Abstract

We develop an efficient numerical method for computing defect modes in two dimensional photonic crystals based on the Dirichlet-to-Neumann (DtN) maps of the defect and normal unit cells. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative. The frequencies of the defect modes are solved from a condition that a small matrix is singular. The size of the matrix is related to the number of points used to discretize the boundary of the defect cell. The matrix is obtained by solving a linear system involving only discrete points on the edges of the unit cells in a truncated domain.

© 2007 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ. 1995.
  2. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
    [CrossRef] [PubMed]
  3. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
    [CrossRef] [PubMed]
  4. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
    [CrossRef]
  5. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
    [CrossRef]
  6. R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [CrossRef]
  7. X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics 36, L120–L123, (1997).
    [CrossRef]
  8. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  9. 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]
  10. D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
    [CrossRef]
  11. W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
    [CrossRef]
  12. H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
    [CrossRef]
  13. C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express 12, 1397–1408 (2004).
    [CrossRef] [PubMed]
  14. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
    [CrossRef]
  15. V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. 23, 1514–1521 (2005).
    [CrossRef]
  16. K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
    [CrossRef]
  17. K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics,  84, 1210–1214 (1998).
    [CrossRef]
  18. V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B 57, 15242–15250 (1998).
    [CrossRef]
  19. N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
    [CrossRef]
  20. S. P. Guo and S. Albin, “Numerical techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express 11, 1080–1089 (2003).
    [CrossRef] [PubMed]
  21. V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. 16, 816–818 (2004).
    [CrossRef]
  22. R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
    [CrossRef]
  23. J. H. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
    [CrossRef]
  24. J. H. Yuan and Y. Y. Lu, “Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice,” Opt. Commun. 273, 114–120 (2007).
    [CrossRef]
  25. Y. X. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol. 24, 3448–3453 (2006).
    [CrossRef]
  26. Y. H. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” Journal of Computational Mathematics 25, 337–349 (2007).
  27. S. J. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A 24, 2438–2442 (2007).
    [CrossRef]
  28. J. H. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” submitted for publication.
  29. Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications 12, 597–609 (1991).
    [CrossRef]
  30. T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. 21, 1793–1807 (2003).
    [CrossRef]
  31. L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057–2059 (2004).
    [CrossRef]
  32. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  33. K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express 15, 4753–4762 (2007).
    [CrossRef] [PubMed]

2007 (5)

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

J. H. 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. H. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” Journal of Computational Mathematics 25, 337–349 (2007).

K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express 15, 4753–4762 (2007).
[CrossRef] [PubMed]

S. J. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A 24, 2438–2442 (2007).
[CrossRef]

2006 (2)

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

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

2005 (2)

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. 23, 1514–1521 (2005).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

2004 (3)

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. 16, 816–818 (2004).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057–2059 (2004).
[CrossRef]

C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express 12, 1397–1408 (2004).
[CrossRef] [PubMed]

2003 (3)

S. P. Guo and S. Albin, “Numerical techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express 11, 1080–1089 (2003).
[CrossRef] [PubMed]

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. 21, 1793–1807 (2003).
[CrossRef]

R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
[CrossRef]

2001 (2)

N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
[CrossRef]

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]

1999 (2)

D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

1998 (2)

K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics,  84, 1210–1214 (1998).
[CrossRef]

V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B 57, 15242–15250 (1998).
[CrossRef]

1997 (2)

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics 36, L120–L123, (1997).
[CrossRef]

1996 (2)

R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

1994 (1)

1993 (1)

1991 (3)

Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications 12, 597–609 (1991).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Albin, S.

Antoine, X.

J. H. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” submitted for publication.

Aourag, H.

R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
[CrossRef]

Arakawa, Y.

X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics 36, L120–L123, (1997).
[CrossRef]

Asatryan, A. A.

Axmann, W.

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

Botten, L. C.

K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express 15, 4753–4762 (2007).
[CrossRef] [PubMed]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Brommer, K. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Chang, H. C.

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express 12, 1397–1408 (2004).
[CrossRef] [PubMed]

Chiang, P. J.

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Ctyroky, J.

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057–2059 (2004).
[CrossRef]

Dalichaouch, R.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

de Fornel, F.

R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
[CrossRef]

de Sterke, C. M.

K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express 15, 4753–4762 (2007).
[CrossRef] [PubMed]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Deng, Y.

N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
[CrossRef]

Dobson, D. C.

D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

Dossou, K. B.

Fan, S. H.

R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Felbacq, D.

Feng, X. P.

X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics 36, L120–L123, (1997).
[CrossRef]

Glimm, J.

N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
[CrossRef]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Guo, S. P.

Haus, J. W.

N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
[CrossRef]

Hernández-Figueroa, H. E.

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. 23, 1514–1521 (2005).
[CrossRef]

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. 16, 816–818 (2004).
[CrossRef]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Huang, Y. H.

Y. H. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” Journal of Computational Mathematics 25, 337–349 (2007).

Huang, Y. X.

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

Hubalek, M.

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057–2059 (2004).
[CrossRef]

Joannopoulos, J. D.

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]

R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

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

Johnson, S. G.

Koshiba, M.

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. 23, 1514–1521 (2005).
[CrossRef]

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. 16, 816–818 (2004).
[CrossRef]

Kroll, N.

Kuchment, P.

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

Kuzmiak, V.

V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B 57, 15242–15250 (1998).
[CrossRef]

Li, S. J.

Lu, T.

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. 21, 1793–1807 (2003).
[CrossRef]

Lu, Y. Y.

J. H. 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. H. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” Journal of Computational Mathematics 25, 337–349 (2007).

S. J. Li and Y. Y. Lu, “Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells,” J. Opt. Soc. Am. A 24, 2438–2442 (2007).
[CrossRef]

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

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

Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications 12, 597–609 (1991).
[CrossRef]

J. H. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” submitted for publication.

Maradudin, A. A.

V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B 57, 15242–15250 (1998).
[CrossRef]

Maystre, D.

McCall, S. L.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

McPhedran, R. C.

K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, “Gap-edge asymptotics of defect modes in two-dimensional photonic crystals,” Opt. Express 15, 4753–4762 (2007).
[CrossRef] [PubMed]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Meade, R. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

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

Moussa, R.

R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
[CrossRef]

Platzman, P. M.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Poulton, C. G.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Prkna, L.

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057–2059 (2004).
[CrossRef]

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Rodríguez-Esquerre, V. F.

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. 23, 1514–1521 (2005).
[CrossRef]

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. 16, 816–818 (2004).
[CrossRef]

Sakoda, K.

K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics,  84, 1210–1214 (1998).
[CrossRef]

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

Salomon, L.

R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
[CrossRef]

Schultz, S.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Shiroma, H.

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Smith, D. R.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Stojíc, N.

N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
[CrossRef]

Tayeb, G.

Villeneuve, R. R.

R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Wilcox, S.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Winn, J. N.

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

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Yang, H. Y. D.

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

Yau, S.-T.

Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications 12, 597–609 (1991).
[CrossRef]

Yevick, D.

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. 21, 1793–1807 (2003).
[CrossRef]

Yu, C. P.

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express 12, 1397–1408 (2004).
[CrossRef] [PubMed]

Yuan, J. H.

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

J. H. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” submitted for publication.

IEEE Photon. Technol. Lett. (2)

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett. 16, 2057–2059 (2004).
[CrossRef]

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element time-domain analysis of 2-D photonic crystal resonant cavities,” IEEE Photon. Technol. Lett. 16, 816–818 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

J. Comput. Phys. (2)

D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

J. Lightw. Technol. (3)

V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-element analysis of photonic crystal cavities: Time and frequency domains,” J. Lightw. Technol. 23, 1514–1521 (2005).
[CrossRef]

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

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightw. Technol. 21, 1793–1807 (2003).
[CrossRef]

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

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

Japanese Journal of Applied Physics (1)

X. P. Feng and Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Japanese Journal of Applied Physics 36, L120–L123, (1997).
[CrossRef]

Journal of Applied Physics (1)

K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” Journal of Applied Physics,  84, 1210–1214 (1998).
[CrossRef]

Journal of Computational Mathematics (1)

Y. H. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” Journal of Computational Mathematics 25, 337–349 (2007).

Opt. Commun. (1)

J. H. 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 (4)

Phys. Rev. B (3)

R. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

V. Kuzmiak and A. A. Maradudin, “Localized defect modes in a two-dimensional triangular photonic crystal,” Phys. Rev. B 57, 15242–15250 (1998).
[CrossRef]

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

Phys. Rev. E (3)

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[CrossRef]

N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, “Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals,” Phys. Rev. E 64, 056614 (2001).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave Propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Physica B - Condensed Matter (1)

R. Moussa, L. Salomon, F. de Fornel, and H. Aourag, “Numerical study on localized defect modes in two-dimensional lattices: a high Q-resonant cavity,” Physica B - Condensed Matter 338, 97–102 (2003).
[CrossRef]

SIAM Journal on Matrix Analysis and Applications (1)

Y. Y. Lu and S.-T. Yau, “Eigenvalues of the Laplacian through boundary integral equations,” SIAM Journal on Matrix Analysis and Applications 12, 597–609 (1991).
[CrossRef]

Other (2)

J. H. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” submitted for publication.

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

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Figures (10)

Fig. 1.
Fig. 1.

Point defect in a triangular lattice of dielectric rods. The defect cell contains a rod with a different radius and/or refractive index.

Fig. 2.
Fig. 2.

Parallelogram and hexagon supercells for a triangular lattice and m = 3.

Fig. 3.
Fig. 3.

Truncated domain with two rings around the defect cell for a triangular lattice.

Fig. 4.
Fig. 4.

Ordering of the edges and the sampling points on the edges of a hexagon unit cell for discretization of the DtN map.

Fig. 5.
Fig. 5.

Three neighboring hexagon unit cells and a possible ordering of the edges.

Fig. 6.
Fig. 6.

The smallest singular value s 1 of the matrix B in Eq. (10) versus the normalized frequency ωa/(2πc) for a triangular lattice of dielectric rods with one missing rod in the defect cell.

Fig. 7.
Fig. 7.

Relative errors of the defect mode frequency calculated using different values of N (the number of points on each edge of the unit cells).

Fig. 8.
Fig. 8.

Relative errors of the defect mode frequency calculated using different values of p (the number of rings around the defect cell).

Fig. 9.
Fig. 9.

Eigenfunction (z-component of the electric field) of the defect mode for a triangular lattice of dielectric rods with one missing rod (E polarization).

Fig. 10.
Fig. 10.

Normalized frequencies of the defect modes versus the ratio Rd/a between the radius of the defect rod and the lattice constant.

Equations (23)

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

ρ x ( 1 ρ u x ) + ρ y ( 1 ρ u y ) + k 0 2 n 2 u = 0 ,
n ( x ) = n ( x + l 1 a 1 + l 2 a 2 )
u ( x ) = e i k x Φ ( x )
B ( ω ) u | Γ d = 0
Λ u | Γ = u ν | Γ ,
u ( x ) = j = 1 K c j ϕ j ( x ) ,
[ u ( x 1 ) u ( x 2 ) u ( x K ) ] = Λ 1 [ c 1 c 2 c K ] ,
[ ν 1 u ( x 1 ) ν 2 u ( x 2 ) ν K u ( x K ) ] = Λ 2 [ c 1 c 2 c K ] ,
Λ = Λ 2 Λ 1 1 .
Λ [ u 1 u 2 u 3 u 4 u 5 u 6 ] = [ Λ 11 Λ 12 Λ 13 Λ 14 Λ 15 Λ 16 Λ 21 Λ 22 Λ 23 Λ 24 Λ 25 Λ 26 Λ 31 Λ 32 Λ 33 Λ 34 Λ 35 Λ 36 Λ 41 Λ 42 Λ 43 Λ 44 Λ 45 Λ 46 Λ 51 Λ 52 Λ 53 Λ 54 Λ 55 Λ 56 Λ 61 Λ 62 Λ 63 Λ 64 Λ 65 Λ 66 ] [ u 1 u 2 u 3 u 4 u 5 u 6 ] = [ ν u 1 ν u 2 ν u 3 ν u 4 ν u 5 ν u 6 ] .
u 1 ν = Λ 11 ( 1 ) u 1 + Λ 12 ( 1 ) u 2 + Λ 13 ( 1 ) u 3 + Λ 14 ( 1 ) u 4 + Λ 15 ( 1 ) u 5 + Λ 16 ( 1 ) u 6
= Λ 41 ( 2 ) u 9 + Λ 42 ( 2 ) u 10 + Λ 43 ( 2 ) u 11 + Λ 44 ( 2 ) u 1 + Λ 45 ( 2 ) u 7 + Λ 46 ( 2 ) u 8 .
u 2 ν = Λ 21 ( 1 ) u 1 + Λ 22 ( 1 ) u 2 + Λ 23 ( 1 ) u 3 + Λ 24 ( 1 ) u 4 + Λ 25 ( 1 ) u 5 + Λ 26 ( 1 ) u 6
= Λ 51 ( 3 ) u 12 + Λ 52 ( 3 ) u 13 + Λ 53 ( 3 ) u 14 + Λ 54 ( 3 ) u 15 + Λ 55 ( 3 ) u 2 + Λ 56 ( 3 ) u 11 .
U = [ u 1 ; u 2 ; ] = [ U d ; U n ; U f ] ,
A U = [ A 11 A 12 A 21 A 22 A 23 A 32 A 33 ] [ U d U n U f ] = 0 ,
[ A 22 A 23 A 32 A 33 ] [ D 1 D 2 ] = [ A 21 0 ] ,
U n = D 1 U d , U f = D 2 U d .
B U d = 0 for B = A 11 + A 12 D 1 .
s 1 ( B ) = 0 ,
ω 2 ( 1 ) = ω 1 ω 1 ω 0 s 1 ( ω 1 ) s 1 ( ω 0 ) s 1 ( ω 1 ) .
ω 2 ( 2 ) = ω 1 ω 1 ω 0 s 1 ( ω 1 ) + s 1 ( ω 0 ) s 1 ( ω 1 ) .
𝓔 N = ω ( N ) ω ( 16 ) ω ( 16 ) .

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