Abstract

The identification of the refractive index and the wave vector for general (possibly active) linear, isotropic, homogeneous, and nonspatially dispersive media is discussed. Correct conditions for negative refraction necessarily include the global properties of the permittivity and permeability functions ε=ε(ω) and μ=μ(ω). On the other hand, a necessary and sufficient condition for left handedness can be identified at a single frequency (Reεε+Reμμ<0). At oblique incidence to semi-infinite, active media, it is explained that the wave vector generally loses its usual interpretation for real frequencies.

© 2006 Optical Society of America

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  1. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  2. S. A. Ramakrishna and J. B. Pendry, Phys. Rev. B 67, 201101(R) (2003).
    [CrossRef]
  3. S. A. Ramakrishna, Opt. Lett. 30, 2626 (2005).
    [CrossRef] [PubMed]
  4. S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
    [CrossRef]
  5. T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
    [CrossRef] [PubMed]
  6. Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
    [CrossRef] [PubMed]
  7. Y.-F. Chen, P. Fischer, and F. W. Wise, J. Opt. Soc. Am. B 23, 45 (2006).
    [CrossRef]
  8. J. Skaar, Phys. Rev. E 73, 026605 (2006).
    [CrossRef]
  9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960), Chap. 9.
  10. Writing epsi(omega)=1+int inf 0 x(t) exp(iwt)dt [and similarly for µ(omega)], we restrict ourselves to stable media in the sense that the response function x(t) is bounded.
  11. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  12. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1.
  13. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  14. M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys. 22, 353 (2002).
    [CrossRef]
  15. R. A. Depine and A. Lakhtakia, Microwave Opt. Technol. Lett. 41, 315 (2004).
    [CrossRef]
  16. For passive media, Im [n(omega)omega-omega]>0 for real frequencies. As the function Im [n(omega)omega-omega] is harmonic, Poisson's integral formula ensures that Im [n(omega)omega-omega]>0 remains valid in the upper halfplane. It follows that n(omega)omega cannot be real there.
  17. If the function kz(omega,kx)=sqrt n2 (omega) omega2/c2-kx2 was discontinuous in kx, we could find a (omega,kx) and a tiny delta>0 such that kz(omega,kx+delta)ap -kz(omega,kx). This leads to a contradiction, since kz(omega,kx) is continuous in omega, and kz(omega,kx+delta)-->kz(omega,kx) as Re omega-->inf.

2006 (3)

T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, J. Opt. Soc. Am. B 23, 45 (2006).
[CrossRef]

J. Skaar, Phys. Rev. E 73, 026605 (2006).
[CrossRef]

2005 (3)

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

S. A. Ramakrishna, Opt. Lett. 30, 2626 (2005).
[CrossRef] [PubMed]

S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
[CrossRef]

2004 (1)

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

2003 (1)

S. A. Ramakrishna and J. B. Pendry, Phys. Rev. B 67, 201101(R) (2003).
[CrossRef]

2002 (1)

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys. 22, 353 (2002).
[CrossRef]

2000 (1)

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

1968 (1)

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

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Chen, Y.-F.

Y.-F. Chen, P. Fischer, and F. W. Wise, J. Opt. Soc. Am. B 23, 45 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Depine, R. A.

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

Fischer, P.

Y.-F. Chen, P. Fischer, and F. W. Wise, J. Opt. Soc. Am. B 23, 45 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
[CrossRef] [PubMed]

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

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys. 22, 353 (2002).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960), Chap. 9.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960), Chap. 9.

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
[CrossRef] [PubMed]

McCall, M. W.

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys. 22, 353 (2002).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1.

Pendry, J. B.

S. A. Ramakrishna and J. B. Pendry, Phys. Rev. B 67, 201101(R) (2003).
[CrossRef]

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

Ramakrishna, S. A.

S. A. Ramakrishna, Opt. Lett. 30, 2626 (2005).
[CrossRef] [PubMed]

S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
[CrossRef]

S. A. Ramakrishna and J. B. Pendry, Phys. Rev. B 67, 201101(R) (2003).
[CrossRef]

Skaar, J.

J. Skaar, Phys. Rev. E 73, 026605 (2006).
[CrossRef]

Veselago, V. G.

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

Weiglhofer, W. S.

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys. 22, 353 (2002).
[CrossRef]

Wise, F. W.

Y.-F. Chen, P. Fischer, and F. W. Wise, J. Opt. Soc. Am. B 23, 45 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Eur. J. Phys. (1)

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys. 22, 353 (2002).
[CrossRef]

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

Microwave Opt. Technol. Lett. (1)

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

Opt. Lett. (1)

Phys. Rev. B (1)

S. A. Ramakrishna and J. B. Pendry, Phys. Rev. B 67, 201101(R) (2003).
[CrossRef]

Phys. Rev. E (1)

J. Skaar, Phys. Rev. E 73, 026605 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

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

T. G. Mackay and A. Lakhtakia, Phys. Rev. Lett. 96, 159701 (2006).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
[CrossRef]

Sov. Phys. Usp. (1)

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

Other (6)

For passive media, Im [n(omega)omega-omega]>0 for real frequencies. As the function Im [n(omega)omega-omega] is harmonic, Poisson's integral formula ensures that Im [n(omega)omega-omega]>0 remains valid in the upper halfplane. It follows that n(omega)omega cannot be real there.

If the function kz(omega,kx)=sqrt n2 (omega) omega2/c2-kx2 was discontinuous in kx, we could find a (omega,kx) and a tiny delta>0 such that kz(omega,kx+delta)ap -kz(omega,kx). This leads to a contradiction, since kz(omega,kx) is continuous in omega, and kz(omega,kx+delta)-->kz(omega,kx) as Re omega-->inf.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960), Chap. 9.

Writing epsi(omega)=1+int inf 0 x(t) exp(iwt)dt [and similarly for µ(omega)], we restrict ourselves to stable media in the sense that the response function x(t) is bounded.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1.

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Equations (5)

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

n = ε μ exp [ i ( arg ε + arg μ ) 2 ] ,
arg ε + arg μ > π .
n = ε μ exp [ i ( φ ε + φ μ ) 2 ] ,
Re ε ε + Re μ μ < 0 .
f ( ω ) = F ω 0 2 ω 0 2 ω 2 i ω Γ .

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