Abstract

A very stable approach for finding optical resonances is to solve an eigenvalue equation that evolves from the linearization of the inverse scattering matrix. In this paper, we show how to use this approach in the Fourier modal method so that advanced coordinate transformation methods such as adaptive spatial resolution and matched coordinates can be included. Furthermore, we present a way that accelerates the computation of the inverse scattering matrix tremendously and allows the derivation of the resonant field distribution inside the structure efficiently.

© 2011 Optical Society of America

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    [PubMed]
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    [CrossRef]
  28. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
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    [CrossRef]
  31. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
    [CrossRef]
  32. G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particles,” Phys. Rev. B 73, 035115 (2006).
    [CrossRef]
  33. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003).
    [CrossRef]

2010 (3)

2009 (6)

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061(2009).
[CrossRef] [PubMed]

H. Yala, B. Guizal, and D. Felbacq, “Fourier modal method with spatial adaptive resolution for structures comprising homogeneous layers,” J. Opt. Soc. Am. A 26, 2567–2570 (2009).
[CrossRef]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. 3, 157–162 (2009).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[CrossRef]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

2008 (2)

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

P. Götz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express 16, 17295–17301 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (3)

G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particles,” Phys. Rev. B 73, 035115 (2006).
[CrossRef]

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

2005 (2)

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 45138 (2005).
[CrossRef]

2004 (1)

2003 (2)

L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003).
[CrossRef]

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

2002 (3)

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A: Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: Application to multilevel profiles,” Opt. Express 10, 24–34 (2002).
[PubMed]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

2001 (1)

1999 (2)

D. M. Whittaker and I. S. Culshaw, “Scattering matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

1997 (1)

1996 (5)

Bai, B.

Bird, D. M.

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Brand, S.

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

Busch, K.

S. Essig and K. Busch, “Generation of adaptive coordinates and their use in the Fourier modal method,” Opt. Express 18, 23258–23274 (2010).
[CrossRef] [PubMed]

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Chigrin, D. N.

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

DeLaRue, R. M.

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

Essig, S.

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Etrich, C.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Felbacq, D.

Feth, N.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Frenner, K.

Gantzounis, G.

G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particles,” Phys. Rev. B 73, 035115 (2006).
[CrossRef]

Giessen, H.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061(2009).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. 3, 157–162 (2009).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Gippius, N. A.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061(2009).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[CrossRef]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 45138 (2005).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Götz, P.

Granet, G.

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061(2009).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A: Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

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

Guizal, B.

Guo, H.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Hedley, T. D.

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Honkanen, M.

Husnik, M.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 45138 (2005).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Kerwien, N.

Klein, M. W.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

König, M.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Krauss, T. F.

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

Kuhl, J.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Lalanne, P.

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Lavrinenko, A. V.

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Lederer, F.

T. Paul, C. Rockstuhl, and F. Lederer, “A numerical approach for analyzing higher harmonic generation in multilayer nanostructures,” J. Opt. Soc. Am. B 27, 1118–1130 (2010).
[CrossRef]

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Li, L.

Linden, S.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Liu, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. 3, 157–162 (2009).
[CrossRef]

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. 3, 157–162 (2009).
[CrossRef]

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Loa, I.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Morris, G. M.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Nevière, M.

Niegemann, J.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Osten, W.

Paul, T.

Pearce, G. J.

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Plumey, J. P.

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A: Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

Popov, E.

Rafler, S.

Rockstuhl, C.

T. Paul, C. Rockstuhl, and F. Lederer, “A numerical approach for analyzing higher harmonic generation in multilayer nanostructures,” J. Opt. Soc. Am. B 27, 1118–1130 (2010).
[CrossRef]

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Ruoff, J.

Schuster, T.

Stefanou, N.

G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particles,” Phys. Rev. B 73, 035115 (2006).
[CrossRef]

Syassen, K.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Tikhodeev, S. G.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061(2009).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[CrossRef]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 45138 (2005).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Torres, C. M. S.

Turunen, J.

Vallius, T.

Wegener, M.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

Weiss, T.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061(2009).
[CrossRef] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, “Scattering matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Yala, H.

Zentgraf, T.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

Zhu, S.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. 3, 157–162 (2009).
[CrossRef]

AIP Conf. Proc. (1)

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Calculation of complex shapes in the Fourier modal method through the concept of coordinate transformations,” AIP Conf. Proc. 1176, 163–165 (2009).
[CrossRef]

Appl. Phys. B (1)

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84, 219–227 (2006).
[CrossRef]

J. Chem. Phys. (1)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

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L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A: Pure Appl. Opt. 4, S145–S149 (2002).
[CrossRef]

J. Opt. Soc. Am. A (10)

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767(1997).
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G. Granet and B. Guizal, “Efficient implementation of the coupled-waved method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
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L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
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[CrossRef]

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H. Yala, B. Guizal, and D. Felbacq, “Fourier modal method with spatial adaptive resolution for structures comprising homogeneous layers,” J. Opt. Soc. Am. A 26, 2567–2570 (2009).
[CrossRef]

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

Nat. Mater. (1)

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Nat. Photon. (2)

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photon. 2, 614–617 (2008).
[CrossRef]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. 3, 157–162 (2009).
[CrossRef]

Nature (London) (1)

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

Opt. Express (6)

Phys. Rev. B (5)

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, “Scattering matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particles,” Phys. Rev. B 73, 035115 (2006).
[CrossRef]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 45138 (2005).
[CrossRef]

Phys. Usp. (1)

N. A. Gippius and S. G. Tikhodeev, “Application of the scattering matrix method for calculating the optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic of a periodic array of gold cylinders in vacuum as test system, including the definition of a coordinate system with matched curvilinear coordinates. The cylinder parameters are radius 250 nm , height 30 nm , period 700 nm .

Fig. 2
Fig. 2

Transmittance (T), reflectance (R), and absorbance (A) of gold cylinders calculated for a truncation order of N G = 21 × 21 harmonics. Thick lines depict normal incidence with the electric field polarized along the x 1 axis. Thin lines in the gray insets show inclined incidence with the incident k vector tilted by 5 ° in the x 1 direction.

Fig. 3
Fig. 3

Convergence map of the iterative mode search procedure for varying starting positions in the case of a truncation order of N G = 15 × 15 . Gray squares converged to 305 108 i THz , right-oriented blue triangles to 332 2 i THz , and left-oriented red triangles to 377 3 i THz . The size of the symbols is proportional to the number of search iterations between 3 and 10. In addition, black crosses mark the determined resonance positions and black thick lines with circles denote exemplarily the trace on the complex plane for the iterative mode search procedure.

Fig. 4
Fig. 4

Electric field distributions calculated for a truncation order of N G = 21 × 21 harmonics at resonance without external excitation. Columns depict from left to right: the fundamental plasmon resonance at 305 108 i THz , an optically inactive higher order mode at 332 2 i THz , and an optically inactive resonance at 378 3 i THz . The lower row shows a top view of one unit cell in the x 1 x 2 plane, 10 nm below the upper interface of the cylinders. The upper row contains side views of planes normal to the x 2 axis and shifted by 100 nm with respect to the center of the unit cell. The locations of the slices are indicated by white dashed lines. Light blue arrows show the typical electric field pattern, the background color denotes the square of the absolute value of the time-averaged electric field in a normalized logarithmic scale.

Fig. 5
Fig. 5

Comparison of numerically exact results with those obtained by approximating the frequency dependence of the eigenvalues in homogeneous layers using Eq. (14): maximum (black dashed-dotted line) and mean (blue dashed line) relative deviation of propagation constants k 3 as well as far-field transmission t (red solid line) calculated for a truncation order of N G = 21 × 21 . The reference eigenvalues have been derived for 300 THz .

Equations (21)

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ε α 3 = μ α 3 = ε 3 α = μ 3 α = 0 , α = 1 , 2.
E α = m , n E α , m n e i ( k 1 + 2 π P 1 m ) x 1 + i ( k 2 + 2 π P 2 n ) x 2 + i k 3 x 3 .
E K 2 = 1 k 0 2 L EH L HE E .
K = ( K 0 0 0 K 0 ) ,
K 0 = ( k 0 2 ε μ 1 κ 1 κ 1 κ 2 κ 2 ) 1 2 .
κ α , m n , p q = δ m p δ n q { ( k 1 + 2 π P 1 m ) for α = 1 ( k 2 + 2 π P 2 n ) for α = 2 .
[ A + ( x b 3 ) A ( x a 3 ) ] = | O = ( S b a ++ S b b + S a a + S a b -- ) = S [ A + ( x a 3 ) A ( x b 3 ) ] = | I .
S 1 ( ω , k ) | O = 0.
S 1 ( ω ) | O [ S 1 ( ω 0 ) + S 1 ω ( ω 0 ) Δ ω ] | O = 0.
[ A ( x b 3 ) A + ( x a 3 ) ] = ( S ˜ b a S ˜ b b + S ˜ a a + S ˜ a b ++ ) = S ˜ [ A ( x a 3 ) A + ( x b 3 ) ] ,
F ( A + A ) = ( E H ) = F ˜ ( A A + ) .
[ A + ( x i 3 ) A + ( x a 3 ) ] = ( S i a + S i i + S a a + S a i + ) [ A ( x a 3 ) A ( x i 3 ) ] .
E ( K 2 k 0 2 ε μ 1 ) = ME .
K 0 2 ( ω ) ω 2 c 2 ε ( ω ) μ ( ω ) 1 = K 0 2 ( ω ref ) ω ref 2 c 2 ε ref μ ref 1 .
L ^ EH = 1 ε D ^ k 0 2 μ G ^ ,
L ^ HE = 1 μ D ^ + k 0 2 ε G ^ ,
D ^ = ( 1 g 1 2 2 1 g 1 2 1 2 g 1 2 2 2 g 1 2 1 ) ,
G ^ = g 1 2 ( g 21 g 22 g 11 g 12 ) .
D ^ 2 = 0 ,
G ^ 2 = 1 .
1 k 0 2 L ^ EH L ^ HE = k 0 2 ε μ 1 + D ^ G ^ + G ^ D ^ ,

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