Abstract

Abstract

The theory of determining the sign of the refractive index in active materials is discussed. Animations of numerical simulations are presented, supporting the claim that negative refractive index may occur in right-handed media. An example of such a medium, in the form of a lumped circuit model with active and passive resonances, is presented.

© 2007 Optical Society of America

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References

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative ε and μ," Sov. Phys. Usp. 10(4), 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
    [CrossRef]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999).
    [CrossRef]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
    [CrossRef]
  5. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50(12), 2702-2712 (2002).
    [CrossRef]
  6. S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67(20), 201101 (2003).
    [CrossRef]
  7. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, "Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium," Opt. Lett. 31, 3022 (2006).
    [CrossRef] [PubMed]
  8. A. K. Popov and V. M. Shalaev, "Compensating losses in negative-index metamaterials by optical parametric amplification," Opt. Lett. 31, 2169 (2006).
    [CrossRef] [PubMed]
  9. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
    [CrossRef]
  10. Y.-F. Chen, P. Fischer, and F.W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95(6), 067402 (2005).
    [CrossRef]
  11. J. Skaar, "Fresnel equations and the refractive index of active media," Phys. Rev. E 73, 026605 (2006).
    [CrossRef]
  12. Y.-F. Chen, P. Fischer, and F. W. Wise, "Sign of the refractive index in a gain medium with negative permittivity and permeability," J. Opt. Soc. Am. B 23, 45-50 (2006).
    [CrossRef]
  13. T. G. Mackay and A. Lakhtakia, "Comment on "Negative refraction at optical frequencies in nonmagnetic twocomponent molecular media," Phys. Rev. Lett. 96(15), 159701 (2006).
    [CrossRef]
  14. Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (2006).
    [CrossRef]
  15. S. A. Ramakrishna, "Comment on "Negative refraction at optical frequencies in nonmagnetic two-component molecular media"," Phys. Rev. Lett. 98(5), 059701 (2007).
    [CrossRef]
  16. Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007).
    [CrossRef]
  17. S. A. Ramakrishna and O. J. F. Martin, "Resolving the wave vector in negative refractive index media," Opt. Lett. 30, 2626 (2005).
    [CrossRef] [PubMed]
  18. A. N. Grigorenko, "Negative refractive index in artificial metamaterials," Opt. Lett. 31, 2483 (2006).
    [CrossRef] [PubMed]
  19. J. Skaar, "On resolving the refractive index and the wave vector," Opt. Lett. 31, 3372 (2006).
    [CrossRef] [PubMed]
  20. L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).
  21. L. D. Landau and E. M. Lifshits, Electrodynamics of continuous media, chap. IX.62 (Pergamon Press, 1960).
  22. In general, we can set © =0+ provided the refractive index is analytic in the upper half-plane and the denominator in the Fresnel equations is nonzero in the upper half-plane. Excluding media with absolute instabilities, the refractive index can always be identified as an analytic function in the upper half-plane.
  23. Poles in the upper half-plane mean that the slab will start lasing.
  24. T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
    [CrossRef]
  25. B. Nistad and J. Skaar, in Photonic metamaterials: From random to periodic, V. M. Shalaev and A. Genack, eds. (OSA, 2006). ISBN:1-55752-808-X.
  26. J.-S. Lee and Y.-S. Kwon, "Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback," Electron. Lett. 34(18), 1758-1760 (1998).
    [CrossRef]
  27. L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985).
    [CrossRef]
  28. D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).
  29. M. Born and E. Wolf, Principles of Optics, chap. 1.6.5, pp. 66-67, 3rd ed. (Pergamon Press, 1965).

2007 (3)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

S. A. Ramakrishna, "Comment on "Negative refraction at optical frequencies in nonmagnetic two-component molecular media"," Phys. Rev. Lett. 98(5), 059701 (2007).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007).
[CrossRef]

2006 (8)

2005 (3)

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

S. A. Ramakrishna and O. J. F. Martin, "Resolving the wave vector in negative refractive index media," Opt. Lett. 30, 2626 (2005).
[CrossRef] [PubMed]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95(6), 067402 (2005).
[CrossRef]

2003 (1)

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67(20), 201101 (2003).
[CrossRef]

2002 (1)

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50(12), 2702-2712 (2002).
[CrossRef]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999).
[CrossRef]

1998 (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
[CrossRef]

J.-S. Lee and Y.-S. Kwon, "Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback," Electron. Lett. 34(18), 1758-1760 (1998).
[CrossRef]

1985 (1)

L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative ε and μ," Sov. Phys. Usp. 10(4), 509-514 (1968).
[CrossRef]

Adegoke, J.

Bahoura, M.

Chen, Y.-F.

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F. W. Wise, "Sign of the refractive index in a gain medium with negative permittivity and permeability," J. Opt. Soc. Am. B 23, 45-50 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95(6), 067402 (2005).
[CrossRef]

Chua, L. O.

L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985).
[CrossRef]

Drachev, V. P.

Economou, E. N.

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

Eleftheriades, G. V.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50(12), 2702-2712 (2002).
[CrossRef]

Fischer, P.

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F. W. Wise, "Sign of the refractive index in a gain medium with negative permittivity and permeability," J. Opt. Soc. Am. B 23, 45-50 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95(6), 067402 (2005).
[CrossRef]

Grigorenko, A. N.

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
[CrossRef]

Iyer, A. K.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50(12), 2702-2712 (2002).
[CrossRef]

Koschny, T.

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

Kremer, P. C.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50(12), 2702-2712 (2002).
[CrossRef]

Kwon, Y.-S.

J.-S. Lee and Y.-S. Kwon, "Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback," Electron. Lett. 34(18), 1758-1760 (1998).
[CrossRef]

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, "Comment on "Negative refraction at optical frequencies in nonmagnetic twocomponent molecular media," Phys. Rev. Lett. 96(15), 159701 (2006).
[CrossRef]

Lee, J.-S.

J.-S. Lee and Y.-S. Kwon, "Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback," Electron. Lett. 34(18), 1758-1760 (1998).
[CrossRef]

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, "Comment on "Negative refraction at optical frequencies in nonmagnetic twocomponent molecular media," Phys. Rev. Lett. 96(15), 159701 (2006).
[CrossRef]

Markos, P.

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

Martin, O. J. F.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

Noginov, M. A.

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

Pendry, J. B.

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67(20), 201101 (2003).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
[CrossRef]

Popov, A. K.

Ramakrishna, S. A.

S. A. Ramakrishna, "Comment on "Negative refraction at optical frequencies in nonmagnetic two-component molecular media"," Phys. Rev. Lett. 98(5), 059701 (2007).
[CrossRef]

S. A. Ramakrishna and O. J. F. Martin, "Resolving the wave vector in negative refractive index media," Opt. Lett. 30, 2626 (2005).
[CrossRef] [PubMed]

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67(20), 201101 (2003).
[CrossRef]

Ritzo, B. A.

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

Shalaev, V. M.

Skaar, J.

J. Skaar, "Fresnel equations and the refractive index of active media," Phys. Rev. E 73, 026605 (2006).
[CrossRef]

J. Skaar, "On resolving the refractive index and the wave vector," Opt. Lett. 31, 3372 (2006).
[CrossRef] [PubMed]

Small, C. E.

Smith, D. R.

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

Soukoulis, C. M.

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative ε and μ," Sov. Phys. Usp. 10(4), 509-514 (1968).
[CrossRef]

Vier, D. C.

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

Wise, F. W.

Wise, F.W.

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95(6), 067402 (2005).
[CrossRef]

Yu, J.

L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985).
[CrossRef]

Yu, Y.

L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985).
[CrossRef]

Zhu, G.

Electron. Lett. (1)

J.-S. Lee and Y.-S. Kwon, "Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback," Electron. Lett. 34(18), 1758-1760 (1998).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999).
[CrossRef]

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50(12), 2702-2712 (2002).
[CrossRef]

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

J. Phys.: Condens. Matter (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998).
[CrossRef]

Nat. Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. B (2)

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67(20), 201101 (2003).
[CrossRef]

T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005).
[CrossRef]

Phys. Rev. E (1)

J. Skaar, "Fresnel equations and the refractive index of active media," Phys. Rev. E 73, 026605 (2006).
[CrossRef]

Phys. Rev. Lett. (6)

Y.-F. Chen, P. Fischer, and F.W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95(6), 067402 (2005).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000).
[CrossRef]

T. G. Mackay and A. Lakhtakia, "Comment on "Negative refraction at optical frequencies in nonmagnetic twocomponent molecular media," Phys. Rev. Lett. 96(15), 159701 (2006).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (2006).
[CrossRef]

S. A. Ramakrishna, "Comment on "Negative refraction at optical frequencies in nonmagnetic two-component molecular media"," Phys. Rev. Lett. 98(5), 059701 (2007).
[CrossRef]

Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative ε and μ," Sov. Phys. Usp. 10(4), 509-514 (1968).
[CrossRef]

Other (7)

L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).

L. D. Landau and E. M. Lifshits, Electrodynamics of continuous media, chap. IX.62 (Pergamon Press, 1960).

In general, we can set © =0+ provided the refractive index is analytic in the upper half-plane and the denominator in the Fresnel equations is nonzero in the upper half-plane. Excluding media with absolute instabilities, the refractive index can always be identified as an analytic function in the upper half-plane.

Poles in the upper half-plane mean that the slab will start lasing.

B. Nistad and J. Skaar, in Photonic metamaterials: From random to periodic, V. M. Shalaev and A. Genack, eds. (OSA, 2006). ISBN:1-55752-808-X.

D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics, chap. 1.6.5, pp. 66-67, 3rd ed. (Pergamon Press, 1965).

Supplementary Material (4)

» Media 1: MOV (2577 KB)     
» Media 2: MOV (1508 KB)     
» Media 3: MOV (3073 KB)     
» Media 4: MOV (3490 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Refractive index vs. frequency for the two choices n 1(ω) and n 2(ω).

Fig. 2.
Fig. 2.

Numerical simulation of the time-domain electric field E(z,t) for a semi-infinite medium with refractive index n(ω)=n1(ω) and excitation frequency ω 1=1.4. The excitation, cos(ω1t), is initiated at t=0. After roughly a time 70/ω 0 the steady-state solution (monochromatic solution) has been built up. The excitation is turned off at t=100/ω 0. Frame grabbed at t=31.6/ω 0. [fig2.mov 2.5MB] [Media 1]

Fig. 3.
Fig. 3.

Numerical simulation of E(z,t) of a semi-infinite medium with n(ω)=n 1(ω) and ω 1=1.4. Frame grabbed at t=0.3/ω 0. Note that fields exist in front of the red line z=ct. The figure does not have the same axes and time scale as for Fig. 2. [fig3.mov 1.5MB] [Media 2]

Fig. 4.
Fig. 4.

Numerical simulation of E(z,t) for a medium with ε(ω)=(1+f(ω))2, µ(ω)=1, finite thickness ω0d/c=35, and ω 1=1.4ω 0. The excitation is initiated at t=0 and turned off at t=100/ω 0. The wave front z=ct is indicated with a vertical red line. Frame grabbed at t=31.6/ω 0. [fig4.mov 3MB] [Media 3]

Fig. 5.
Fig. 5.

A general transmission line representation.

Fig. 6.
Fig. 6.

Transmission line with additional shunt admittance Ys .

Fig. 7.
Fig. 7.

Refractive index vs. frequency for the transmission-line effective medium.

Fig. 8.
Fig. 8.

R vs. frequency for total length d=20Λ.

Fig. 9.
Fig. 9.

|S 11-R| vs frequency for Λ=λ 0/8,λ0/16,λ0/30,λ0/60. λ0=25cm is the vacuum wavelength where |R| is maximal.

Fig. 10.
Fig. 10.

Electrical field vs. distance and time, for a causal excitation [u(t)-u(t-20ns)]cos(2πf 1 t); f1=1.22GHz, d=150Λ, and Λ=0.83cm. Frame grabbed at t=3.1ns. [fig10.mov 3.4 MB] [Media 4]

Equations (26)

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R = η 1 η + 1 ,
S = 2 η η + 1 exp ( i ω n z c ) .
f ( ω ) = F ω 0 2 ω 0 2 ω 2 i ω Γ .
E ( z , t ) = i 2 π i γ i γ + ω S exp ( i ω t ) ω 1 2 ω 2 d ω .
R = ( η 2 1 ) exp ( ikd ) ( η 2 1 ) exp ( ikd ) ( η + 1 ) 2 exp ( ikd ) ( η 1 ) 2 exp ( ikd ) ,
S + = 2 η ( η + 1 ) ( η + 1 ) 2 ( η 1 ) 2 exp ( 2 ikd ) ,
S = 2 η ( η 1 ) ( η 1 ) 2 ( η + 1 ) 2 exp ( 2 ikd ) ,
T = 4 η ( η + 1 ) 2 exp ( ikd ) ( η 1 ) 2 exp ( ikd ) .
n = ε μ exp [ i ( φ ε + φ μ ) 2 ] ,
d V d z = IZ ,
d I d z = VY .
d 2 V d z 2 + β 2 V = 0 , β 2 = ZY ,
d E y d z = i ω μ μ 0 H x ,
d H x d z = i ω ε μ 0 E y ,
μ = Z i ω μ 0 ,
ε = Y i ω ε 0 .
β 2 = ZY = ω 2 c 2 ε μ .
Y = i ω C l 1 i ω L n i ω L p 1 L p C p ω 2 i ω R p L p + R a + R n R a R n 1 L a C a ω 2 i ω ( R a + R n ) C a 1 L a C a ω 2 i ω R a C a .
ε ( ω ) = 1 ε 0 Λ [ C l 1 ω 2 L n + 1 L p 1 L p C p ω 2 i ω R p L p R a + R n i ω R a R n 1 L a C a ω 2 i ω ( R a + R n ) C a 1 L a C a ω 2 i ω R a C a ] ,
μ ( ω ) = L l μ 0 Λ .
T tl = [ cos ( β 0 Λ ) i sin ( β 0 Λ ) Z 0 i sin ( β 0 Λ ) Z 0 cos ( β 0 Λ ) ] ,
T s = [ 1 0 Y s 1 ] .
T = T uc N = [ A B C D ] .
S = [ S 11 S 12 S 21 S 22 ] = [ A + B Z 0 C Z 0 D A + B Z 0 + C Z 0 + D 2 ( AD BC ) A + B Z 0 + C Z 0 + D 2 A + B Z 0 + C Z 0 + D A + B Z 0 C Z 0 + D A + B Z 0 + C Z 0 + D ] .
ϒ = [ 0 i β 0 Z 0 i β 0 Z 0 + Y s Λ 0 ] .
[ V 2 ( ω ) I 2 ( ω ) ] = T uc 1 [ V 1 ( ω ) I 1 ( ω ) ] ,

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