Abstract

The effect of spatial dispersion on the electromagnetic properties of a metamaterial consisting of a three-dimensional mesh of crossing metallic wires is reported. The effective dielectric permittivity tensor ϵij(ω,k) of the wire mesh is calculated in the limit of small wavenumbers. The procedure for extracting the spatial dispersion from the ω versus k dependence for electromagnetic waves propagating in the bulk of the metamaterial is developed. These propagating modes are identified as similar to the longitudinal (plasmon) and transverse (photon) waves in a plasma. Spatial dispersion is found to have the most dramatic effect on the surface waves that exist at the wire mesh–vacuum interface.

© 2006 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  2. D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
    [CrossRef] [PubMed]
  3. D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
    [CrossRef]
  4. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
    [CrossRef] [PubMed]
  5. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  6. G. Shvets, Acoust. Phys. 647, 371 (2002).
  7. E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
    [CrossRef] [PubMed]
  8. J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
    [CrossRef] [PubMed]
  9. G. Shvets, A. K. Sarychev, and V. M. Shalaev, in Proc. SPIE 5218, 156 (2003).
    [CrossRef]
  10. P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
    [CrossRef]
  11. High Frequency Structure Simulator, Ver. 10 (Ansoft Corp., Pittsburgh, Pa., 2005).
  12. V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley, 1966).
  13. Th. Koschny, L. Zhang, and C. M. Soukoulis, Phys. Rev. B 71, 121103(R) (2005).
    [CrossRef]
  14. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  15. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
    [CrossRef] [PubMed]

2005 (2)

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

Th. Koschny, L. Zhang, and C. M. Soukoulis, Phys. Rev. B 71, 121103(R) (2005).
[CrossRef]

2004 (1)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

2003 (2)

G. Shvets, A. K. Sarychev, and V. M. Shalaev, in Proc. SPIE 5218, 156 (2003).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

2002 (2)

G. Shvets, Acoust. Phys. 647, 371 (2002).

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2001 (1)

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

1999 (1)

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

1996 (2)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
[CrossRef] [PubMed]

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley, 1966).

Belov, P. A.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley, 1966).

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Joannopoulos, J. D.

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

Kesar, A. S.

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

Koschny, Th.

Th. Koschny, L. Zhang, and C. M. Soukoulis, Phys. Rev. B 71, 121103(R) (2005).
[CrossRef]

Kreischer, K. E.

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

Machuzak, J.

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Marques, R.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Martin-Moreno, L.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

Maslovski, S. I.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Mastovsky, I.

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

Meade, R. D.

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

Nefedov, I. S.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

Padilla, W.

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

Pendry, J. B.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Sarychev, A. K.

G. Shvets, A. K. Sarychev, and V. M. Shalaev, in Proc. SPIE 5218, 156 (2003).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

Shalaev, V. M.

G. Shvets, A. K. Sarychev, and V. M. Shalaev, in Proc. SPIE 5218, 156 (2003).
[CrossRef]

Shapiro, M. A.

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

Shvets, G.

G. Shvets, A. K. Sarychev, and V. M. Shalaev, in Proc. SPIE 5218, 156 (2003).
[CrossRef]

G. Shvets, Acoust. Phys. 647, 371 (2002).

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
[CrossRef] [PubMed]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
[CrossRef] [PubMed]

Silveirinha, M.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Simovski, C. R.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Sirigiri, J. R.

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

Smirnova, E. I.

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

Soukoulis, C. M.

Th. Koschny, L. Zhang, and C. M. Soukoulis, Phys. Rev. B 71, 121103(R) (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Temkin, R. J.

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

Tretyakov, S. A.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Vier, D. C.

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

Winn, J. N.

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

Yablonovitch, E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
[CrossRef] [PubMed]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

Zhang, L.

Th. Koschny, L. Zhang, and C. M. Soukoulis, Phys. Rev. B 71, 121103(R) (2005).
[CrossRef]

Acoust. Phys. (1)

G. Shvets, Acoust. Phys. 647, 371 (2002).

Appl. Phys. Lett. (1)

D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 75, 1425 (1999).
[CrossRef]

Phys. Rev. B (3)

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Th. Koschny, L. Zhang, and C. M. Soukoulis, Phys. Rev. B 71, 121103(R) (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
[CrossRef] [PubMed]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
[CrossRef] [PubMed]

E. I. Smirnova, A. S. Kesar, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 95, 074801 (2005).
[CrossRef] [PubMed]

J. R. Sirigiri, K. E. Kreischer, J. Machuzak, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. 86, 5628 (2001).
[CrossRef] [PubMed]

Proc. SPIE (1)

G. Shvets, A. K. Sarychev, and V. M. Shalaev, in Proc. SPIE 5218, 156 (2003).
[CrossRef]

Science (1)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004).
[CrossRef] [PubMed]

Other (3)

High Frequency Structure Simulator, Ver. 10 (Ansoft Corp., Pittsburgh, Pa., 2005).

V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley, 1966).

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

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Figures (5)

Fig. 1
Fig. 1

Metamaterial consisting of a mesh of crossing metallic wires.

Fig. 2
Fig. 2

Unit cell of the wire mesh. Square wire cross section of width w = 0.02 cm , period d = 0.58 cm . Left, plasmon (longitudinal) mode. E-vectors are plotted in the symmetry planes. Amplitude ranges from 1 (red) to 0.1 (blue). Right, photon (transverse) mode.

Fig. 3
Fig. 3

Brillouin diagram for the wire mesh metamaterial. Mesh parameters: same as in Fig. 2. The free space dispersion lines a are shown as blue dashed lines. The green line b is the dispersion curve of the surface wave at the metamaterial–vacuum interface. Dashed red lines c are dispersion curves of the medium with SD.

Fig. 4
Fig. 4

SD parameters α i defined by Eqs. (3) as functions of the ratio of wire width w to wire spacing d.

Fig. 5
Fig. 5

Surface mode at the wire mesh–vacuum interface. The electric field distribution is shown in one period along x. Three time snapshots indicate opposite signs of the Poynting fluxes in the mesh and vacuum.

Tables (1)

Tables Icon

Table 1 Coefficient A in Eq. (4)

Equations (8)

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ϵ p ( ω ) = 1 ω p 2 ω 2 ,
ω p = ( c d ) [ 2 π ln ( d w ) ] 1 2
ϵ x x = ϵ p + α 1 ( k x 2 c 2 ω 2 ) + α 2 ( k z 2 c 2 ω 2 ) ,
ϵ z z = ϵ p + α 2 ( k x 2 c 2 ω 2 ) + α 1 ( k z 2 c 2 ω 2 ) ,
ϵ x z = ϵ z x = 2 α 3 ( k x k z c 2 ω 2 ) ,
ω 2 = ω p 2 + A k 2 c 2 ,
E x ( 1 ) = E x ( 2 ) , H y ( 1 ) 2 ( k x c ω ) α 3 E z ( 1 ) = H y ( 2 ) ,
ϵ p 2 ( k x 2 ω 2 c 2 ) [ 1 + 4 ( α 3 + α 3 2 ) k x 2 c 2 ω 2 ϵ p ] = k x 2 ω 2 c 2 ϵ p .

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