Abstract

Using homogenization theory, we derive the effective permittivity and permeability of a wire-mesh photonic crystal from first principles. We show that this structure does not lead to a left-handed medium when it is embedded in a matrix with a negative μ, but that resonators in a medium with a negative ε do form a left-handed medium in the correct range of frequencies.

© 2005 Optical Society of America

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References

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  1. J. P. Powling, “Photonic & Sonic Band-Gap Bibliography” (Louisiana State University, Baton Rouge, La., December 1, 2004), http://phys.lsu.edu/?jdowling/pbgbib.html.
  2. D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, Phys. Rev. Lett. 76, 2480 (1996).
    [CrossRef] [PubMed]
  3. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 (1996).
    [CrossRef] [PubMed]
  4. J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
    [CrossRef]
  5. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
    [CrossRef] [PubMed]
  7. A. L. Pokrovsky and A. L. Efros, Phys. Rev. Lett. 89, 093901 (2002).
    [CrossRef]
  8. D. Felbacq and G. Bouchitté, Waves Random Media 7, 245 (1997).
    [CrossRef]
  9. D. Felbacq, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
    [CrossRef]
  10. M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
    [CrossRef]
  11. S. O’Brien and J. B. Pendry, J. Phys. (Paris) 14, 4035 (2002).
    [CrossRef]
  12. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2002).
  13. G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. 339, 377 (2004).
    [CrossRef]

2004 (1)

G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. 339, 377 (2004).
[CrossRef]

2002 (2)

S. O’Brien and J. B. Pendry, J. Phys. (Paris) 14, 4035 (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

1997 (1)

D. Felbacq and G. Bouchitté, Waves Random Media 7, 245 (1997).
[CrossRef]

1996 (2)

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

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

1994 (1)

1968 (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Bayindir, M.

M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
[CrossRef]

Bouchitté, G.

G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. 339, 377 (2004).
[CrossRef]

D. Felbacq and G. Bouchitté, Waves Random Media 7, 245 (1997).
[CrossRef]

Bulu, I.

M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
[CrossRef]

Cubukcu, E.

M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
[CrossRef]

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

Felbacq, D.

G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. 339, 377 (2004).
[CrossRef]

D. Felbacq and G. Bouchitté, Waves Random Media 7, 245 (1997).
[CrossRef]

D. Felbacq, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

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

Maystre, D.

Nevière, M.

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2002).

O’Brien, S.

S. O’Brien and J. B. Pendry, J. Phys. (Paris) 14, 4035 (2002).
[CrossRef]

Ozbay, E.

M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
[CrossRef]

Pendry, J. B.

S. O’Brien and J. B. Pendry, J. Phys. (Paris) 14, 4035 (2002).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

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

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

Popov, E.

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2002).

Powling, J. P.

J. P. Powling, “Photonic & Sonic Band-Gap Bibliography” (Louisiana State University, Baton Rouge, La., December 1, 2004), http://phys.lsu.edu/?jdowling/pbgbib.html.

Robins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

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]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

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

Tayeb, G.

Veselago, V. G.

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

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]

C. R. Math. Acad. Sci. (1)

G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. 339, 377 (2004).
[CrossRef]

Europhys. Lett. (1)

M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Europhys. Lett. 56, 41 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

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

J. Phys. (Paris) (1)

S. O’Brien and J. B. Pendry, J. Phys. (Paris) 14, 4035 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

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

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

A. L. Pokrovsky and A. L. Efros, Phys. Rev. Lett. 89, 093901 (2002).
[CrossRef]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Waves Random Media (1)

D. Felbacq and G. Bouchitté, Waves Random Media 7, 245 (1997).
[CrossRef]

Other (2)

J. P. Powling, “Photonic & Sonic Band-Gap Bibliography” (Louisiana State University, Baton Rouge, La., December 1, 2004), http://phys.lsu.edu/?jdowling/pbgbib.html.

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2002).

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Figures (4)

Fig. 1
Fig. 1

Sketch of the device along with the basic cell. We chose a d = 1 2 .

Fig. 2
Fig. 2

Transmission spectrum of the finite photonic crystal.

Fig. 3
Fig. 3

Real and imaginary parts of the effective permeability of the photonic crystal.

Fig. 4
Fig. 4

Map of the field above and below the photonic crystal for an incident Gaussian beam.

Equations (10)

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

div ( μ 1 u η ) + k 2 ε u η = 0 ,
Δ u η + k 2 u η = k 0 2 d η χ η ,
Δ u η + k 2 u η = Σ j P j δ j .
Σ j P j δ j = 2 π γ η 2 Σ j ( u j v j ) δ j .
Δ u + k 2 u = 2 π γ ( u v ) ,
Δ u + k 2 u = i k 0 2 κ ̃ v .
Δ u + ( k 2 2 π γ k 0 2 k 0 2 + 2 i π γ κ ̃ 1 ) u = 0 .
ε eff = ε m 2 π γ μ m ( k 0 2 + 2 i π γ κ ̃ 1 )
μ eff = μ m .
μ eff ( k ) = 1 + 64 a 2 π 4 ( n , m ) odd k 2 n 2 m 2 ( k ̃ n m 2 k 2 ) ,

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