Abstract

We propose a method to obtain the resonance frequencies of coupled optical modes for a stack of two periodically corrugated slabs. The method is based on the modes in each slab, which are derived by the Fourier modal method in combination with the optical scattering matrix theory. We then use the resonant mode approximation of the scattering matrices to develop a linear eigenvalue problem with dimensions equal to the number of resonant modes. Its solutions are the resonance frequencies of the coupled system and exhibit a good agreement with exact solutions. We demonstrate the capabilities of this method for pairs of planar waveguides and gratings of one-dimensional wires.

© 2010 Optical Society of America

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  1. J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
    [CrossRef]
  2. N. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
    [CrossRef]
  3. N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, "Plasmonic electromagnetically induced transparency at the Drude damping limit," Nat. Mater. 8, 758-762 (2009).
    [CrossRef] [PubMed]
  4. D. M. Whittaker and I. S. Culshaw, "Scattering matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
    [CrossRef]
  5. 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, 451021 (2002).
    [CrossRef]
  6. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of photonic crystal slabs with an asymmetrical unit cell," Phys. Rev. B 72, 451381 (2005).
    [CrossRef]
  7. 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]
  8. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  9. 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, 145-149 (2002).
    [CrossRef]
  10. G. Gantzounis and N. Stefanou, "Layer-multiple-scattering method for photonic crystals of nonspherical particles," Phys. Rev. B 73, 035115 (2006).
    [CrossRef]
  11. L. Landau and E. M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory (Pergamon, 1966).
  12. V. Podolskiy, A. Sarychev, and V. Shalaev, "Plasmon modes and negative refraction in metal nanowire composites," Opt. Express 11, 735-745 (2003).
    [CrossRef] [PubMed]
  13. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, "An analytic model for the optical properties of gold," J. Chem. Phys.  125, 164705-1-3 (2006).
    [CrossRef] [PubMed]

2009 (3)

N. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kastel, 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]

2008 (1)

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

2006 (2)

G. Gantzounis and N. Stefanou, "Layer-multiple-scattering method for photonic crystals of nonspherical particles," Phys. Rev. B 73, 035115 (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, 164705-1-3 (2006).
[CrossRef] [PubMed]

2005 (1)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of photonic crystal slabs with an asymmetrical unit cell," Phys. Rev. B 72, 451381 (2005).
[CrossRef]

2003 (1)

2002 (2)

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, 145-149 (2002).
[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, 451021 (2002).
[CrossRef]

1999 (1)

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

1997 (1)

Chipouline, A.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

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]

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, 164705-1-3 (2006).
[CrossRef] [PubMed]

Fleischhauer, M.

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

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. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
[CrossRef]

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

Gippius, N. A.

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, 451381 (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, 451021 (2002).
[CrossRef]

Granet, G.

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, 145-149 (2002).
[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, 451381 (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, 451021 (2002).
[CrossRef]

Kastel, J.

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

Langguth, L.

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

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, 164705-1-3 (2006).
[CrossRef] [PubMed]

Lederer, F.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

Li, L.

Liu, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
[CrossRef]

Liu, N.

N. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
[CrossRef]

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

Menzel, C.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[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, 164705-1-3 (2006).
[CrossRef] [PubMed]

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, 451021 (2002).
[CrossRef]

Pertsch, T.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

Petschulat, J.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kastel, 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, 145-149 (2002).
[CrossRef]

Podolskiy, V.

Rockstuhl, C.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

Sarychev, A.

Shalaev, V.

Stefanou, N.

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

Tikhodeev, S. G.

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, 451381 (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, 451021 (2002).
[CrossRef]

Tunnermann, A.

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

Weiss, T.

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

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, 451021 (2002).
[CrossRef]

Zhu, S.

N. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
[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, 164705-1-3 (2006).
[CrossRef] [PubMed]

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

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, 145-149 (2002).
[CrossRef]

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

Nat. Mater. (1)

N. Liu, L. Langguth, T. Weiss, J. Kastel, 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. Photonics (1)

N. Liu, H. Liu, S. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photonics 3, 157-162 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, "Multipole approach to metamaterials," Phys. Rev. A 78, 438111 (2008).
[CrossRef]

Phys. Rev. B (4)

G. Gantzounis and N. Stefanou, "Layer-multiple-scattering method for photonic crystals of nonspherical particles," Phys. Rev. B 73, 035115 (2006).
[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, 451021 (2002).
[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, 451381 (2005).
[CrossRef]

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]

Other (1)

L. Landau and E. M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory (Pergamon, 1966).

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Figures (3)

Figure 1.
Figure 1.

Structures of interest: (a) Dielectric slab waveguides. (b) Dielectric gratings. (c) Gold wires embedded in quartz.

Figure 2.
Figure 2.

Coupled resonances in stacked dielectric structures derived by the coupling model (solid blue lines) and the full calculation of the coupled system (red crosses) for TM polarization. The antisymmetric and symmetric branches of the coupled resonances are indicated by the letters a and s, respectively. (a) Planar waveguides. (b) Optically inactive modes in coupled gratings. (c) Real and (d) imaginary parts of optically active resonances in the coupled gratings. The inset in (c) shows a magnification of the region with larger distances, where weak coupling to Fabry-Perot modes can be observed.

Figure 3.
Figure 3.

Comparison of the coupled mode approximation (solid blue line) with the full calculation of the coupled system (red crosses) for (a) the real and (b) the imaginary parts of a localized resonance in gold wires. The letters a and s indicate the antisymmetric and the symmetric branches of the coupled resonances, respectively.

Equations (16)

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

d b u t O = ( S d d S d u S u d S u u ) S d t u b I ,
S 1 ( ω ) O [ S 1 ( ω 0 ) + ( ω ω 0 ) S 1 ω | ω 0 ] O = ! 0 .
S 1 ( ω 0 ) X = S 1 ω | ω 0 S 1 ( ω 0 ) = S 1 ω | ω 0 X Δ X 1 .
S X [ ( ω ω 0 ) 𝕀 Δ ] 1 Y = S ˜ + n = 1 N O n 1 ω ω n I n ,
X = ( O 1 , O 2 , ) , Y = ( S 1 ω | ω 0 X ) 1 = ( I 1 I 2 ) ,
| d 2 | u 1 = [ ( S ˜ d d a S ˜ d u a S ˜ u d a S ˜ u u a ) + n = 1 N O n a 1 ω ω n a I n a ] | d 1 | u 2 ,
| d 3 | u 2 = [ ( S ˜ d d b S ˜ d u b S ˜ u d b S ˜ u u b ) + n = 1 M O n b 1 ω ω n b I n b ] | d 2 | u 3 .
d 2 = S d u a u 2 = ( S ˜ d u a + n = 1 N O d , n a 1 ω ω n a I u , n a ) u 2 ,
u 2 = S u d b d 2 = ( S ˜ u d b + n = 1 M O u , n b 1 ω ω n b I d , n b ) d 2 .
α n 1 ω ω n a I u , n a u 2 d 2 = S ˜ d u a u 2 + n = 1 N α n O d , n a ,
β n 1 ω ω n b I d , n b d 2 u 2 = S ˜ u d b d 2 + n = 1 M β n O u , n b .
d 2 = n = 1 N α n O d , n a + n = 1 M β n S ˜ d u a O u , n b ,
u 2 = n = 1 N α n S ˜ u d b O d , n a + n = 1 M β n O u , n b ,
O d , n a = ( S ˜ d u a S ˜ u d b ) 1 O d , n a , O u , n b = ( S ˜ u d b S ˜ d u a ) 1 O u , n b .
( ω ω n a ) α n = m = 1 N I u , n a S ˜ u d b O d , m a α m + m = 1 M I u , n a O u , m b β m ,
( ω ω n b ) β n = m = 1 N I d , n b O d , m a α m + m = 1 M I d , n b S ˜ d u a O u , m b β m .

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