Abstract

An all-optical refractive index switch is proposed based on coherent control of a doped double-negative metamaterial slab. By embedding the three-level atoms through the left-handed slab, the effective permittivity of the slab becomes a function of the doped atom’s susceptibility. Absorption properties manipulation of the doped atoms leads to active control over the total loss of the metamaterial slab. This way the refractive index of the slab can be switched between negative and positive by the change of the relative phase of the applied fields. This may offer active control over the transmission in a wide range from 0 to 0.58.

© 2014 Optical Society of America

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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. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  3. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
    [CrossRef]
  4. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
    [CrossRef]
  5. W. Cai and V. M. Šalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).
  6. S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
    [CrossRef]
  7. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
    [CrossRef]
  8. 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, 3022–3024 (2006).
    [CrossRef]
  9. 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, 201101 (2003).
    [CrossRef]
  10. T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
    [CrossRef]
  11. A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75, 085436 (2007).
    [CrossRef]
  12. M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  13. P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101, 167401 (2008).
    [CrossRef]
  14. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
    [CrossRef]
  15. J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
    [CrossRef]
  16. Z. Huang, Th. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012).
    [CrossRef]
  17. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
    [CrossRef]
  18. O. A. Kocharovskaia and I. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without population inversion,” Pis ma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki 48, 581–584 (1988).
  19. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
    [CrossRef]
  20. A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
    [CrossRef]
  21. A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” arXiv preprint nlin/0506006 (2005).
  22. L.-G. Wang, H. Chen, and S.-Y. Zhu, “Superluminal pulse reflection and transmission in a slab system doped with dispersive materials,” Phys. Rev. E 70, 066602 (2004).
    [CrossRef]
  23. N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
    [CrossRef]
  24. S. Chakrabarti, S. A. Ramakrishna, and H. Wanare, “Coherently controlling metamaterials,” Opt. Express 16, 19504–19511 (2008).
    [CrossRef]
  25. A. K. Popov, S. A. Myslivets, T. F. George, and V. M. Shalaev, “Four-wave mixing, quantum control, and compensating losses in doped negative-index photonic metamaterials,” Opt. Lett. 32, 3044–3046 (2007).
    [CrossRef]
  26. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  27. R. Marqués, F. Martín, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).
  28. J.-H. Wu and J.-Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
    [CrossRef]
  29. J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407–412 (1992).
    [CrossRef]
  30. D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  31. Th. Koschny, P. Markoš, E. 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, 245105 (2005).
    [CrossRef]
  32. P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
    [CrossRef]
  33. S. D. Gupta, R. Arun, and G. S. Agarwal, “Subluminal to superluminal propagation in a left-handed medium,” Phys. Rev. B 69, 113104 (2004).
    [CrossRef]
  34. J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
    [CrossRef]
  35. E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).
  36. E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
    [CrossRef]

2012 (1)

Z. Huang, Th. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012).
[CrossRef]

2011 (1)

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef]

2010 (3)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef]

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

2009 (1)

J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
[CrossRef]

2008 (2)

S. Chakrabarti, S. A. Ramakrishna, and H. Wanare, “Coherently controlling metamaterials,” Opt. Express 16, 19504–19511 (2008).
[CrossRef]

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101, 167401 (2008).
[CrossRef]

2007 (4)

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75, 085436 (2007).
[CrossRef]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

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

A. K. Popov, S. A. Myslivets, T. F. George, and V. M. Shalaev, “Four-wave mixing, quantum control, and compensating losses in doped negative-index photonic metamaterials,” Opt. Lett. 32, 3044–3046 (2007).
[CrossRef]

2006 (3)

2005 (3)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[CrossRef]

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

2004 (2)

S. D. Gupta, R. Arun, and G. S. Agarwal, “Subluminal to superluminal propagation in a left-handed medium,” Phys. Rev. B 69, 113104 (2004).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Superluminal pulse reflection and transmission in a slab system doped with dispersive materials,” Phys. Rev. E 70, 066602 (2004).
[CrossRef]

2003 (2)

P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef]

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, 201101 (2003).
[CrossRef]

2002 (4)

D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef]

J.-H. Wu and J.-Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

1999 (2)

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[CrossRef]

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

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[CrossRef]

1992 (1)

J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407–412 (1992).
[CrossRef]

1988 (1)

O. A. Kocharovskaia and I. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without population inversion,” Pis ma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki 48, 581–584 (1988).

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]

Adegoke, J.

Agarwal, G. S.

S. D. Gupta, R. Arun, and G. S. Agarwal, “Subluminal to superluminal propagation in a left-handed medium,” Phys. Rev. B 69, 113104 (2004).
[CrossRef]

Arun, R.

S. D. Gupta, R. Arun, and G. S. Agarwal, “Subluminal to superluminal propagation in a left-handed medium,” Phys. Rev. B 69, 113104 (2004).
[CrossRef]

Atwater, H. A.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef]

Bahoura, M.

Baldit, E.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Bencheikh, K.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Brentenaker, F.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Briaudeau, S.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Burgos, S. P.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef]

Cai, W.

W. Cai and V. M. Šalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

Chakrabarti, S.

Chen, H.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Superluminal pulse reflection and transmission in a slab system doped with dispersive materials,” Phys. Rev. E 70, 066602 (2004).
[CrossRef]

N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Clark, S. M.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” arXiv preprint nlin/0506006 (2005).

Crozatier, V.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Dawes, A.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” arXiv preprint nlin/0506006 (2005).

de Waele, R.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef]

Dolling, G.

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

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, 3022–3024 (2006).
[CrossRef]

Economou, E.

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

Enkrich, C.

Gao, J.-Y.

J.-H. Wu and J.-Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

Gauthier, D. J.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” arXiv preprint nlin/0506006 (2005).

George, T. F.

Goldner, Ph.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Guillot-Noël, O.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Gupta, S. D.

S. D. Gupta, R. Arun, and G. S. Agarwal, “Subluminal to superluminal propagation in a left-handed medium,” Phys. Rev. B 69, 113104 (2004).
[CrossRef]

Hamm, J. M.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef]

Harris, S. E.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[CrossRef]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[CrossRef]

Hau, L. V.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[CrossRef]

Hess, O.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef]

Holden, A. J.

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

Huang, Z.

Z. Huang, Th. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012).
[CrossRef]

Illing, L.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” arXiv preprint nlin/0506006 (2005).

Javanainen, J.

J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407–412 (1992).
[CrossRef]

Khanin, I. I.

O. A. Kocharovskaia and I. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without population inversion,” Pis ma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki 48, 581–584 (1988).

Kildishev, A. V.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

Kinsler, P.

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101, 167401 (2008).
[CrossRef]

Klar, T. A.

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

Kocharovskaia, O. A.

O. A. Kocharovskaia and I. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without population inversion,” Pis ma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki 48, 581–584 (1988).

Koschny, Th.

Z. Huang, Th. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012).
[CrossRef]

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

Le Gouët, J. L.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Levenson, J. A.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Linden, S.

Litvak, A. G.

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef]

Liu, N.

N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Lorgeré, I.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

Luo, H.

J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
[CrossRef]

Markoš, P.

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef]

D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Marqués, R.

R. Marqués, F. Martín, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).

Martín, F.

R. Marqués, F. Martín, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).

McCall, M. W.

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101, 167401 (2008).
[CrossRef]

Monnier, P.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Myslivets, S. A.

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Noginov, M. A.

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, 201101 (2003).
[CrossRef]

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

Polman, A.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef]

Popov, A. K.

Ramakrishna, S. A.

S. Chakrabarti, S. A. Ramakrishna, and H. Wanare, “Coherently controlling metamaterials,” Opt. Express 16, 19504–19511 (2008).
[CrossRef]

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[CrossRef]

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, 201101 (2003).
[CrossRef]

Ritzo, B. A.

Robbins, D. J.

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

Rouget, V.

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Šalaev, V. M.

W. Cai and V. M. Šalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

Sarychev, A. K.

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75, 085436 (2007).
[CrossRef]

Schultz, P.

D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

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

A. K. Popov, S. A. Myslivets, T. F. George, and V. M. Shalaev, “Four-wave mixing, quantum control, and compensating losses in doped negative-index photonic metamaterials,” Opt. Lett. 32, 3044–3046 (2007).
[CrossRef]

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, 3022–3024 (2006).
[CrossRef]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

Small, C. E.

Smith, D. R.

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

Smith, D. R. S.

D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sorolla, M.

R. Marqués, F. Martín, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).

Soukoulis, C. M.

Z. Huang, Th. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

P. Markoš and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef]

D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Stewart, W. J.

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

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

Tartakovsky, G.

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75, 085436 (2007).
[CrossRef]

Tokman, M. D.

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef]

Tsakmakidis, K. L.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[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]

Vier, D. C.

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

Wanare, H.

Wang, L.-G.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Superluminal pulse reflection and transmission in a slab system doped with dispersive materials,” Phys. Rev. E 70, 066602 (2004).
[CrossRef]

Wegener, M.

Wen, S.

J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
[CrossRef]

Wu, J.-H.

J.-H. Wu and J.-Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

Wu, X.

N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Wuestner, S.

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Yuan, H.-K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Zeng, Y.

J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
[CrossRef]

Zhou, J.

J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
[CrossRef]

Zhu, G.

Zhu, S.-Y.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Superluminal pulse reflection and transmission in a slab system doped with dispersive materials,” Phys. Rev. E 70, 066602 (2004).
[CrossRef]

N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Europhys. Lett. (1)

J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407–412 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

Nat. Mater. (1)

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef]

Nat. Photonics (1)

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

Nature (1)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Opt. Commun. (1)

J. Zhou, H. Luo, S. Wen, and Y. Zeng, “ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system,” Opt. Commun. 282, 2670–2675 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

J.-H. Wu and J.-Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

Phys. Rev. B (6)

D. R. S. Smith, P. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Th. Koschny, P. Markoš, E. 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, 245105 (2005).
[CrossRef]

S. D. Gupta, R. Arun, and G. S. Agarwal, “Subluminal to superluminal propagation in a left-handed medium,” Phys. Rev. B 69, 113104 (2004).
[CrossRef]

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Brentenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010).

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, 201101 (2003).
[CrossRef]

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75, 085436 (2007).
[CrossRef]

Phys. Rev. E (2)

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Superluminal pulse reflection and transmission in a slab system doped with dispersive materials,” Phys. Rev. E 70, 066602 (2004).
[CrossRef]

N. Liu, S.-Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Phys. Rev. Lett (1)

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett 95, 143601 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[CrossRef]

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101, 167401 (2008).
[CrossRef]

J. M. Hamm, S. Wuestner, K. L. Tsakmakidis, and O. Hess, “Theory of light amplification in active fishnet metamaterials,” Phys. Rev. Lett. 107, 167405 (2011).
[CrossRef]

Z. Huang, Th. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012).
[CrossRef]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[CrossRef]

Pis ma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki (1)

O. A. Kocharovskaia and I. I. Khanin, “Coherent amplification of an ultrashort pulse in a three-level medium without population inversion,” Pis ma Zhurnal Eksperimental noi i Teoreticheskoi Fiziki 48, 581–584 (1988).

Rep. Prog. Phys. (1)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[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 (4)

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” arXiv preprint nlin/0506006 (2005).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

R. Marqués, F. Martín, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).

W. Cai and V. M. Šalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Schematic of the DNM slab and energy diagram of doped atoms to the slab. (b) Arrangement of polarized fields Ep and Ec interacting with induced dipole moments μ12 and μ13, respectively.

Fig. 2.
Fig. 2.

(a) Transmission and reflection and, (b) real and imaginary pars of the refractive index of the DNM slab versus normalized probe field detuning for |P|=0 and any arbitrary relative phase φ. Selected parameters are |G˜|=1γ, r=0.5γ, |g˜|=0.1γ, and γ1=γ2=γ.

Fig. 3.
Fig. 3.

Transmission and reflection of a probe pulse propagating through the active slab versus normalized probe field detuning for (a) φ=0, (b) φ=π/2, and (c) φ=π/2. Susceptibility of the doped atoms to the DNM slab for (d) φ=0, (e) φ=π/2, and (f) φ=π/2. Switching between a gain and almost fully absorption regime is apparent. Selected parameters are |P|=0.99, |G˜|=1γ, r=0.5γ, |g˜|=0.1γ, and γ1=γ2=γ.

Fig. 4.
Fig. 4.

Real and imaginary parts of the refractive index of the active slab versus normalized probe field detuning for (a) φ=0, (b) φ=π/2, and (c) φ=π. Switching from positive to negative refractive index and vice versa is achievable. Selected parameters are as Fig. 3.

Fig. 5.
Fig. 5.

Imaginary part of the permittivity of the active slab with respect to the normalized probe field detuning for (a) φ=0, (b) φ=π/2, and (c) φ=π. Critical points are the points that the imaginary part of the permittivity takes the zero value. Selected parameters are as Fig. 3.

Equations (10)

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

n2n2=εμεμ,
nn=εμ+εμ2.
n=±{(εμεμ)+(εμεμ)2+(εμ+εμ)22}12,
n=±12(εμ+εμ){(εμεμ)+(εμεμ)2+(εμ+εμ)22}12.
n=±ε{μ+μ2+μ22}12,
n=±μ{ε+ε2+ε22}12.
t=4μ(ω)μ0kkeikd[μ0k+μ(ω)k]2[μ0kμ(ω)k]2e2ikd,
r=[μ(ω)kμ0k][μ0k+μ(ω)k](1e2ikd)[μ0k+μ(ω)k]2[μ0kμ(ω)k]2e2ikd,
ρ.11=2(γ1+γ2)ρ11+2r(ρ33ρ11)+igρ31+iGρ21iG*ρ12ig*ρ13,ρ.22=2γ2ρ11iGρ21+iG*ρ12,ρ.33=2γ1ρ11+2r(ρ11ρ33)igρ31+ig*ρ13,ρ.12=(γ1+γ2+r+γC+iΔC)ρ12+igρ32iG(ρ11ρ22),ρ.23=[γC+r+i(ΔPΔC)]ρ23+2Pγ1γ2ρ11+iG*ρ13igρ21,ρ.13=(γ1+γ2+2r+iΔP)ρ13+iGρ23ig(ρ11ρ33),
χ(ωp)=2Nμ13ε0Epρ˜13(ωp).

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