Abstract

The rigorous coupled-wave method formulated by N. Chateau and J. P. Hugonin [J. Opt. Soc. Am. A 11, 1321 (1994)] and revisited by S. Peng and G. M. Morris [J. Opt. Soc. Am. A 12, 1087 (1995)] for one-dimensional (1D) diffraction gratings is used for the modeling of diffraction properties of photonic bandgap (PBG) structures. A two-dimensional (2D)-PBG structure is considered as a stack of 1D gratings. An original S-matrix algorithm is formulated for the modeling of any 1D grating, formed by rods that have a symmetry plane in the grating plane. Many examples—dealing with stacks of infinite rods of square (circular) sections, whose intersection with a perpendicular plane forms square, triangular, or hexagonal lattices—are studied. Particular attention is devoted to TM polarization in lossless (lossy) dielectric and metallic materials. For this polarization we take advantage of the convergence improvement formulated for 1D metallic gratings by P. Lalanne and G. M. Morris [J. Opt. Soc. Am. A 13, 779 (1996)] and G. Granet and R. Guizal [J. Opt. Soc. Am. A 13, 1019 (1996)]. The introduction of a periodic defect—made of dielectric material that has linear (nonlinear) optical properties—in a 2D-PBG structure and the feasibility of optical filters and switches in the 1.3–1.55 µm wavelength range are briefly studied. Limitations for the use of the modeling tool are illustrated through an example of a cubic three-dimensional (3D)-PBG structure of cubes.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. T. H. Krauss, R. M. De La Rue, “Optical characterization of waveguide based photonic microstructure,” Appl. Phys. Lett. 68, 1613–1615 (1996).
    [CrossRef]
  4. D. Cassagne, C. Jouanin, D. Bertho, “Optical properties of two-dimensional photonic crystals with graphite structure,” Appl. Phys. Lett. 70, 289–291 (1997).
    [CrossRef]
  5. D. Cassagne, C. Jouanin, D. Bertho, “Photonic band gaps in a two-dimensional graphite structure,” Phys. Rev. B 52, R2217–R2230 (1995).
    [CrossRef]
  6. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
    [CrossRef] [PubMed]
  7. S. Y. Lin, V. M. Hiettala, S. K. Lyo, “Photonic band gap quantum well and quantum box structures: a high-Q resonant cavity,” Appl. Phys. Lett. 68, 3233–3235 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. G. Granet, R. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  28. V. Kuzmiak, A. A. Maradudin, F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic elements,” Phys. Rev. B 50, 16835–16844 (1994).
    [CrossRef]
  29. P. Dansas, N. Paraire, S. Laval, “Feasibility of optical filters and switches using plastic photonic band gap structures,” in Precision Plastic Optics for Optical Storage, Displays, Imaging and Communications, W. F. Frank, ed., Proc. SPIE3135, pp. 219–229 (1997).
    [CrossRef]
  30. S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  31. E. Neoponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  32. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  33. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  34. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]

1997 (3)

1996 (10)

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

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]

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

G. Granet, R. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
[CrossRef]

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

T. H. Krauss, R. M. De La Rue, “Optical characterization of waveguide based photonic microstructure,” Appl. Phys. Lett. 68, 1613–1615 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
[CrossRef] [PubMed]

S. Y. Lin, V. M. Hiettala, S. K. Lyo, “Photonic band gap quantum well and quantum box structures: a high-Q resonant cavity,” Appl. Phys. Lett. 68, 3233–3235 (1996).
[CrossRef]

1995 (6)

D. Cassagne, C. Jouanin, D. Bertho, “Photonic band gaps in a two-dimensional graphite structure,” Phys. Rev. B 52, R2217–R2230 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

K. Sakoda, “Transmittance and Bragg reflectivity of two-dimensional photonic lattices,” Phys. Rev. B 52, 8992–9002 (1995).
[CrossRef]

P. Dansas, N. Paraire, F. Lederer, “Fast modeling of light beam diffraction by multilayer structures including a grating coupler,” Pure Appl. Opt. 4, 139–160 (1995).
[CrossRef]

J. M. Elson, P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

1994 (8)

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, K. M. Ho. “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

P. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic elements,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

E. Neoponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

D. Maystre, “Electromagnetic theory of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

J. B. Pendry, “Photonic structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

P. R. Villeneuve, M. Piché, “Photonic bandgaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
[CrossRef]

P. R. Villeneuve, M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structure?” J. Mod. Phys. 41, 241–256 (1994).

1993 (3)

M. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

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

1992 (1)

J. B. Pendry, A. Mac Kinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Bertho, D.

D. Cassagne, C. Jouanin, D. Bertho, “Optical properties of two-dimensional photonic crystals with graphite structure,” Appl. Phys. Lett. 70, 289–291 (1997).
[CrossRef]

D. Cassagne, C. Jouanin, D. Bertho, “Photonic band gaps in a two-dimensional graphite structure,” Phys. Rev. B 52, R2217–R2230 (1995).
[CrossRef]

Brand, S.

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Cassagne, D.

D. Cassagne, C. Jouanin, D. Bertho, “Optical properties of two-dimensional photonic crystals with graphite structure,” Appl. Phys. Lett. 70, 289–291 (1997).
[CrossRef]

D. Cassagne, C. Jouanin, D. Bertho, “Photonic band gaps in a two-dimensional graphite structure,” Phys. Rev. B 52, R2217–R2230 (1995).
[CrossRef]

Chan, C. T.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, K. M. Ho. “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Chateau, P.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
[CrossRef] [PubMed]

Dalichaouch, R.

Dansas, P.

P. Dansas, N. Paraire, F. Lederer, “Fast modeling of light beam diffraction by multilayer structures including a grating coupler,” Pure Appl. Opt. 4, 139–160 (1995).
[CrossRef]

P. Dansas, N. Paraire, S. Laval, “Feasibility of optical filters and switches using plastic photonic band gap structures,” in Precision Plastic Optics for Optical Storage, Displays, Imaging and Communications, W. F. Frank, ed., Proc. SPIE3135, pp. 219–229 (1997).
[CrossRef]

De La Rue, R. M.

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

T. H. Krauss, R. M. De La Rue, “Optical characterization of waveguide based photonic microstructure,” Appl. Phys. Lett. 68, 1613–1615 (1996).
[CrossRef]

Economou, E. N.

M. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Elson, J. M.

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
[CrossRef] [PubMed]

Granet, G.

Grann, E. B.

Guizal, R.

Hiettala, V. M.

S. Y. Lin, V. M. Hiettala, S. K. Lyo, “Photonic band gap quantum well and quantum box structures: a high-Q resonant cavity,” Appl. Phys. Lett. 68, 3233–3235 (1996).
[CrossRef]

Ho, K. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, K. M. Ho. “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Hugonin, J. P.

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

See, for instance, J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Princeton U. Press, Princeton, N.J., 1995 and special issues on photonic bandgap structures. J. Mod. Opt. 41, 2 (1994) and J. Opt. Soc. Am. B 10, 283–413 (1993).

Jouanin, C.

D. Cassagne, C. Jouanin, D. Bertho, “Optical properties of two-dimensional photonic crystals with graphite structure,” Appl. Phys. Lett. 70, 289–291 (1997).
[CrossRef]

D. Cassagne, C. Jouanin, D. Bertho, “Photonic band gaps in a two-dimensional graphite structure,” Phys. Rev. B 52, R2217–R2230 (1995).
[CrossRef]

Krauss, T. H.

T. H. Krauss, R. M. De La Rue, “Optical characterization of waveguide based photonic microstructure,” Appl. Phys. Lett. 68, 1613–1615 (1996).
[CrossRef]

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Kroll, N.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
[CrossRef] [PubMed]

Kuzmiak, V.

V. Kuzmiak, A. A. Maradudin, F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic elements,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Lalanne, P.

Laval, S.

P. Dansas, N. Paraire, S. Laval, “Feasibility of optical filters and switches using plastic photonic band gap structures,” in Precision Plastic Optics for Optical Storage, Displays, Imaging and Communications, W. F. Frank, ed., Proc. SPIE3135, pp. 219–229 (1997).
[CrossRef]

Lederer, F.

P. Dansas, N. Paraire, F. Lederer, “Fast modeling of light beam diffraction by multilayer structures including a grating coupler,” Pure Appl. Opt. 4, 139–160 (1995).
[CrossRef]

Li, L.

Lin, S. Y.

S. Y. Lin, V. M. Hiettala, S. K. Lyo, “Photonic band gap quantum well and quantum box structures: a high-Q resonant cavity,” Appl. Phys. Lett. 68, 3233–3235 (1996).
[CrossRef]

Lyo, S. K.

S. Y. Lin, V. M. Hiettala, S. K. Lyo, “Photonic band gap quantum well and quantum box structures: a high-Q resonant cavity,” Appl. Phys. Lett. 68, 3233–3235 (1996).
[CrossRef]

Mac Call, S. L.

Mac Kinnon, A.

J. B. Pendry, A. Mac Kinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Maradudin, A. A.

V. Kuzmiak, A. A. Maradudin, F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic elements,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Maystre, D.

D. Maystre, “Electromagnetic theory of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

See, for instance, J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals, Princeton U. Press, Princeton, N.J., 1995 and special issues on photonic bandgap structures. J. Mod. Opt. 41, 2 (1994) and J. Opt. Soc. Am. B 10, 283–413 (1993).

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996);see also Ref. 7 herein.
[CrossRef] [PubMed]

Moharam, M. G.

Morris, G. M.

Neoponen, E.

Paraire, N.

P. Dansas, N. Paraire, F. Lederer, “Fast modeling of light beam diffraction by multilayer structures including a grating coupler,” Pure Appl. Opt. 4, 139–160 (1995).
[CrossRef]

P. Dansas, N. Paraire, S. Laval, “Feasibility of optical filters and switches using plastic photonic band gap structures,” in Precision Plastic Optics for Optical Storage, Displays, Imaging and Communications, W. F. Frank, ed., Proc. SPIE3135, pp. 219–229 (1997).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Photonic structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

J. B. Pendry, A. Mac Kinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Peng, S.

Piché, M.

P. R. Villeneuve, M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structure?” J. Mod. Phys. 41, 241–256 (1994).

P. R. Villeneuve, M. Piché, “Photonic bandgaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
[CrossRef]

Pincemin, F.

V. Kuzmiak, A. A. Maradudin, F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic elements,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Platzman, P. M.

Pommet, D. A.

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Sakoda, K.

K. Sakoda, “Transmittance and Bragg reflectivity of two-dimensional photonic lattices,” Phys. Rev. B 52, 8992–9002 (1995).
[CrossRef]

Schultz, S.

Sievenpiper, D. F.

E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
[CrossRef]

Sigalas, M. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, K. M. Ho. “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Smith, D. R.

Soukoulis, C. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, K. M. Ho. “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Tran, P.

Turunen, J.

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[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Abstract of multilayered structure or layered grating structure.

Fig. 2
Fig. 2

Convergences of the transmission efficiency T of a grating of square rods versus the number N of diffraction orders. Normal incidence, TM polarization. Here, ng=3.69, n1=1, c/a=0.025, λ/a=1.818. Open squares, calculations using Ref. 11, and filled squares, Refs. 22 and 23.

Fig. 3
Fig. 3

Variations of the transmission efficiencies T as a function of λ/a for a SQ-SQR structure. Normal incidence, TM polarization. Here, ng=3.69, n1=1, F=0.15, NG=12, and N=21. Dashed and continuous curves, calculations using Refs. 11 and 22 and 23, respectively.

Fig. 4
Fig. 4

Variation of the transmission efficiency T as a function of λ/a for a triangular structure of circular rods. Normal incidence, TM polarization. Here, ng=2, n1=1, F=0.35, NG=15, N=21, and NC=41.

Fig. 5
Fig. 5

Variation of the transmission efficiency T as a function of a/λ for an hexagonal structure of circular rods. Normal incidence, TE and TM polarizations. Here, g=ng2=13.6, n1=1, F=0.15, NG=12 (six elementary cells), NC=41, and N=31.

Fig. 6
Fig. 6

Variation of the transmission efficiency T as a function of λ/a for a square structure of metallic square rods. Normal incidence, TM polarization. Here, g (see text), n1=1, F=0.001, NG=20, and N=101.

Fig. 7
Fig. 7

Variation of the absorption efficiencies A as a function of λ/a for a square structure of metallic square rods, for three values of the dissipation factor λ0/a. Normal incidence, TM polarization. Here, g (see text), n1=1, F=0.001, NG=20, and N=101.

Fig. 8
Fig. 8

Details of Fig. 7 in the 1.371.45 λ/a range.

Fig. 9
Fig. 9

Convergence of the normalized wavelength λm/a of maximum absorption for the band depicted in Fig. 10 for λ0/a=102.

Fig. 10
Fig. 10

Square structure of square rods with a central periodic defect. Incident, reflected, and transmitted plane waves (I, R and T).

Fig. 11
Fig. 11

Variations of transmission efficiencies T as a function of λ/a for square PBG-D structures of square rods, for three values of defect index nd. Normal incidence, TE polarization. F=0.15, N=31, NA=5, ng=3.69, n1=1, nd=2.89, 3.09, and 3.39.

Fig. 12
Fig. 12

Same as Fig. 11 except for TM polarization.

Fig. 13
Fig. 13

Variations of transmission efficiencies T as a function of λ for two square PBG structures of square rods. Normal incidence, TE polarization. Here, F=0.125, N=11, ng=2.85, n1=1, and a=0.6 µm. Thick curve, structure without defect, NG=11; thin curve, PBG-D structure, NA=5 and nd=2.85.

Fig. 14
Fig. 14

Variations of transmission efficiencies T as a function of λ for square PBG-D structures of square rods for different values of nd and NA. Normal incidence, TE polarization. Here, F=0.125, N=11, ng=2.85, n1=1, and a=0.6 µm.

Fig. 15
Fig. 15

Variations of the maximum absorption efficiencies Am and the corresponding transmission efficiencies Tm as a function of the imaginary part of the defect index for two PBG-D structures of square rods. Normal incidence, TE polarization. Here, F=0.125, N=11, ng=2.85, nd=3.045, n1=1, a=0.6 µm, and NA=5, 6.

Fig. 16
Fig. 16

Convergence of the transmission efficiency in the (1, 1) order for a checkerboard grating as a function of the total number of diffraction orders. Here, n1=1.5 and ng=1. Normal incidence from medium with refractive index n1. Thickness h=λ, grating period=2.5λ, as in Fig. 6 of Ref. 33. Filled circles, Lalanne’s method; open squares, Ho’s method; open triangles, Neoponen’s method.

Fig. 17
Fig. 17

Variations of transmission efficiencies T as a function of λ for a cubic PBG structure of cubes for two values of N. Normal incidence. Both TE and TM polarizations. Here, g=13, n1=1, a=0.44 µm, F=(0.43)3, and NG=5. Dotted curve, Ho’s method; filled circles, N=625; open circles, N=441. Thin solid curve, Lalanne’s method, N=441; thick solid curve, Neoponen’s method, N=441.

Equations (22)

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

Xk-1Yk-1=[Tk-1]·XkYk,[Tk-1]=Ak-100Ak-101Ak-110Ak-111
XkYk-1=[Sk-1]·Xk-1Yk,[Sk-1]=Qk-1Nk-1Pk-1Mk-1,
Qk-1=[Ak-100]-1,Nk-1=-[Ak-100]-1·Ak-101
Pk-1=Ak-110·[Ak-100]-1,
Mk-1=Ak-111-Ak-110·[Ak-100]-1·Ak-101.
T=Tk-1·Tk.
S=Sk-1 * Sk=QNPM,
P=Pk-1+Mk-1·Pk·[I-Nk-1·Pk]-1·Qk-1,
Q=Qk·[I-Nk-1·Pk]-1·Qk-1,
N=Nk+Qk·Nk-1·[I-Pk·Nk-1]-1·Mk,
M=Mk-1·[I-Pk·Nk-1]-1·Mk,
Ak-110·[Ak-100]-1=-[Ak-100]-1·Ak-101
[Ak-100]-1=Ak-111-Ak-110·[Ak-100]-1·Ak-101.
Pk-1=Nk-1,Qk-1=Mk-1.
S0T=SC * SC+1 * SC=SC+2 * SC+1 * SC,
SC+1-p=SC+1+p.
SqT=SC-q * Sq-1T * SC-q.
g(ω)=ng2(ω)=1-ωpω2
g(λ)=ng2(λ)=1-λλp2,
ωp·a=2πcorλp=a.
g(ω)=ng2(ω)=1-ωp2ω(ω+iγ)
g(λ)=ng2(λ)=1-(λ/a)21+(λ/λ0)2+i 1/λ0(λ/a)21+(λ/λ0)2,

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