Abstract

Enhanced reflections and transmissions by slabs of periodic structures with strongly dispersive materials have recently received significant attention because of their unusual physical phenomena and potential engineering applications. To simulate such phenomena for design prototyping with high efficiency, a spectral element method is developed to calculate the electromagnetic fields in a slab of periodic three-dimensional photonic crystal consisting of dispersive or nondispersive materials. The method of moments with the spectral-domain periodic Green’s function is used to truncate the computational domain above and below the photonic crystal slabs. The accuracy of the method is verified. The method is used to calculate the scattering properties of an array of air holes in a dispersive metallic film in optical frequencies. The surface plasmon polariton and local surface plasmon modes are identified, with excellent correlation with experimental results.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Lin, R. J. Reeves, and R. J. Blaikie, “Surface-plasmon-enhanced light transmission through planar metallic films,” Phys. Rev. B 74, 155407 (2006).
    [CrossRef]
  2. A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
    [CrossRef]
  3. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
    [CrossRef] [PubMed]
  4. I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
    [CrossRef]
  5. T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
    [CrossRef]
  6. D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  7. C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N.-A. P. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15, 8163–8169 (2007).
    [CrossRef] [PubMed]
  8. K. M. Byun, S. J. Kim, and D. Kim, “Design study of highly sensitive nanowire-enhanced surface plasmon resonance biosensors using rigorous coupled wave analysis,” Opt. Express 13, 3737–3742 (2005).
    [CrossRef] [PubMed]
  9. K. M. Byun, M. L. Shuler, S. J. Kim, S. J. Yoon, and D. Kim, “Sensitivity enhancement of surface plasmon resonance imaging using periodic metallic nanowires,” J. Lightwave Technol. 26, 1472–1478 (2008).
    [CrossRef]
  10. J. Chen, Z. Li, S. Yue, and Q. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17, 23603–23609(2009).
    [CrossRef]
  11. A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).
  12. B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
    [CrossRef]
  13. P. Sotirelis and J. D. Albrecht, “Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions,” Phys. Rev. B 76, 075123 (2007).
    [CrossRef]
  14. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  15. G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).
  16. J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1848–1858(2005).
    [CrossRef]
  17. J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437–444 (2006).
    [CrossRef]
  18. A. T. Patera, “A spectral element method for fluid dynamics: laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
    [CrossRef]
  19. J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
    [CrossRef]
  20. Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15, 158–165 (1997).
    [CrossRef]
  21. Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antenn. Wireless Propag. Lett. 1, 131–134 (2002).
    [CrossRef]
  22. M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
    [CrossRef]
  23. M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
    [CrossRef]
  24. M. Luo, Q. H. Liu, and J. Guo, “A spectral element method calculation of extraordinary light transmission through periodic subwavelength slits,” J. Opt. Soc. Am. B 27, 560–566(2010).
    [CrossRef]
  25. M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
    [CrossRef]
  26. M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
    [CrossRef]
  27. M. M. Botha and J.-M. Jin, “On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis,” IEEE Trans. Antenn. Propag. 52, 3037–3047 (2004).
    [CrossRef]
  28. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  29. G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
    [CrossRef]
  30. D. Lockau, L. Zschiedrich, and S. Burger, “Accurate simulation of light transmission through subwavelength apertures in metal films,” J. Opt. A Pure Appl. Opt. 11, 114013 (2009).
    [CrossRef]

2010 (1)

2009 (6)

M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
[CrossRef]

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

D. Lockau, L. Zschiedrich, and S. Burger, “Accurate simulation of light transmission through subwavelength apertures in metal films,” J. Opt. A Pure Appl. Opt. 11, 114013 (2009).
[CrossRef]

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
[CrossRef]

J. Chen, Z. Li, S. Yue, and Q. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17, 23603–23609(2009).
[CrossRef]

2008 (2)

B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
[CrossRef]

K. M. Byun, M. L. Shuler, S. J. Kim, S. J. Yoon, and D. Kim, “Sensitivity enhancement of surface plasmon resonance imaging using periodic metallic nanowires,” J. Lightwave Technol. 26, 1472–1478 (2008).
[CrossRef]

2007 (4)

C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N.-A. P. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15, 8163–8169 (2007).
[CrossRef] [PubMed]

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

P. Sotirelis and J. D. Albrecht, “Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions,” Phys. Rev. B 76, 075123 (2007).
[CrossRef]

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[CrossRef]

2006 (3)

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437–444 (2006).
[CrossRef]

L. Lin, R. J. Reeves, and R. J. Blaikie, “Surface-plasmon-enhanced light transmission through planar metallic films,” Phys. Rev. B 74, 155407 (2006).
[CrossRef]

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

2005 (4)

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1848–1858(2005).
[CrossRef]

K. M. Byun, S. J. Kim, and D. Kim, “Design study of highly sensitive nanowire-enhanced surface plasmon resonance biosensors using rigorous coupled wave analysis,” Opt. Express 13, 3737–3742 (2005).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

2004 (2)

M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
[CrossRef]

M. M. Botha and J.-M. Jin, “On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis,” IEEE Trans. Antenn. Propag. 52, 3037–3047 (2004).
[CrossRef]

2002 (2)

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antenn. Wireless Propag. Lett. 1, 131–134 (2002).
[CrossRef]

D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

1997 (1)

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

1986 (1)

1984 (1)

A. T. Patera, “A spectral element method for fluid dynamics: laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Albrecht, J. D.

P. Sotirelis and J. D. Albrecht, “Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions,” Phys. Rev. B 76, 075123 (2007).
[CrossRef]

Alleyne, C. J.

Blaikie, R. J.

L. Lin, R. J. Reeves, and R. J. Blaikie, “Surface-plasmon-enhanced light transmission through planar metallic films,” Phys. Rev. B 74, 155407 (2006).
[CrossRef]

Botha, M. M.

M. M. Botha and J.-M. Jin, “On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis,” IEEE Trans. Antenn. Propag. 52, 3037–3047 (2004).
[CrossRef]

Burger, S.

D. Lockau, L. Zschiedrich, and S. Burger, “Accurate simulation of light transmission through subwavelength apertures in metal films,” J. Opt. A Pure Appl. Opt. 11, 114013 (2009).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Byun, K. M.

Chen, J.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Cohen, G. C.

G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).

Economou, E. N.

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

Enkrich, C.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Frimmer, M.

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
[CrossRef]

Garca-Vidal, F. J.

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

Gaylord, T. K.

Gong, Q.

Granet, G.

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

Guo, J.

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Jin, J.-M.

M. M. Botha and J.-M. Jin, “On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis,” IEEE Trans. Antenn. Propag. 52, 3037–3047 (2004).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kim, D.

Kim, S. J.

Kirk, A. G.

Koenderink, A. F.

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
[CrossRef]

Koschny, T.

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Lee, J.-F.

M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
[CrossRef]

Lee, J.-H.

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[CrossRef]

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437–444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1848–1858(2005).
[CrossRef]

Lee, S.-C.

M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
[CrossRef]

Li, L.

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

Li, Z.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

J. Chen, Z. Li, S. Yue, and Q. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17, 23603–23609(2009).
[CrossRef]

Lin, L.

L. Lin, R. J. Reeves, and R. J. Blaikie, “Surface-plasmon-enhanced light transmission through planar metallic films,” Phys. Rev. B 74, 155407 (2006).
[CrossRef]

Linden, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Liu, Q. H.

M. Luo, Q. H. Liu, and J. Guo, “A spectral element method calculation of extraordinary light transmission through periodic subwavelength slits,” J. Opt. Soc. Am. B 27, 560–566(2010).
[CrossRef]

M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
[CrossRef]

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[CrossRef]

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437–444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1848–1858(2005).
[CrossRef]

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antenn. Wireless Propag. Lett. 1, 131–134 (2002).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

Lockau, D.

D. Lockau, L. Zschiedrich, and S. Burger, “Accurate simulation of light transmission through subwavelength apertures in metal films,” J. Opt. A Pure Appl. Opt. 11, 114013 (2009).
[CrossRef]

Luo, M.

M. Luo, Q. H. Liu, and J. Guo, “A spectral element method calculation of extraordinary light transmission through periodic subwavelength slits,” J. Opt. Soc. Am. B 27, 560–566(2010).
[CrossRef]

M. Luo and Q. H. Liu, “Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method,” J. Opt. Soc. Am. A 26, 1598–1605 (2009).
[CrossRef]

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Marko, P.

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Martn-Moreno, L.

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

Mary, A.

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Moharam, M. G.

Nicorovici, N.-A. P.

Patera, A. T.

A. T. Patera, “A spectral element method for fluid dynamics: laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
[CrossRef]

Reeves, R. J.

L. Lin, R. J. Reeves, and R. J. Blaikie, “Surface-plasmon-enhanced light transmission through planar metallic films,” Phys. Rev. B 74, 155407 (2006).
[CrossRef]

Rodrigo, S. G.

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sersic, I.

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
[CrossRef]

Shuler, M. L.

Smith, D. R.

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sotirelis, P.

P. Sotirelis and J. D. Albrecht, “Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions,” Phys. Rev. B 76, 075123 (2007).
[CrossRef]

Soukoulis, C. M.

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Verhagen, E.

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
[CrossRef]

Vier, D. C.

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

Vouvakis, M. N.

M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
[CrossRef]

Ward, B. G.

B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
[CrossRef]

Wegener, M.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Xiao, T.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437–444 (2006).
[CrossRef]

Yoon, S. J.

Yue, S.

Zhao, K.

M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
[CrossRef]

Zhou, J. F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Zschiedrich, L.

D. Lockau, L. Zschiedrich, and S. Burger, “Accurate simulation of light transmission through subwavelength apertures in metal films,” J. Opt. A Pure Appl. Opt. 11, 114013 (2009).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

IEEE Antenn. Wireless Propag. Lett. (1)

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antenn. Wireless Propag. Lett. 1, 131–134 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. G. Ward, “Finite element analysis of photonic crystal rods with inhomogeneous anisotropic refractive index tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
[CrossRef]

IEEE Trans. Antenn. Propag. (2)

M. N. Vouvakis, S.-C. Lee, K. Zhao, and J.-F. Lee, “A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems,” IEEE Trans. Antenn. Propag. 52, 3060–3070 (2004).
[CrossRef]

M. M. Botha and J.-M. Jin, “On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis,” IEEE Trans. Antenn. Propag. 52, 3037–3047 (2004).
[CrossRef]

IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. (1)

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1848–1858(2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437–444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “A 3-D spectral-element time-domain method for electromagnetic simulation,” IEEE Trans. Microwave Theory Tech. 55, 983–991 (2007).
[CrossRef]

J. Comput. Phys. (1)

A. T. Patera, “A spectral element method for fluid dynamics: laminar flow in a channel expansion,” J. Comput. Phys. 54, 468–488 (1984).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A (1)

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

D. Lockau, L. Zschiedrich, and S. Burger, “Accurate simulation of light transmission through subwavelength apertures in metal films,” J. Opt. A Pure Appl. Opt. 11, 114013 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Microw. Opt. Technol. Lett. (1)

Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

Opt. Express (3)

Phys. Rev. B (6)

T. Koschny, P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Marko, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

L. Lin, R. J. Reeves, and R. J. Blaikie, “Surface-plasmon-enhanced light transmission through planar metallic films,” Phys. Rev. B 74, 155407 (2006).
[CrossRef]

A. Mary, S. G. Rodrigo, L. Martn-Moreno, and F. J. Garca-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

P. Sotirelis and J. D. Albrecht, “Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions,” Phys. Rev. B 76, 075123 (2007).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. E (2)

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Phys. Rev. Lett. (2)

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103, 213902(2009).
[CrossRef]

Other (2)

A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometry and mesh of the scattering problem in (a) a 3D view, (b) vertical side view, and (c) horizontal top view. The structure is periodic in the x and y directions at the unit–cell boundaries marked by the dashed lines in (c). The nodal points of the fourth-order SEM are shown in (c). The vertical boundaries marked by the dashed lines in (b) are radiation boundaries. The period of the lattice L x and L y , the elliptic diameters of the hole D x and D y , and the thickness of each layer h 1 , w, and h 2 are shown in (b) and (c).

Fig. 2
Fig. 2

Relative error of the reflection rate (open circle) and transmission rate (solid circle) of the SEM results compared with the analytical solution versus the order of the SEM. The calculated system is a 30 nm -thick silver film on the surface of glass substrate.

Fig. 3
Fig. 3

Structure of one unit cell of the silver film with a square hole arrangement and the SEM mesh.

Fig. 4
Fig. 4

Results for the structure in Fig. 3. (a) Transmission rate versus the number of the SEM unknowns (solid circle) against the reference result from [29, 30]. (b) Relative error between the SEM result and the reference versus the order of the SEM.

Fig. 5
Fig. 5

Transmission rate through the air hole array in a 35 nm -thick silver film on the surface of glass substrate versus the wavelength of the incident plane wave, with the incidence angle varying from 0 ° to 80 ° with a 10 ° increment. For clarity, the plotting of every incidence angle is shifted by 1 in the vertical scale. The result is given by the fourth-order SEM. Four transmission peaks are marked from “a” to “d” for further investigation in Fig. 6. These peaks correspond to the three resonance bands marked as I, II, and III.

Fig. 6
Fig. 6

(a)–(d) Electric field magnitude patterns of the four transmission peaks “a” to “d” shown in Fig. 5. The observation plane is located 15 nm below the bottom surface of the silver film. The dark lines show the edge of the air holes.

Fig. 7
Fig. 7

(a) Transmission rate of the system of a double silver film system with an array of elliptical air holes on the top film versus the wavelength of a normally incident plane wave with polarization along the x axis (blue dashed curve) and the y axis (red solid curve). The detail of the system is the same as in Fig. 2 in [1]. (b) Relative error of the reflection and transmission rates at the 559 nm incident wave versus the order of the SEM.

Equations (14)

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

Φ i = J Φ ^ i , × Φ i = det ( J ) J T ^ × Φ ^ i ,
E = i E i Φ i ,
H ¯ = i up ( down ) η 0 H i Φ i ,
Ω [ ( × Φ i ) · μ r 1 · ( × E ) k 0 2 Φ i · ε r · E ] d V j k 0 Ω ( Φ i ) · ( n ^ × H ¯ ) d S = 0 ,
1 2 n ^ × [ E + L p ( J ¯ s ) K ˜ p ( M s ) ] = n ^ × E p inc
1 2 n ^ × [ H ¯ + L p ( M s ) + K ˜ p ( J ¯ s ) ] = n ^ × H ¯ p inc ,
L p ( X ) = j k p Ω [ X ( r ) g p ( r , r ) + 1 k p 2 · X ( r ) g p ( r , r ) ] d S ,
K p ( X ) = Ω X ( r ) × g p ( r , r ) d S ,
Ω [ ( × Φ i ) · μ r 1 · ( × E ) k p 2 Φ i · ε r · E ] d V j k p Ω ( n ^ × Φ i ) · [ 1 2 H ¯ + L p ( M s ) + K ˜ p ( J ¯ s ) ] d S = j k p Ω ( n ^ × Φ i ) · H ¯ p inc .
j k p Ω ( n ^ × Φ i ) · [ 1 2 E + L p ( J ¯ s ) K ˜ p ( M s ) ] d S = j k p Ω ( n ^ × Φ i ) · E p inc .
g p ( r , r ) = m = + n = + e j q x m ( x x ) e j q y n ( y y ) I m , n ( k x , k y , k p ) ,
I m , n ( k x , k y , k p ) = 2 π L x 2 π L y 1 4 π j k p 2 ( q x m ) 2 ( q y n ) 2 .
g p ( r , r ) = m = + n = + e j Q m , n · Δ r I m , n ( Q m , n , k p ) ,
I m , n ( Q m , n , k p ) = ( 2 π ) 2 A 1 4 π j k p 2 | Q m , n | 2 ,

Metrics