Abstract

An efficient and accurate computational method is developed for analyzing finite layers of crossed arrays of circular cylinders, including woodpile structures as special cases. The method relies on marching a few operators (approximated by matrices) from one side of the structure to another. The marching step makes use of the Dirichlet-to-Neumann (DtN) maps for two-dimensional unit cells in each layer where the structure is invariant in the direction of the cylinder axes. The DtN map is an operator that maps two wave field components to their normal derivatives on the boundary of the unit cell, and they can be easily constructed by vector cylindrical waves. Unlike existing numerical methods for crossed gratings, our method does not require a discretization of the structure. Compared with the multipole method that uses vector cylindrical wave expansions and scattering matrices, our method is relatively simple since it does not need sophisticated lattice sums techniques.

© 2009 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: Modeling the Flow of Light (Princeton Univ. Press, 1995).
  2. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
    [CrossRef]
  3. H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231-239 (1994).
    [CrossRef]
  4. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
    [CrossRef]
  5. 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]
  6. 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]
  7. D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
    [CrossRef]
  8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time Domain Method, 2nd ed. (Artech House, 2000).
  9. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  10. E. Popov and M. Nevière, “Maxwell equations in Fourier space: a fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886-2894 (2001).
    [CrossRef]
  11. M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).
  12. J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
    [CrossRef]
  13. 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]
  14. G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).
  15. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  16. 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]
  17. K. Yasumoto and H. Jia, “Electromagnetic scattering from multilayered crossed-arrays of circular cylinders,” SPIE 5445, 200-205 (2004).
  18. 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]
  19. J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217-3222 (2006).
    [CrossRef]
  20. S. 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]
  21. J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
    [CrossRef]
  22. H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26, 1606-1614 (2009).
    [CrossRef]
  23. 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]
  24. 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]
  25. S. Li and Y. Y. Lu, “Computing photonic crystal defect modes by Dirichlet-to-Neumann maps,” Opt. Express 15, 14454-14466 (2007).
    [CrossRef] [PubMed]
  26. 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).
  27. 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]
  28. 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]
  29. Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921-932 (2008).
    [CrossRef]
  30. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442-1449 (2009).
    [CrossRef]

2010

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).

2009

2008

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]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921-932 (2008).
[CrossRef]

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

2007

2006

2005

2003

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

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

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

2001

2000

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

1998

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

1997

1994

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231-239 (1994).
[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]

Antoine, X.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

Bao, G.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).

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]

Biswas, R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Botten, L. C.

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]

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

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]

Chen, Z. M.

Dobson, D. C.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Dowling, J. P.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231-239 (1994).
[CrossRef]

Elschner, J.

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

Fleming, J. G.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Gopalakrishnan, J.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Gralak, B.

Hagness, S. C.

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

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[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]

Ho, K. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

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]

Hu, Z.

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]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921-932 (2008).
[CrossRef]

Huang, Y.

Jia, H.

K. Yasumoto and H. Jia, “Electromagnetic scattering from multilayered crossed-arrays of circular cylinders,” SPIE 5445, 200-205 (2004).

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: Modeling the Flow of Light (Princeton Univ. Press, 1995).

Johnson, S. G.

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Li, L.

Li, P.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).

Li, S.

Lin, S. Y.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Lu, Y. Y.

H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26, 1606-1614 (2009).
[CrossRef]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442-1449 (2009).
[CrossRef]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921-932 (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]

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

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

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]

Meade, R. D.

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

Nevière, M.

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]

Pasciak, J. E.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Popov, E.

Schmidt, G.

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

Sigalas, M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Sigalas, M. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[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]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

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]

Sözüer, H. S.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231-239 (1994).
[CrossRef]

Taflove, A.

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

Tayeb, G.

Winn, J. N.

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

Wu, H.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).

Wu, H. J.

Wu, Y.

Xie, H.

Yasumoto, K.

K. Yasumoto and H. Jia, “Electromagnetic scattering from multilayered crossed-arrays of circular cylinders,” SPIE 5445, 200-205 (2004).

Yuan, J.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 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 23, 3217-3222 (2006).
[CrossRef]

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Adv. Comput. Math.

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

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equation and Dirichlet-to-Neumann maps,” J. Comput. Phys. 9, 4617-4629 (2008).
[CrossRef]

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, “An efficient method for band structure calculations in 3D photonic crystals,” J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231-239 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Math. Comput.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1-34 (2010).

Nature

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

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

Opt. Quantum Electron.

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921-932 (2008).
[CrossRef]

Phys. Rev. E

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]

Phys. Rev. Lett.

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]

Solid State Commun.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Other

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

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).

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

K. Yasumoto and H. Jia, “Electromagnetic scattering from multilayered crossed-arrays of circular cylinders,” SPIE 5445, 200-205 (2004).

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Figures (4)

Fig. 1
Fig. 1

Woodpile structure as special crossed arrays of cylinders.

Fig. 2
Fig. 2

Transmission spectra of a four-layer regular crossed array structure (top plot) and a four-layer woodpile structure (bottom plot).

Fig. 3
Fig. 3

Reflection spectra of a 32-layer regular crossed array structure (top plot) and a 32-layer woodpile structure (bottom plot).

Fig. 4
Fig. 4

Relative errors of the power reflectance R 00 for regular crossed arrays (two left plots) and woodpile structures (two right plots).

Equations (86)

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× E = i k 0 μ H ̃ , × H ̃ = i k 0 ε E ,
z [ u v ] = [ A 11 A 12 A 21 A 22 ] [ u v ] ,
u = [ E x H ̃ x ] , v = [ E y H ̃ y ] ,
A 11 = 1 i k 0 [ 0 x ( ε 1 y ) x ( μ 1 y ) 0 ] ,
A 12 = 1 i k 0 [ 0 k 0 2 μ x ( ε 1 x ) k 0 2 ε + x ( μ 1 x ) 0 ] ,
A 21 = 1 i k 0 [ 0 k 0 2 μ + y ( ε 1 y ) k 0 2 ε y ( μ 1 y ) 0 ] ,
A 22 = 1 i k 0 [ 0 y ( ε 1 x ) y ( μ 1 x ) 0 ] .
[ u ( i ) v ( i ) ] = [ a ( i ) b ( i ) ] exp [ i ( α 0 x + β 0 y γ 00 ( 1 ) z ) ] ,
γ 00 ( 1 ) = k 0 2 ε ( 1 ) μ ( 1 ) α 0 2 β 0 2 > 0 .
a ( i ) + B 00 ( i ) b ( i ) = 0 ,
B 00 ( i ) = 1 α 0 2 + ( γ 00 ( 1 ) ) 2 [ α 0 β 0 k 0 μ ( 1 ) γ 00 ( 1 ) k 0 ε ( 1 ) γ 00 ( 1 ) α 0 β 0 ] .
( B 00 ( i ) ) 1 = 1 β 0 2 + ( γ 00 ( 1 ) ) 2 [ α 0 β 0 k 0 μ ( 1 ) γ 00 ( 1 ) k 0 ε ( 1 ) γ 00 ( 1 ) α 0 β 0 ] .
[ u ( r ) v ( r ) ] = exp [ i ( α 0 x + β 0 y ) ] [ Φ ( r ) Ψ ( r ) ] ,
[ u ( r ) ( r ) v ( r ) ( r ) ] = j , k = [ a j k ( r ) b j k ( r ) ] exp [ i ( α j x + β k y + γ j k ( 1 ) z ) ]
[ u ( t ) ( r ) v ( t ) ( r ) ] = j , k = [ a j k ( t ) b j k ( t ) ] exp [ i ( α j x + β k y γ j k ( 2 ) z ) ]
α j = α 0 + 2 π j L , β k = β 0 + 2 π k L ,
γ j k ( l ) = k 0 2 ε ( l ) μ ( l ) α j 2 β k 2 , l = 1 , 2 .
a j k ( r ) + B j k ( r ) b j k ( r ) = 0 ,
a j k ( t ) + B j k ( t ) b j k ( t ) = 0 ,
B j k ( r ) = 1 α j 2 + ( γ j k ( 1 ) ) 2 [ α j β k k 0 μ ( 1 ) γ j k ( 1 ) k 0 ε ( 1 ) γ j k ( 1 ) α j β k ] ,
B j k ( t ) = 1 α j 2 + ( γ j k ( 2 ) ) 2 [ α j β k k 0 μ ( 2 ) γ j k ( 2 ) k 0 ε ( 2 ) γ j k ( 2 ) α j β k ] .
Ω = { ( x , y , z ) | 0 < x < L , 0 < y < L , 0 < z < D } .
[ u ( L , y , z ) v ( L , y , z ) ] = e i α 0 L [ u ( 0 , y , z ) v ( 0 , y , z ) ] ,
[ u ( x , L , z ) v ( x , L , z ) ] = e i β 0 L [ u ( x , 0 , z ) v ( x , 0 , z ) ] .
B ( r ) { e i ( α j x + β k y ) e l } = B j k ( r ) { e i ( α j x + β k y ) e l } ,
B ( t ) { e i ( α j x + β k y ) e l } = B j k ( t ) { e i ( α j x + β k y ) e l } ,
u ( r ) + B ( r ) v ( r ) = 0 , z > D ,
u ( t ) + B ( t ) v ( t ) = 0 , z < 0 .
u + B ( r ) v = u ( i ) + B ( r ) v ( i ) , z > D .
u + B ( r ) v = [ u ( i ) + B ( r ) v ( i ) ] z = D + , z = D ,
u + B ( t ) v = 0 , z = 0 .
| P ( z m ± ) u | z = z m = | u z | z = z m ± ,
| X ( z m ) u | z = z m = | u | z = z 0 ,
| Q ( z m ± ) v | z = z m = | v z | z = z m ± ,
| Y ( z m ) v | z = z m = | v | z = z 0 ,
T { e i ( α j x + β k y ) e s } = l , m = T l m j k { e i ( α l x + β m y ) e s }
g ( x , y ) = j , k = g ̂ j k e i ( α j x + β k y ) ,
h ( x , y ) = j , k = h ̂ j k e i ( α j x + β k y ) ,
j , k T l m j k g ̂ j k = h ̂ l m .
g ̂ = [ g ̂ 1 g ̂ 0 g ̂ 1 ] , for g ̂ k = [ g ̂ 1 , k g ̂ 0 k g ̂ 1 k ] .
g ̃ = [ g ̃ 1 g ̃ 0 g ̃ 1 ] , where g ̃ j = [ g ̂ j , 1 g ̂ j 0 g ̂ j 1 ] .
g ̂ = S g ̃ , T ̂ = S T ̃ S 1 .
A 11 { e i ( α j x + β k y ) e s } = α j β k i k 0 [ 0 ε 1 μ 1 0 ] { e i ( α j x + β k y ) e s }
( A 12 ) j k j k = k 0 2 ε μ α j 2 i k 0 [ 0 ε 1 μ 1 0 ] ,
( A 21 ) j k j k = k 0 2 ε μ β k 2 i k 0 [ 0 ε 1 μ 1 0 ] ,
( A 22 ) j k j k = α j β k i k 0 [ 0 ε 1 μ 1 0 ] = ( A 11 ) j k j k .
v ( r ) = j , k = b j k ( t ) exp [ i ( α j x + β k y γ j k ( 2 ) z ) ]
v z ( r ) = i j , k = γ j k ( 2 ) I 2 b j k ( t ) exp [ i ( α j x + β k y γ j k ( 2 ) z ) ]
[ S ( 2 ) ] l m j k = { i γ j k ( 2 ) I 2 , if ( l , m ) = ( j , k ) , 0 , if ( l , m ) ( j , k ) , }
v z = S ( 2 ) v , z < z 0 .
v ( i ) z = S ( 1 ) v ( 1 ) , v ( r ) z = S ( 1 ) v ( r ) , z > z M .
v z + S ( 1 ) v = 2 S ( 1 ) v ( i ) , z > z M .
[ Q ( z M + ) + S ( 1 ) ] | v | z = z M = | 2 S ( 1 ) v ( i ) | z = z M + .
[ Q ̂ ( z M + ) + S ̂ ( 1 ) ] v ̂ ( z M ) = 2 S ̂ ( 1 ) v ̂ ( i ) ( z M + ) .
Y ̂ ( z M ) v ̂ ( z M ) = v ̂ ( z 0 ) .
Q ( z m ± ) v = A 21 ( z m ± ) u + A 22 ( z m ± ) v ,
Q ( z m + ) = A 21 ( z m + ) A 21 1 ( z m ) [ Q ( z m ) A 22 ( z m ) ] + A 22 ( z m + ) .
G m { e i ( α j x + β k y ) e s } = k 0 2 ε + μ + β k 2 k 0 2 ε μ β k 2 [ ε ε + 0 0 μ μ + ] { e i ( α j x + β k y ) e s }
Q ̂ ( z m + ) = G ̂ m [ Q ̂ ( z m ) A ̂ 22 ( z m ) ] + A ̂ 22 ( z m + ) .
v ( r ) = k v k ( r ) = k Ψ k ( x , z ) e i β k y
Ω 1 = { ( x , z ) | 0 < x < L , z 0 < z < z 1 } .
Λ k v k = v k ν on Ω 1 ,
M k [ v k ( 0 ) v k ( 1 ) ] = [ z v k ( 0 + ) z v k ( 1 ) ] .
M ̂ k [ v ̂ k ( 0 ) v ̂ k ( 1 ) ] = [ z v ̂ k ( 0 + ) z v ̂ k ( 1 ) ] ,
g ( x l ) = j e i α j x l g ̂ j , 1 l N ,
M k = [ M k , 11 M k , 12 M k , 21 M k , 22 ] , M ̂ k = [ M ̂ k , 11 M ̂ k , 12 M ̂ k , 21 M ̂ k , 22 ] ,
M ̂ k , p q = F 1 M k , p q F
[ z v ̂ ( z 0 + ) z v ̂ ( z 1 ) ] = M ̂ [ v ̂ ( z 0 ) v ̂ ( z 1 ) ] = [ M ̂ 11 M ̂ 12 M ̂ 21 M ̂ 22 ] [ v ̂ ( z 0 ) v ̂ ( z 1 ) ] ,
M ̂ p q = [ M ̂ 1 , p q M ̂ 0 , p q M ̂ 1 , p q ]
Q ̂ ( z m ± ) v ̂ ( z m ) = z v ̂ ( z m ± ) , Y ̂ ( z m ) v ̂ ( z m ) = v ̂ ( z 0 ) .
[ Q ̂ ( z 0 + ) 0 0 Q ̂ ( z 1 ) ] [ v ̂ ( z 0 ) v ̂ ( z 1 ) ] = [ M ̂ 11 M ̂ 12 M ̂ 21 M ̂ 22 ] [ v ̂ ( z 0 ) v ̂ ( z 1 ) ] .
Q ̂ ( z 1 ) = M ̂ 22 + M ̂ 21 [ Q ̂ ( z 0 + ) M ̂ 11 ] 1 M ̂ 12 ,
Y ̂ ( z 1 ) = Y ̂ ( z 0 ) [ Q ̂ ( z 0 + ) M ̂ 11 ] 1 M ̂ 12 .
Ω 3 = { ( x , z ) | 0 < x < L , z 2 < z < z 3 }
Ω 3 = { ( x , z ) | L 2 < x < 3 L 2 , z 2 < z < z 3 } .
x l = L 2 + x l = L 2 + ( l 0.5 ) L N , 1 l N .
[ P 0 0 Q ] [ u v ] = [ A 11 A 12 A 21 A 22 ] [ u v ] .
v = ( Q A 22 ) 1 A 21 u ,
P = A 11 + A 12 ( Q A 22 ) 1 A 21 .
P ̂ = A ̂ 11 + A ̂ 12 ( Q ̂ A ̂ 22 ) 1 A ̂ 21 .
| W ( z m ) u | z = z m = | v | z = z 0 ,
X = B ( t ) W ,
W = Y ( Q A 22 ) 1 A 21 .
W ̂ = Y ̂ ( Q ̂ A ̂ 22 ) 1 A ̂ 21 .
a ( i ) = [ 1 0 ] , b ( i ) = [ 0 1 ] ,
R 00 = a 00 ( r ) 2 = b 00 ( r ) 2 , T 00 = a 00 ( t ) 2 = b 00 ( t ) 2 ,

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