

Transmission spectra of seven interpenetrating arrays of air holes in a dielectric layer for oblique incident waves in the E (top) and H (bottom) polarizations (example 4).

Equations (50)

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

[ E x E y ] = i η [ γ 0 E z + k 0 μ J H ̃ z ] ,
[ H ̃ x H ̃ y ] = i η [ γ 0 H ̃ z k 0 ε J E z ] ,
η = k 0 2 ε μ γ 0 2 ,     J = [ 0 1 1 0 ] .
( ε η E z ) + ( γ 0 k 0 η J H ̃ z ) + ε E z = 0 ,
( μ η H ̃ z ) ( γ 0 k 0 η J E z ) + μ H ̃ z = 0 ,
( 1 μ E z ) + k 0 2 ε E z = 0 ,
( 1 ε H ̃ z ) + k 0 2 μ H ̃ z = 0.
Δ ϕ + η ϕ = 0 ,
ε η ν E z + γ 0 k 0 η τ H ̃ z ,     μ η ν H ̃ z γ 0 k 0 η τ E z
α 0 2 + [ β 0 ( t ) ] 2 = η t = k 0 2 ε t μ t γ 0 2 .
α 0 = k 0 ε t μ t sin   θ 0   cos   φ 0 ,
β 0 ( t ) = k 0 ε t μ t sin   θ 0   sin   φ 0 ,
γ 0 = k 0 ε t μ t cos   θ 0 .
u ( i ) ( r ) = A   exp [ i ( α 0 x β 0 ( t ) y ) ] ,     y > D ,
u ( x + L , y ) = exp ( i α 0 L ) u ( x , y ) .
u ( r ) ( r ) = j = B j   exp [ i ( α j x + β j ( t ) y ) ] ,     y > D ,
u ( t ) ( r ) = j = C j   exp [ i ( α j x β j ( b ) y ) ] ,     y < 0 ,
α j = α 0 + 2 π j / L ,     β j ( t ) = η t α j 2 ,     β j ( b ) = η b α j 2 ,
η b = k 0 2 ε b μ b γ 0 2 .
A = [ A ( e ) A ( h ) ] ,     B j = [ B j ( e ) B j ( h ) ] ,     C j = [ C j ( e ) C j ( h ) ] ,
β 0 ( t ) η t ( ε t | A ( e ) | 2 + μ t | A ( h ) | 2 ) = 1 ,
R j = Re ( β j ( t ) ) η t ( ε t | B j ( e ) | 2 + μ t | B j ( h ) | 2 ) ,
T j = Re ( β j ( b ) ) η b ( ε b | C j ( e ) | 2 + μ b | C j ( h ) | 2 ) .
u ( L , y ) = exp ( i α 0 L ) u ( 0 , y ) ,     x u ( L + , y ) = exp ( i α 0 L ) x u ( 0 + , y ) .
y u = S b u ,     y = 0 ,
y u = S t u 2 S t u ( i ) ,     y = D + .
S b b   exp ( i α j x ) = i β j ( b )   exp ( i α j x ) ,     j = 0 , ± 1 , ± 2 , .
S b b [ j = c j   exp ( i α j x ) ] = i j = c j β j ( b )   exp ( i α j x ) .
ϕ ( r ) = j = c j   exp [ i ( α j x β j ( b ) y ) ] ,     y < 0
S b = [ S b b 0 0 S b b ] .
Q j + u j = ν u j + ,     Q j u j = ν u j ,     Y j u j = u 0 ,
M [ u j 1 u j ] = [ M 11 M 12 M 21 M 22 ] [ u j 1 u j ] = [ ν u j 1 + ν u j ] ,
Q j = M 22 + M 21 ( Q j 1 + M 11 ) 1 M 12 ,
Y j = Y j 1 ( Q j 1 + M 11 ) 1 M 12 .
Q j + = [ t 1 0 0 t 2 ] Q j + [ 0 t 3 τ t 4 τ 0 ] ,
t 1 = ε η + ε + η ,     t 2 = μ η + μ + η ,     t 3 = γ 0 ( η + η ) k 0 η ε + ,
t 4 = γ 0 ( η + η ) k 0 η μ + .
x   exp ( i α j x ) = i α j   exp ( i α j x ) ,     j = 0 , ± 1 , ± 2 , ,
Λ [ u j 1 w 0 w 1 u j ] = [ Λ 11 Λ 12 Λ 13 Λ 14 Λ 21 Λ 22 Λ 23 Λ 24 Λ 31 Λ 32 Λ 33 Λ 34 Λ 41 Λ 42 Λ 43 Λ 44 ] [ u j 1 w 0 w 1 u j ] = [ ν u j 1 + x w 0 + x w 1 ν u j ] ,
u ( r ) = { j = a j J j ( ρ 1 r ) e i j θ , r < a j = [ b j J j ( ρ 2 r ) + c j H j ( 1 ) ( ρ 2 r ) ] e i j θ , r > a , }
ρ 1 = η 1 = k 0 2 ε 1 μ 1 γ 0 2 ,     ρ 2 = η 2 = k 0 2 ε 2 μ 2 γ 0 2 ,
J j ( ρ 1 a ) a j = J j ( ρ 2 a ) b j + H j ( 1 ) ( ρ 2 a ) c j .
F 1 a j = F 2 b j + F 3 c j ,
F 2 = [ ε 2 ρ 2 J j ( ρ 2 a ) i j γ 0 k 0 a ρ 2 2 J j ( ρ 2 a ) i j γ 0 k 0 a ρ 2 2 J j ( ρ 2 a ) μ 2 ρ 2 J j ( ρ 2 a ) ] ,
c j = T j b j ,
u ( r ) = j = Φ j ( r ) b j
Φ j ( r ) = e i j θ [ J j ( ρ 2 r ) I + H j ( 1 ) ( ρ 2 r ) T j ] ,     r > a ,
ν [ ϕ ( r ) e i j θ ] = ϕ ( r ) e i j θ ν [ cos   θ sin   θ ] + i j ϕ ( r ) e i j θ r ν [ sin   θ cos   θ ] .
u j 1 = [ E z ( r 1 ) , H ̃ z ( r 1 ) , E z ( r 2 ) , H ̃ z ( r 2 ) , ] T .
u j 1 = [ E z ( r 1 ) , E z ( r 2 ) , , H ̃ z ( r 1 ) , H ̃ z ( r 2 ) , ] T .

Metrics