Abstract

An efficient numerical method is developed for computing the transmission and reflection spectra of finite two-dimensional photonic crystals composed of circular cylinders in a triangular lattice. Our method manipulates a pair of operators defined on a set of curves, and it remains effective when the radius of the cylinders is larger than 34 of the lattice constant—a condition where different arrays of cylinders cannot be separated by planes without intersecting the cylinders. The method is efficient since it never calculates the wave field in the interiors of the (hexagon) unit cells and it approximates the operators by small matrices. This is achieved by using the Dirichlet-to-Neumann (DtN) maps of the unit cells, which map the wave field on the boundaries of the unit cells to its normal derivative.

© 2008 Optical Society of America

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References

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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, 1995).
  2. G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029-1042 (1995).
    [CrossRef]
  3. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  4. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  5. G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
    [CrossRef]
  6. 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]
  7. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
    [CrossRef]
  8. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
    [CrossRef]
  9. T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69-85 (2000).
    [CrossRef]
  10. 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]
  11. 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]
  12. K. Yasumoto, H. Jia, and H. Toyama, “Analysis of two-dimensional electromagnetic crystals consisting of multilayered periodic arrays of circular cylinders,” Electron. Commun. Jpn., Part 2: Electron. 88, 19-28 (2005).
    [CrossRef]
  13. 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]
  14. 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).
  15. 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]
  16. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
    [CrossRef]
  17. 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]
  18. K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672-4675 (1995).
    [CrossRef]
  19. 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]

2007

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 and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 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]

2006

2005

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

K. Yasumoto, H. Jia, and H. Toyama, “Analysis of two-dimensional electromagnetic crystals consisting of multilayered periodic arrays of circular cylinders,” Electron. Commun. Jpn., Part 2: Electron. 88, 19-28 (2005).
[CrossRef]

2004

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]

2001

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 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]

2000

T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

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]

1999

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

1996

1995

G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029-1042 (1995).
[CrossRef]

K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672-4675 (1995).
[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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Bao, G.

Botten, L. C.

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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Chen, Z. M.

Cox, J. A.

de Sterke, C. M.

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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Dobson, D. C.

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]

Huang, Y.

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).

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]

Jia, H.

K. Yasumoto, H. Jia, and H. Toyama, “Analysis of two-dimensional electromagnetic crystals consisting of multilayered periodic arrays of circular cylinders,” Electron. Commun. Jpn., Part 2: Electron. 88, 19-28 (2005).
[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]

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

Johnson, S. G.

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]

T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

Lalanne, P.

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.

Li, S. J.

Lu, Y. Y.

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]

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

McPhedran, R. C.

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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Meade, R. D.

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

Morris, G. M.

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

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]

Robinson, P. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165-2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Sakoda, K.

K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

Toyama, H.

K. Yasumoto, H. Jia, and H. Toyama, “Analysis of two-dimensional electromagnetic crystals consisting of multilayered periodic arrays of circular cylinders,” Electron. Commun. Jpn., Part 2: Electron. 88, 19-28 (2005).
[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]

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 University Press, 1995).

Wu, H. J.

Yasumoto, K.

K. Yasumoto, H. Jia, and H. Toyama, “Analysis of two-dimensional electromagnetic crystals consisting of multilayered periodic arrays of circular cylinders,” Electron. Commun. Jpn., Part 2: Electron. 88, 19-28 (2005).
[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]

T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

Yuan, J.

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]

Electron. Commun. Jpn., Part 2: Electron.

K. Yasumoto, H. Jia, and H. Toyama, “Analysis of two-dimensional electromagnetic crystals consisting of multilayered periodic arrays of circular cylinders,” Electron. Commun. Jpn., Part 2: Electron. 88, 19-28 (2005).
[CrossRef]

IEEE Trans. Antennas Propag.

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.

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. Lightwave Technol.

J. Opt. Soc. Am. A

Opt. Commun.

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

Phys. Rev. B

K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

Phys. Rev. E

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603, 2001.
[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]

Prog. Electromagn. Res.

T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69-85 (2000).
[CrossRef]

SIAM J. Appl. Math.

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]

Other

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

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Figures (12)

Fig. 1
Fig. 1

Finite 2D PhC with five interpenetrating arrays of circular cylinders. The refractive indices of the cylinders, the background medium, and the media above and below are n 1 , n 2 , n top , and n bot , respectively. The domain S covering one period in the horizontal direction is divided into five cells.

Fig. 2
Fig. 2

Hexagon unit cell for a triangular lattice (left) and polygon cells (middle and right) near the edges of a finite PhC, each containing one circular cylinder. A unit normal vector is also shown on each edge of the cells.

Fig. 3
Fig. 3

Shifted cells corresponding to the hexagon unit cell and the polygon cells in Fig. 2, each containing two half cylinders.

Fig. 4
Fig. 4

Shifted hexagon cell Ω j containing two half cylinders and the nearby regular hexagon cell Ω ̃ j . Here, Ω ̃ j is obtained by translating the left half of Ω j horizontally by the distance a.

Fig. 5
Fig. 5

Four layers of air holes in a dielectric medium with n = 2.72 . The structure is surrounded by air. The domain S covering one period of the structure in the horizontal direction is divided into six cells.

Fig. 6
Fig. 6

Transmission spectrum obtained using the improved DtN map method for 14 arrays of air holes in a dielectric medium. Here, T 0 is the transmission coefficient defined in Eq. (3).

Fig. 7
Fig. 7

Transmission spectrum of 15 layers of dielectric cylinders with radius r = 0.45 a for the E polarization. The bandgaps of the bulk PhC are shown as vertical stripes. The vertical dashed line indicates the starting point of the first partial gap obtained by assuming that the Bloch wave vector is restricted to the Γ M edge of the first Brillouin zone.

Fig. 8
Fig. 8

Relative errors of the reflection coefficient R 0 versus N for 15 layers of dielectric cylinders with radius r = 0.45 a and the E polarization. The frequencies are ω a ( 2 π c ) = 0.3 (left) and ω a ( 2 π c ) = 0.8 (right).

Fig. 9
Fig. 9

Transmission spectrum of 15 layers of dielectric cylinders with radius r = 0.5 a for the E polarization.

Fig. 10
Fig. 10

Zeroth order reflection coefficient R 0 versus N for 15 layers of cylinders and ω a ( 2 π c ) = 0.3 . Left, E polarization and r = 0.5 a ; right, H polarization and r = 0.49 a .

Fig. 11
Fig. 11

Reflection spectra of one layer of PEC cylinders for the E polarization and different values of the cylinder radius r.

Fig. 12
Fig. 12

Transmission spectrum of 15 layers of PEC cylinders in a triangular lattice for normal incident plane waves in the H polarization. The radius of the cylinders is r = 0.45 a .

Equations (31)

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

p x ( 1 p u x ) + p y ( 1 p u y ) + k 0 2 n 2 u = 0 ,
u ( i ) ( x , y ) = e i [ α 0 x β 0 ( y D ) ] ,
u ( r ) ( x , y ) = j = R j e i [ α j x + β j ( y D ) ] , y > D ,
u ( t ) ( x , y ) = j = T j e i ( α j x γ j y ) , y < 0 ,
α j = α 0 + 2 π j a , β j = k 0 2 n top 2 α j 2 , γ j = k 0 2 n bot 2 α j 2 .
u ( a , y ) = ρ u ( 0 , y ) , u x ( a , y ) = ρ u x ( 0 , y ) .
u y = i S ̃ bot u , y = 0 ,
u y = i S ̃ top u 2 i β 0 e i α 0 x , y = D + .
Q j u Γ j = u ν Γ j , Y j u Γ j = u Γ 0 .
Q 0 = i δ 0 S ̃ bot , Y 0 = I ,
1 n bot 2 u y y = 0 = i n bot 2 S ̃ bot u y = 0 = 1 n 2 2 u y y = 0 + = 1 n 2 2 Q 0 u y = 0 .
[ δ m Q m i S ̃ top ] u ( x , D ) = 2 i β 0 e i α 0 x ,
1 n 2 2 u y y = D = 1 n 2 2 Q m u y = D = 1 n top 2 u y y = D + = 1 n top 2 [ i S ̃ top u y = D 2 i β 0 e i α 0 x ] .
u ( r ) ( x , D + ) = u ( x , D ) u ( i ) ( x , D + ) = u ( x , D ) e i α 0 x .
u ( t ) ( x , 0 ) = u ( x , 0 ) = Y m u ( x , D ) .
M j [ 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 1 M 11 ) u j 1 = M 12 u j , M 21 u j 1 = ( Q j M 22 ) u j .
Q j = M 22 + M 21 ( Q j 1 M 11 ) 1 M 12 ,
Y j = Y j 1 ( Q j M 11 ) 1 M 12 .
Λ j [ 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 ( x ) k = 1 K c k ϕ k ( x ) , for x = ( x , y ) .
Λ j = B A 1 .
M j = [ Λ 11 Λ 14 Λ 41 Λ 44 ] + [ C 1 D 1 C 1 D 2 C 2 D 1 C 2 D 2 ] ,
C 1 = Λ 12 + ρ Λ 13 , C 2 = Λ 42 + ρ Λ 43 ,
D 0 = ρ Λ 22 + ρ 2 Λ 23 Λ 32 ρ Λ 33 ,
D 1 = D 0 1 ( Λ 31 ρ Λ 21 ) , D 2 = D 0 1 ( Λ 34 ρ Λ 24 ) .
u j = [ u j ( 1 ) u j ( 2 ) ] , u ̃ j = [ u j ( 2 ) ρ u j ( 1 ) ] = T j u j , T j = [ 0 I ρ I 0 ] ,
M ̃ j [ 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 ] .
M j = [ T j 1 T j ] 1 M ̃ j [ T j 1 T j ] .
M 11 = T j 1 1 M ̃ 11 T j 1 , M 12 = T j 1 1 M ̃ 12 T j ,
M 21 = T j 1 M ̃ 21 T j 1 , M 22 = T j 1 M ̃ 22 T j .

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