Abstract

We address the general issue of resolving the wave vector in complex electromagnetic media including negative refractive media. This requires us to make a physical choice of the sign of a square root imposed merely by conditions of causality. By considering the analytic behavior of the wave vector in the complex plane, it is shown that there are a total of eight physically distinct cases in the four quadrants of two Riemann sheets.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Butterworth, 1984).
  2. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
    [CrossRef] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
    [CrossRef] [PubMed]
  5. A. Grbic and G. V. Eleftheriades, J. Appl. Phys. 92, 5930 (2002).
    [CrossRef]
  6. C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef]
  7. A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 (2003).
    [CrossRef]
  8. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  9. D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000)
    [CrossRef] [PubMed]
  10. P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002).
    [CrossRef]
  11. A. L. Pokrovsky and A. L. Efros, Solid State Commun. 124, 283 (2002).
    [CrossRef]
  12. J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
    [CrossRef]
  13. S. O’Brien and J. B. Pendry, J. Phys.: Condens. Matter 14, 4035 (2002).
    [CrossRef]
  14. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
    [CrossRef]
  15. R. V. Churchill, Complex Variables and Applications, 3rd ed. (McGraw-Hill, 1974).
  16. R. A. Depine and A. Lakhtakia, Microwave Opt. Technol. Lett. 41, 315 (2004).
    [CrossRef]

2004

R. A. Depine and A. Lakhtakia, Microwave Opt. Technol. Lett. 41, 315 (2004).
[CrossRef]

2003

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

2002

A. Grbic and G. V. Eleftheriades, J. Appl. Phys. 92, 5930 (2002).
[CrossRef]

P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, Solid State Commun. 124, 283 (2002).
[CrossRef]

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

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

2001

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

2000

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

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

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000)
[CrossRef] [PubMed]

1968

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

Brock, J. B.

A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

Chuang, I. L.

A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

Churchill, R. V.

R. V. Churchill, Complex Variables and Applications, 3rd ed. (McGraw-Hill, 1974).

Depine, R. A.

R. A. Depine and A. Lakhtakia, Microwave Opt. Technol. Lett. 41, 315 (2004).
[CrossRef]

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, Solid State Commun. 124, 283 (2002).
[CrossRef]

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, J. Appl. Phys. 92, 5930 (2002).
[CrossRef]

Grbic, A.

A. Grbic and G. V. Eleftheriades, J. Appl. Phys. 92, 5930 (2002).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

Grzegorczyk, T. M.

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

Houck, A. A.

A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

Kong, J. A.

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

Koschny, T.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
[CrossRef]

Kotenbah, B. E.C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000)
[CrossRef] [PubMed]

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, Microwave Opt. Technol. Lett. 41, 315 (2004).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Butterworth, 1984).

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Butterworth, 1984).

Markos, P.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

O’Brien, S.

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

Pacheco, J.

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

Pendry, J. B.

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

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

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, Solid State Commun. 124, 283 (2002).
[CrossRef]

Schultz, S.

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Shelby, R. A.

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

Smith, D. R.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
[CrossRef]

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000)
[CrossRef] [PubMed]

Soukoulis, C. M.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
[CrossRef]

Tanielan, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

Valanju, A. P.

P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002).
[CrossRef]

Valanju, P. M.

P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002).
[CrossRef]

Veselago, V. G.

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

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Walser, R. M.

P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002).
[CrossRef]

Wu, B.-I.

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

Zhang, Y.

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

J. Appl. Phys.

A. Grbic and G. V. Eleftheriades, J. Appl. Phys. 92, 5930 (2002).
[CrossRef]

J. Phys.: Condens. Matter

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

Microwave Opt. Technol. Lett.

R. A. Depine and A. Lakhtakia, Microwave Opt. Technol. Lett. 41, 315 (2004).
[CrossRef]

Phys. Rev. E

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, Phys. Rev. E 68, 065602(R) (2003).
[CrossRef]

Phys. Rev. Lett.

J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Kotenbah, and M. Tanielan, Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

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

D. R. Smith and N. Kroll, Phys. Rev. Lett. 85, 2933 (2000)
[CrossRef] [PubMed]

P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[CrossRef] [PubMed]

Science

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

Solid State Commun.

A. L. Pokrovsky and A. L. Efros, Solid State Commun. 124, 283 (2002).
[CrossRef]

Sov. Phys. Usp.

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

Other

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Butterworth, 1984).

R. V. Churchill, Complex Variables and Applications, 3rd ed. (McGraw-Hill, 1974).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Complex plane for k z 2 , showing the different regions for the propagating and evanescent waves. The branch cut along the negative imaginary axis for the square root is shown.

Fig. 2
Fig. 2

Complex plane for k z = Z with the corresponding eight regions in the two Riemann sheets for k z 2 . The evanescent regions could be absorbing or amplifying as explained in the text, depending on the signs of ϵ and μ . The sign of the square root for each region is shown enclosed in a small circle.

Equations (4)

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

V d 3 r ω [ ϵ ( ω ) E ( r , ω ) 2 + μ ( ω ) H ( r , ω ) 2 ] d ω 2 π ,
k × E = ω c μ H , k × H = ω c ϵ E ,
k 2 = k x 2 + k y 2 + k z 2 = ϵ μ ω 2 c 2 ,
k z = ± ϵ μ ω 2 c 2 k x 2 ,

Metrics