Abstract

Initial experiments on wedge samples composed of isotropic metamaterials with simultaneously negative permittivity and permeability have indicated that electromagnetic radiation can be negatively refracted. In more recently reported experiments [Phys. Rev. Lett. 90, 1074011 (2003)], indefinite metamaterial samples, for which the permittivity and permeability tensors are negative along only certain of the principal axes of the metamaterial, have also been used to demonstrate negative refraction. We present here a detailed analysis of the refraction and reflection behavior of electromagnetic waves at an interface between an indefinite medium and vacuum. We conclude that certain classes of indefinite media have identical refractive properties as isotropic negative index materials. However, there are limits to this correspondence, and other complicating phenomena may occur when indefinite media are substituted for isotropic negative index materials. We illustrate the results of our analysis with finite-element-based numerical simulations on planar slabs and wedges of negative index and indefinite media.

© 2004 Optical Society of America

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 4184–4187 (2000).
    [CrossRef]
  3. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 05662501–05662515 (2001).
    [CrossRef]
  4. See the special issue on negative refraction in Opt. Express 11, (2003), http://www.opticsexpress.org.
  5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 79–81 (2001).
    [CrossRef]
  6. P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 1874011–1874014 (2002).
    [CrossRef]
  7. N. Garcia and M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27, 885–887 (2002).
    [CrossRef]
  8. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
    [CrossRef]
  9. A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 1374011–1374014 (2003).
    [CrossRef]
  10. J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
    [CrossRef]
  11. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 1074021–1074024 (2003).
    [CrossRef]
  12. D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. 81, 2713–2715 (2002).
    [CrossRef]
  13. R. A. Silin, “Derivation of refraction and reflection laws by the isofrequency method,” Radiotekh. Elektron. (Moscow) 47, 186–191 (2002) [Translated version: R. A. Silin, J. Commun. Technol. Electron. 47, 169–174 (2002)].
  14. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
    [CrossRef]
  15. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 0774051–0774054 (2003).
    [CrossRef]
  16. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
    [CrossRef]
  17. R. K. Fisher and R. W. Gould, “Resonance cones in the field pattern of a short antenna in an anisotropic plasma,” Phys. Rev. Lett. 22, 1093–1095 (1969).
    [CrossRef]
  18. K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wireless Propag. Lett. 1, 146–149 (2002).
    [CrossRef]
  19. L. Hu and S. T. Chui, “Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials,” Phys. Rev. B 66, 0851081–0851087 (2002).
    [CrossRef]
  20. J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 391–394.
  21. A. Lakhtakia, “On perfect lenses and nihility,” Int. J. Infrared Millim. Waves 23, 339–343 (2002).
    [CrossRef]
  22. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  23. P. Kolinko and D. R. Smith, “Numerical study of electromagnetic waves interacting with negative index materials,” Opt. Express 11, 640–648 (2003), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  24. C. P. Parazzoli and K. Li, Phantom Works, The Boeing Company (personal communication, 2003).
  25. W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), p. 58.

2003 (5)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 1374011–1374014 (2003).
[CrossRef]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 1074021–1074024 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 0774051–0774054 (2003).
[CrossRef]

P. Kolinko and D. R. Smith, “Numerical study of electromagnetic waves interacting with negative index materials,” Opt. Express 11, 640–648 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

2002 (8)

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 1874011–1874014 (2002).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27, 885–887 (2002).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
[CrossRef]

D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. 81, 2713–2715 (2002).
[CrossRef]

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wireless Propag. Lett. 1, 146–149 (2002).
[CrossRef]

L. Hu and S. T. Chui, “Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials,” Phys. Rev. B 66, 0851081–0851087 (2002).
[CrossRef]

A. Lakhtakia, “On perfect lenses and nihility,” Int. J. Infrared Millim. Waves 23, 339–343 (2002).
[CrossRef]

2001 (3)

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
[CrossRef]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 05662501–05662515 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 79–81 (2001).
[CrossRef]

2000 (2)

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 4184–4187 (2000).
[CrossRef]

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

1969 (1)

R. K. Fisher and R. W. Gould, “Resonance cones in the field pattern of a short antenna in an anisotropic plasma,” Phys. Rev. Lett. 22, 1093–1095 (1969).
[CrossRef]

1968 (1)

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

Balmain, K. G.

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wireless Propag. Lett. 1, 146–149 (2002).
[CrossRef]

Brock, J. B.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 1374011–1374014 (2003).
[CrossRef]

Chuang, I. L.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 1374011–1374014 (2003).
[CrossRef]

Chui, S. T.

L. Hu and S. T. Chui, “Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials,” Phys. Rev. B 66, 0851081–0851087 (2002).
[CrossRef]

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 1074021–1074024 (2003).
[CrossRef]

Fisher, R. K.

R. K. Fisher and R. W. Gould, “Resonance cones in the field pattern of a short antenna in an anisotropic plasma,” Phys. Rev. Lett. 22, 1093–1095 (1969).
[CrossRef]

Foteinopoulou, S.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 1074021–1074024 (2003).
[CrossRef]

Garcia, N.

Gould, R. W.

R. K. Fisher and R. W. Gould, “Resonance cones in the field pattern of a short antenna in an anisotropic plasma,” Phys. Rev. Lett. 22, 1093–1095 (1969).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

Grzegorczyk, T. M.

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 05662501–05662515 (2001).
[CrossRef]

Houck, A. A.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 1374011–1374014 (2003).
[CrossRef]

Hu, L.

L. Hu and S. T. Chui, “Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials,” Phys. Rev. B 66, 0851081–0851087 (2002).
[CrossRef]

Ilvonen, S.

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
[CrossRef]

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
[CrossRef]

Kolinko, P.

Koltenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

Kong, J. A.

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

Kremer, P. C.

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wireless Propag. Lett. 1, 146–149 (2002).
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 4184–4187 (2000).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, “On perfect lenses and nihility,” Int. J. Infrared Millim. Waves 23, 339–343 (2002).
[CrossRef]

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

Lindell, I. V.

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
[CrossRef]

Luttgen, A. A. E.

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wireless Propag. Lett. 1, 146–149 (2002).
[CrossRef]

Nieto-Vesperinas, M.

Nikoskinen, K. I.

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
[CrossRef]

Pacheco Jr., J.

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

Pendry, J. B.

D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. 81, 2713–2715 (2002).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
[CrossRef]

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

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 79–81 (2001).
[CrossRef]

Schurig, D.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 0774051–0774054 (2003).
[CrossRef]

D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. 81, 2713–2715 (2002).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 79–81 (2001).
[CrossRef]

Smith, D. R.

P. Kolinko and D. R. Smith, “Numerical study of electromagnetic waves interacting with negative index materials,” Opt. Express 11, 640–648 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 0774051–0774054 (2003).
[CrossRef]

D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. 81, 2713–2715 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 79–81 (2001).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 4184–4187 (2000).
[CrossRef]

Soukoulis, C. M.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 1074021–1074024 (2003).
[CrossRef]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

Tretyakov, S. A.

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
[CrossRef]

Valanju, A. P.

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 1874011–1874014 (2002).
[CrossRef]

Valanju, P. M.

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 1874011–1874014 (2002).
[CrossRef]

Veselago, V. G.

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

Walser, R. M.

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 1874011–1874014 (2002).
[CrossRef]

Wu, B.-I.

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

Zhang, Y.

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 05662501–05662515 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. 81, 2713–2715 (2002).
[CrossRef]

IEEE Antennas Wireless Propag. Lett. (1)

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wireless Propag. Lett. 1, 146–149 (2002).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

A. Lakhtakia, “On perfect lenses and nihility,” Int. J. Infrared Millim. Waves 23, 339–343 (2002).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (2)

L. Hu and S. T. Chui, “Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials,” Phys. Rev. B 66, 0851081–0851087 (2002).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 2011041–2011044 (2002).
[CrossRef]

Phys. Rev. E (1)

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 05662501–05662515 (2001).
[CrossRef]

Phys. Rev. Lett. (9)

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

R. K. Fisher and R. W. Gould, “Resonance cones in the field pattern of a short antenna in an anisotropic plasma,” Phys. Rev. Lett. 22, 1093–1095 (1969).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 0774051–0774054 (2003).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 1074011–1074014 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 1374011–1374014 (2003).
[CrossRef]

J. Pacheco, Jr., T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 2574011–2574014 (2002).
[CrossRef]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 1074021–1074024 (2003).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 4184–4187 (2000).
[CrossRef]

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 1874011–1874014 (2002).
[CrossRef]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 79–81 (2001).
[CrossRef]

Sov. Phys. Usp. (1)

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

Other (5)

See the special issue on negative refraction in Opt. Express 11, (2003), http://www.opticsexpress.org.

J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 391–394.

R. A. Silin, “Derivation of refraction and reflection laws by the isofrequency method,” Radiotekh. Elektron. (Moscow) 47, 186–191 (2002) [Translated version: R. A. Silin, J. Commun. Technol. Electron. 47, 169–174 (2002)].

C. P. Parazzoli and K. Li, Phantom Works, The Boeing Company (personal communication, 2003).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), p. 58.

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Figures (20)

Fig. 1
Fig. 1

In a typical negative refraction experiment, a wave from free space impinges on the flat side of a prism sample. The wave passes through the material and undergoes refraction at the second interface as shown. In this diagram, arrows represent the direction of energy flow. In the ensuing analysis, waves are assumed polarized such that the electric field points along the z axis and the magnetic field lies in the plane of propagation (s polarization).

Fig. 2
Fig. 2

Isofrequency curves corresponding to free space (solid circle) and an isotropic medium with |n|=2 (dashed circle). Arrows indicate the graphical solution of an interface matching problem, showing both positive and negative refracting solutions.

Fig. 3
Fig. 3

Isofrequency curves corresponding to free space (solid circle) and an indefinite medium (dashed curves). Arrows indicate the graphical solution of an interface matching problem. The direction of energy flow within the indefinite medium is indicated by the arrows drawn normal to the hyperbolic isofrequency surface. The dark gray arrows indicate the case for negatively refracting indefinite media, and the light gray arrows indicate the case for positively refracting indefinite media.

Fig. 4
Fig. 4

Isofrequency curves corresponding to free space (solid circle) and an indefinite medium (dashed curves). The interface here is assumed to cut the indefinite medium along a nonprincipal axis. Arrows indicate the graphical solution of the interface matching problem and show that the two wave-vector solutions in the medium do not have the same magnitudes or directions.

Fig. 5
Fig. 5

Shown is a wave, incident from within an indefinite medium, that reflects and refracts from an interface with free space. The principal axes within the medium are indicated by the lighter lines, which show that the interface lies along one of the principal axes. The analysis in Section 4 follows this geometry. Note that even though the phase diagram indicates negative refraction, the group velocity may not in general exhibit negative refraction.

Fig. 6
Fig. 6

Reflectance for an isotropic medium with μ=2 and =1 (solid curve) and for an isotropic medium with μ=1 and =2 (dashed curve). The gray curve shows the angle of refraction (right axis).

Fig. 7
Fig. 7

Reflectance (black curve) and angle of refraction (gray curve) for an isotropic medium with μ=-2 and =-1. Note that the angle of refraction is for the phase, which is neither parallel nor antiparallel to the direction of energy flow.

Fig. 8
Fig. 8

Reflectance (black curve) and angle of refraction (gray curve) for an indefinite medium with μ=(-2, 1, 1) and z=-1. In this case, the refraction angle for the energy is actually positive.

Fig. 9
Fig. 9

Reflectance (black curve) and angle of refraction (gray curve) for an indefinite medium with μ=(-1, 1, 1) and z=-1.

Fig. 10
Fig. 10

Shown is a wave, incident from within an indefinite medium, that reflects and refracts from an interface with free space. The principal axes within the medium, depicted by the lighter lines, indicate that the interface does not lie along one of the principal axes. The analysis in Section 5 follows this geometry.

Fig. 11
Fig. 11

Reflectance (black curves) and angle of refraction (gray curves) for an indefinite medium with μ=(-1, 1, 1) and z=-1. The principal axes of the medium have been rotated by -10° (leftmost dashed curve), 0° (solid curve), and +10° (rightmost dashed curve). Note that the rotation is with respect to the surface normal, as defined in Fig. 10.

Fig. 12
Fig. 12

Reflectance (black curves) and angle of refraction (gray curves) for an indefinite medium with μx=-8, -6, -4, -2, -1, and -0.5, with z=-1.

Fig. 13
Fig. 13

Isofrequency curves for an indefinite medium (upper and lower dashed curves) and for free space (solid circle). The incident wave is assumed to propagate along the principal axis in the material, for which the phase and group velocities are antiparallel. The black arrow indicates the wave vector corresponding to the indefinite material considered. The short dashed line is plotted along the cut of the interface; a line perpendicular to the interface line and intersecting the tip of the incident wave vector defines the outgoing wave in free space. In this case, the wave undergoes negative refraction at the interface.

Fig. 14
Fig. 14

Negative refracting indefinite medium. The wave vector emerging from the wedge always lies on the same side of the surface normal as the incident wave vector, in the same manner as if the wedge had an isotropic negative index of refraction. The medium parameters are μ=(-1, 1, 1) with z=-1.

Fig. 15
Fig. 15

Ray diagrams showing the possibilities of subsidiary peaks due to multiple reflections in the sample. These ray diagrams take into account the anisotropic properties of the medium, with μ=(-2, 1, 1) for the wedge on the left and μ=(-1.5, 1, 1) for the wedge on the right. z=-1 in both cases.

Fig. 16
Fig. 16

Numerically computed spatial maps of the magnitude of the electric field for an incident wave refracting from a wedge. The incident wave is guided to the first interface of the wedge by an absorber, for which Re(μ)=Re()=1 and Im()=0.5. The material parameters of the wedge are (A) μ=(-1.4, 1, 1) and zz=-1.4, (B) μ=(-2, 1, 1) and zz=-2, (C) μ=(-2, 1, 1) and zz=-0.5, (D) μ=(-4, 1, 1) and zz=-0.25. In (A) and (B), the index varies but the impedance is matched to free space, whereas in (C) and (D) the index is fixed at n=1, but the impedance varies.

Fig. 17
Fig. 17

Angular plot of the log of the field intensity shown in Fig. 16(A), taken along the circumference of the circle depicted in Fig. 16(B). The dashed line corresponds to the angle of the surface normal, defined here as 0°.

Fig. 18
Fig. 18

Angular plot of the log of the field intensity shown in Fig. 16(D), taken along the circumference of the circle shown in Fig. 16(B). The dashed line corresponds to the angle of the surface normal, defined here as 0°.

Fig. 19
Fig. 19

Line source placed next to (A) an isotropic slab with =-1 and μ=-1, (B) a slab of negative refracting indefinite media with μx=-1 and z=-1, (C) a slab of positively refracting indefinite medium with μz=-1.

Fig. 20
Fig. 20

Spatial map of the computed fields for a converging negative index lens of (A) isotropic material with =μ=-1 and (B) indefinite media for which μx=-1 and z=-1. A plane wave is incident from the left.

Equations (18)

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=10001000zz μ=μxxμyx0μxyμyy0001.
kx2+ky2=ω2c2,
αqx2+βqy2+γqxqy=ω2c2,
α=μxxμxxμyy-μxyμyx 1z,
β=μyyμxxμyy-μxyμyx 1z,
γ=μyx+μxyμxxμyy-μxyμyx 1z,
qy=-γ2βqx±γ2βqx2-1β αqx2-ω2c21/2.
θI=tan-1 kxqy,
θT=tan-1 kxky,
θR=tan-1 kxqy.
EI=exp[i(qxx+qyy)]+r exp[i(qxx+qyy)],
ET=t exp[i(kxx+kyy)].
1+r=t.
H=-i cω μ-1×E.
xˆ·μ-1·qyxˆ-qxyˆky+rxˆ·μ-1·qyxˆ-qxyˆky=t.
r=-1-xˆ·μ-1·-qxxˆ+qyyˆky1-xˆ·μ-1·-qxxˆ+qyyˆky.
S=-c28πω E*×[μ-1(q×E)]=18π c2ω|E|2zˆ×[μ-1(q×zˆ)],
S=μz8π c2ω μqdet(μ)|E|2.

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