Abstract

In a recent paper, Pendry [Phys. Rev. Lett. 86, 3966 (2000)] mentioned the possibility of making perfect lenses by using a slab of left-handed material with relative permeability and permittivity equal to -1, a property first stated by Veselago [Sov. Phys. Usp. 10, 509 (1968)]. Pendry gave a demonstration of the vital effect of the evanescent waves in this process, arguing that these waves are amplified inside the slab. We present first a very simple theoretical demonstration that a homogeneous material with both relative permittivity and permeability equal to -1 cannot exist, even for a unique frequency. This demonstration shows that the perfect lens proposed by Pendry can be interpreted as a means to move in real space the virtual perfect image of a point source given by a plane mirror. We show that, owing to evanescent waves, the concept of effective medium for heterogeneous materials is questionable, even when the wavelength of the incident light is much larger than the size of the heterogeneities. The effect of heterogeneities is compared with that of absorption. We conclude that a material able to focus the light more efficiently than the current devices (but not perfectly) could exist.

© 2004 Optical Society of America

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  2. G. W. ’t Hooft, “Comment on ‘Negative refraction makes a perfect lens,’ ” Phys. Rev. Lett. 87, 249701 (2001);J. B. Pendry, “Reply,” p. 249702.
    [CrossRef]
  3. J. M. Williams, “Some problems with negative refraction,” Phys. Rev. Lett. 87, 249703 (2001);J. B. Pendry, Reply, Phys. Rev. Lett. 87, 249704 (2001).
    [CrossRef] [PubMed]
  4. D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).
    [CrossRef]
  5. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
    [CrossRef] [PubMed]
  6. J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
    [CrossRef]
  7. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
    [CrossRef]
  8. S. Y. Lin, V. M. Hietala, L. Wang, E. D. Jones, “Highly dispersive photonic band-gap prism,” Opt. Lett. 21, 1771–1773 (1996).
    [CrossRef] [PubMed]
  9. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
    [CrossRef]
  10. C. M. Soukoulis, Photonic Band-Gap Materials (Kluwer Academic, Dordrecht, The Netherlands1996).
  11. S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
    [CrossRef]
  12. B. Gralak, S. Enoch, G. Tayeb, “Anolalous refractive properties of photonic crsytals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
    [CrossRef]
  13. V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ∊ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  14. P. M. Valanju, R. M. Walser, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401-1–4 (2002).
    [CrossRef] [PubMed]
  15. R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–15 (2001).
    [CrossRef]
  16. N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403-1–4 (2002).
    [CrossRef] [PubMed]
  17. J. B. Pendry, “Comment on ‘Left-handed materials do not make a perfect lens,’ ” Phys. Rev. Lett. 91, 99701-1 (2003).
    [CrossRef]
  18. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  19. J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
    [CrossRef] [PubMed]
  20. R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  21. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 53.
  22. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2, p. 501.
  23. John Pendry, Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, UK, 2003 (personal communication).
  24. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), p. 823.
  25. D. Maystre, “Rigorous vector theories of gratings,” in Progress in Optics, Vol. XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), p. 22.
  26. H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. 21, 657–662 (1973).
    [CrossRef]
  27. R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  28. D. W. Pohl, D. Courjon, Near Field Optics (Kluwer Academic, Dordrecht, The Netherlands, 1993).
  29. D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibres,” Waves Random Media 7, 245–256 (1997).
    [CrossRef]
  30. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals ((Princeton U. Press, Princeton, N.J., 1995).

2003

D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).
[CrossRef]

J. B. Pendry, “Comment on ‘Left-handed materials do not make a perfect lens,’ ” Phys. Rev. Lett. 91, 99701-1 (2003).
[CrossRef]

2002

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403-1–4 (2002).
[CrossRef] [PubMed]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

P. M. Valanju, R. M. Walser, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401-1–4 (2002).
[CrossRef] [PubMed]

2001

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–15 (2001).
[CrossRef]

G. W. ’t Hooft, “Comment on ‘Negative refraction makes a perfect lens,’ ” Phys. Rev. Lett. 87, 249701 (2001);J. B. Pendry, “Reply,” p. 249702.
[CrossRef]

J. M. Williams, “Some problems with negative refraction,” Phys. Rev. Lett. 87, 249703 (2001);J. B. Pendry, Reply, Phys. Rev. Lett. 87, 249704 (2001).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

2000

1999

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1998

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

1997

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibres,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

1996

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

S. Y. Lin, V. M. Hietala, L. Wang, E. D. Jones, “Highly dispersive photonic band-gap prism,” Opt. Lett. 21, 1771–1773 (1996).
[CrossRef] [PubMed]

1994

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

1987

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

1973

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. 21, 657–662 (1973).
[CrossRef]

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

1968

V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ∊ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

’t Hooft, G. W.

G. W. ’t Hooft, “Comment on ‘Negative refraction makes a perfect lens,’ ” Phys. Rev. Lett. 87, 249701 (2001);J. B. Pendry, “Reply,” p. 249702.
[CrossRef]

Bouchitté, G.

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibres,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

Bowden, C. M.

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Cadilhac, M.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 53.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2, p. 501.

Courjon, D.

D. W. Pohl, D. Courjon, Near Field Optics (Kluwer Academic, Dordrecht, The Netherlands, 1993).

Dowling, J. P.

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Enoch, S.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

B. Gralak, S. Enoch, G. Tayeb, “Anolalous refractive properties of photonic crsytals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
[CrossRef]

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Felbacq, D.

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibres,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), p. 823.

Garcia, N.

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403-1–4 (2002).
[CrossRef] [PubMed]

Gralak, B.

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Heyman, E.

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–15 (2001).
[CrossRef]

Hietala, V. M.

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. 2, p. 501.

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

Ikuno, H.

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. 21, 657–662 (1973).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals ((Princeton U. Press, Princeton, N.J., 1995).

Jones, E. D.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Lin, S. Y.

Maystre, D.

D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).
[CrossRef]

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

D. Maystre, “Rigorous vector theories of gratings,” in Progress in Optics, Vol. XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), p. 22.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals ((Princeton U. Press, Princeton, N.J., 1995).

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (Mc-Graw Hill, New York, 1953), p. 823.

Nieto-Vesperinas, M.

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403-1–4 (2002).
[CrossRef] [PubMed]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Comment on ‘Left-handed materials do not make a perfect lens,’ ” Phys. Rev. Lett. 91, 99701-1 (2003).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

Pendry, John

John Pendry, Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, UK, 2003 (personal communication).

Pohl, D. W.

D. W. Pohl, D. Courjon, Near Field Optics (Kluwer Academic, Dordrecht, The Netherlands, 1993).

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, Photonic Band-Gap Materials (Kluwer Academic, Dordrecht, The Netherlands1996).

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

B. Gralak, S. Enoch, G. Tayeb, “Anolalous refractive properties of photonic crsytals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
[CrossRef]

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Valanju, A. P.

P. M. Valanju, R. M. Walser, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401-1–4 (2002).
[CrossRef] [PubMed]

Valanju, P. M.

P. M. Valanju, R. M. Walser, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401-1–4 (2002).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ∊ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Walser, R. M.

P. M. Valanju, R. M. Walser, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401-1–4 (2002).
[CrossRef] [PubMed]

Wang, L.

Williams, J. M.

J. M. Williams, “Some problems with negative refraction,” Phys. Rev. Lett. 87, 249703 (2001);J. B. Pendry, Reply, Phys. Rev. Lett. 87, 249704 (2001).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals ((Princeton U. Press, Princeton, N.J., 1995).

Yasuura, K.

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. 21, 657–662 (1973).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

Zengerle, R.

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–15 (2001).
[CrossRef]

IEEE Trans. Antennas Propag.

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. 21, 657–662 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Mod. Opt.

D. Maystre, “Electromagnetic analysis of ultra-refraction and negative refraction,” J. Mod. Opt. 50, 1431–1444 (2003).
[CrossRef]

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap material,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Opt. Lett.

Phys. Rev. B

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Phys. Rev. E

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–15 (2001).
[CrossRef]

Phys. Rev. Lett.

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403-1–4 (2002).
[CrossRef] [PubMed]

J. B. Pendry, “Comment on ‘Left-handed materials do not make a perfect lens,’ ” Phys. Rev. Lett. 91, 99701-1 (2003).
[CrossRef]

P. M. Valanju, R. M. Walser, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401-1–4 (2002).
[CrossRef] [PubMed]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

G. W. ’t Hooft, “Comment on ‘Negative refraction makes a perfect lens,’ ” Phys. Rev. Lett. 87, 249701 (2001);J. B. Pendry, “Reply,” p. 249702.
[CrossRef]

J. M. Williams, “Some problems with negative refraction,” Phys. Rev. Lett. 87, 249703 (2001);J. B. Pendry, Reply, Phys. Rev. Lett. 87, 249704 (2001).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, “Extremely low frequency plasmon in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

Radio Sci.

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

Science

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ∊ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Waves Random Media

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibres,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

Other

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals ((Princeton U. Press, Princeton, N.J., 1995).

D. W. Pohl, D. Courjon, Near Field Optics (Kluwer Academic, Dordrecht, The Netherlands, 1993).

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Figures (2)

Fig. 1
Fig. 1

Presentation of the problem.

Fig. 2
Fig. 2

Different regions of the transverse wave vector α.

Equations (60)

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ES=H0(1)(k0SM),
2ES+k02ES=4iδ(x)δ(y-yS),
2E+k02E=4iδ(x)δ(y-yS)ify>0,
2E+k02E=0,if-e<y<0,
2E+k02E=0ify<-e.
U(x, y)=E(x, y)ify>0E(x,-y)if-e<y<0.
E˜S=H0(1)(k0S˜M).
Er(x, y)=-H0(1)(k0S˜M).
V(x, y)=ES(x, y)ify>0H0(1)(k0S˜M)=-Er(x, y)=ES(x,-y)ify<0
H0(1)(k0SM)=1π α=-+ 1β(α) exp[iαx+iβ(α)|y-yS|]dα,
β(α)=k02-α2 oriα2-k02.
H0(1)(k0SM)=1π α=-+ 1β(α)×exp[iαx-iβ(α)(y-yS)]dα=1π α=-+×exp[iβ(α)yS]β(α) exp[iαx-iβ(α)y]dα.
Ei=exp[iαx-iβ(α)y].
E=exp[iαx-iβ(α)y]+r exp[iαx+iβ(α)y]if0<y<yS.
E=a exp[iαx-iβ(α)y]+b exp[iαx+iβ(α)y]if-e<y<0.
E=t exp[iαx-iβ(α)y]ify<-e.
1+r=a+b,
a exp[iβ(α)e]+b exp[-iβ(α)e]=t exp[iβ(α)e],
-1+r=-(-a+b)=a-b,
-a exp[iβ(α)e]+b exp[-iβ(α)e]=-{-t exp[iβ(α)e]}=t exp[iβ(x)e].
r=a=0,
b=1,
t=exp[-2iβ(α)e].
E=1π α=-+ 1β(α) exp[iαx+iβ(α)×(y+yS)]dα,if-e<y<0,
E=1π α=-+exp[iβ(α)yS]β(α) exp[-2iβ(α)e]×exp[iαx-iβ(α)y]dα=1π α=-+ 1β(α) exp[iαx-iβ(α)(α)(y-yF)]dα ify<-e,
yF=yS-2e.
E=1π α=-+ 1β(α) exp[iαx+iβ(α)|y+yS|]dα=H0(1)(k0S˜M)if-yS<y<0.
I|α|1iπ|α| exp[iαx-|α|(y+yS)],
Iα±±1iπα exp[iαxα(y+yS)].
Iα+1iπα exp(iαu)=1iπαwα,
Iα--1iπα exp(iαu¯)=-1iπαw¯-α.
J+=α=Δα 1iπ wαα dα=1iπ n=1 wnΔαnΔαΔα=1iπ n=1 tnn,
J+=iπ ln(1-t)
limΔα0[J+]=iπ ln(-i)+iπ ln(1-t)=12+iπ ln(Δα)+iπ ln(u).
FPα=0 1iπ wαα dα=limΔα0α=Δα 1iπ wαα dα-iπ ln(Δα)=12+iπ ln[x+i(y+yS)].
FPα=-0 -1iπ w¯-αα dα=12+iπ ln[-x+i(y+yS)].
E=1π α=-+ 1β(α) exp[iαx-iβ(α)(y-yF)]dα
y<-e,E=H0(1)(k0FM).
Er=n=-+BnΦn(x),
ENr=n=-N+NBn(N)Φn(x).
E(x, y)=α=-+Eˆ(α, y)exp(iαx)dα.
E(x, y)=α=-K/2+K/2Eα(x, y)dα,
Eα(x, y)=m=--Eˆ(α+mK, y)exp[i(α+mK)x]
Eα(x+d, y)=exp(iαd)Eα(x, y).
E(x, y)=α=-K/h+K/hEˆ(α, y)exp[i(αx)]dα.
E=a exp[iαx-iγ(α)y]+b exp[iαx+iγ(α)y]if-d<y<0,
r=(1-p2)[1-exp(2iγe)]D,
t=4p exp[i(γ-β)e]D,
a=2(1+p)D,
b=-2(1-p)exp(2iγe)D,
D=(1+p)2-(1-p)2 exp(2iγe),
p=1μ γβ.
γβ-k02 iδ2β,p-1+k02 iδ2β2, (1+p)2-k04 δ24β4.
D-k04 δ24β4-4 exp(2iβe).
f=11+u,
u=k02δ4β22 exp(-2iβe).
ify=yF=yS-2e,E=α=-+f exp(-iβyS)exp(iαx)dα.
u=k02δ4β22 exp(-2iβe)=1.
k02δ4|βl|2 exp(|βl|e)=1.
πu-ln(u)=1.15n.

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