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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    [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]
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    [CrossRef]
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    [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," Jpn. J. Appl. Phys. 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]
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    [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]
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    [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 and 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. Rodr´ıguez-Esquerre, M. Koshiba and H. E. Hern´andez-Figueroa, "Finite-element analysis of photonic crystal cavities: Time and frequency domains," J. Lightwave Technol. 23, 1514-1521 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [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]
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    [CrossRef]
  22. R. Moussa, L. Salomon, F. de Fornel and H. Aourag, "Numerical study on localized defect modes in twodimensional lattices: a high Q-resonant cavity," Physica B 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. Lightwave 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," J. Comput. Math. 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, 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 J. Matrix Anal. Appl. 12, 597-609 (1991).
    [CrossRef]
  30. T. Lu and D. Yevick, "A vectorial boundary element method analysis of integrated optical waveguides," J. Lightwave 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

P. J. Chiang and 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," J. Comput. Math. 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

2005

V. F. Rodr´ıguez-Esquerre, M. Koshiba and H. E. Hern´andez-Figueroa, "Finite-element analysis of photonic crystal cavities: Time and frequency domains," J. Lightwave 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

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]

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]

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]

2003

2001

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

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

K. Sakoda, "Numerical study on localized defect modes in two-dimensional triangular photonic crystals," J. Appl. Phys. 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

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," Jpn. J. Appl. Phys. 36, L120-L123, (1997).
[CrossRef]

1996

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

1993

1991

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]

Y. Y. Lu and S.-T. Yau, "Eigenvalues of the Laplacian through boundary integral equations," SIAM J. Matrix Anal. Appl. 12, 597-609 (1991).
[CrossRef]

1990

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]

IEEE Photon. Technol. Lett.

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]

IEEE Trans. Microwave Theory Tech.

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

J. Appl. Phys.

K. Sakoda, "Numerical study on localized defect modes in two-dimensional triangular photonic crystals," J. Appl. Phys. 84, 1210-1214 (1998).
[CrossRef]

J. Comput. Math.

Y. H. 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.

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. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

X. P. Feng and Y. Arakawa, "Defect modes in two-dimensional triangular photonic crystals," Jpn. J. Appl. Phys. 36, L120-L123, (1997).
[CrossRef]

Opt. Commun.

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

Phys. Rev. B

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

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]

P. J. Chiang and 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]

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.

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

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

SIAM J. Matrix Anal. Appl.

Y. Y. Lu and S.-T. Yau, "Eigenvalues of the Laplacian through boundary integral equations," SIAM J. Matrix Anal. Appl. 12, 597-609 (1991).
[CrossRef]

Other

J. H. Yuan, Y. Y. Lu, 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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