Abstract

We study numerically the propagation of two-color light pulses through a metamaterial doped with active atoms such that the carrier frequencies of the pulses are in resonance with two atomic transitions in the Λ configuration and that one color propagates in the regime of positive refraction and the other in the regime of negative refraction. In such a metamaterial, one resonant color of light propagates with positive and the other with negative group velocity. We investigate nonlinear interaction of these forward- and backward-propagating waves and find self-trapped waves, counterpropagating radiation waves, and hot spots of medium excitation.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
    [CrossRef]
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    [CrossRef]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef]
  7. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
    [CrossRef]
  8. V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
    [CrossRef]
  9. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
    [CrossRef]
  10. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–47 (2007).
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  11. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
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  12. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett 85, 3966–3969 (2000).
    [CrossRef]
  13. V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
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  14. J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco, B.-I. Wu, J. A. Kong, and M. Chen, “Čerenkov radiation in materials with negative permittivity and permeability,” Opt. Express 11, 723–734 (2003).
    [CrossRef]
  15. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
  16. L. I. Mandelshtam, “Group velocity in crystalline arrays,” Zh. Eksp. Teor. Fiz. 15, 475–478 (1945).
  17. V. G. Veselago, “Properties of materials having simultaneously negative values of dielectric (ξ) and magnetic (μ) susceptibilities,” Sov. Phys. Solid State 8, 2854 (1967).
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  20. N. Lazarides, and G. P. Tsironis, “Coupled nonlinear Schrodinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
    [CrossRef]
  21. A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
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  22. I. R. Gabitov, R. Indik, N. M. Litchinitser, A. I. Maimistov, V. M. Shalaev, and J. Soneson, “Double-resonant optical materials with embedded metal nanostructures,” J. Opt. Soc. Am. B 23, 535–542 (2006).
    [CrossRef]
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    [CrossRef]
  24. N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
    [CrossRef]
  25. A. I. Maimistov and I. R. Gabitov, “Nonlinear optical effects in artificial materials,” Eur. Phys. J. Spec. Top. 147, 265–286 (2007).
    [CrossRef]
  26. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  27. I. R. Gabitov, A. I. Maimistov, A. Korotkevich, and J. B. McMahon, “Solitary waves in plasmonic Bragg gratings,” Appl. Phys. A 89, 277–281 (2007).
    [CrossRef]
  28. M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567–2583 (1981).
    [CrossRef]
  29. J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
    [CrossRef]
  30. T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E 69, 026602 (2004).
    [CrossRef]
  31. A. I. Maimistov, I. R. Gabitov, and E. V. Kazantseva, “Quadratic solitons in negative refractive index medium,” Opt. Spectrosc. 102, 90–97 (2007).
    [CrossRef]
  32. A. Maimistov, I. Gabitov, and N. Litchinitser, “Solitary waves in nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
    [CrossRef]
  33. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
    [CrossRef]
  34. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Nat. Acad. Sci. 109, 8834–8838 (2012).
  35. 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]
  36. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).
  37. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  38. I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16, 401–409 (2010).
    [CrossRef]
  39. L. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys. Rev. B 66, 155411 (2002).
    [CrossRef]
  40. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. 23, 391–403 (2006).
    [CrossRef]
  41. 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]
  42. 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]
  43. P. Markoš and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622 (2002).
    [CrossRef]
  44. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977).
  45. A. I. Maimistov and Y. M. Sklyarov, “Coherent interaction of light pulses with a three-level medium,” Opt. Spectrosc. 59, 459–461 (1985).
  46. J. A. Byrne, I. R. Gabitov, and G. Kovačič, “Polarization switching of light interacting with a degenerate two-level optical medium,” Physica D 186, 69–92 (2003).
    [CrossRef]
  47. K. A. Newhall, “Three level atom interaction with external electric field,” (2007) (paper in preparation).

2012 (3)

F. Bilotti and L. Sevgi, “Metamaterials: definitions, properties, applications, and FDTD-based modeling and simulation (invited paper),” Int. J. RF Microwave Computer-Aided Eng. 22, 422–438 (2012).
[CrossRef]

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[CrossRef]

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Nat. Acad. Sci. 109, 8834–8838 (2012).

2010 (2)

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]

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16, 401–409 (2010).
[CrossRef]

2008 (1)

A. Maimistov, I. Gabitov, and N. Litchinitser, “Solitary waves in nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

2007 (7)

A. I. Maimistov, I. R. Gabitov, and E. V. Kazantseva, “Quadratic solitons in negative refractive index medium,” Opt. Spectrosc. 102, 90–97 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

A. I. Maimistov and I. R. Gabitov, “Nonlinear optical effects in artificial materials,” Eur. Phys. J. Spec. Top. 147, 265–286 (2007).
[CrossRef]

I. R. Gabitov, A. I. Maimistov, A. Korotkevich, and J. B. McMahon, “Solitary waves in plasmonic Bragg gratings,” Appl. Phys. A 89, 277–281 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–47 (2007).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Effect of an optical negative index thin film on optical bistability,” Opt. Lett. 32, 151–153 (2007).
[CrossRef]

2006 (5)

I. R. Gabitov, R. Indik, N. M. Litchinitser, A. I. Maimistov, V. M. Shalaev, and J. Soneson, “Double-resonant optical materials with embedded metal nanostructures,” J. Opt. Soc. Am. B 23, 535–542 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. 23, 391–403 (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]

2005 (6)

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

N. Lazarides, and G. P. Tsironis, “Coupled nonlinear Schrodinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

2004 (5)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef]

T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E 69, 026602 (2004).
[CrossRef]

2003 (3)

J. A. Byrne, I. R. Gabitov, and G. Kovačič, “Polarization switching of light interacting with a degenerate two-level optical medium,” Physica D 186, 69–92 (2003).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco, B.-I. Wu, J. A. Kong, and M. Chen, “Čerenkov radiation in materials with negative permittivity and permeability,” Opt. Express 11, 723–734 (2003).
[CrossRef]

2002 (2)

P. Markoš and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622 (2002).
[CrossRef]

L. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys. Rev. B 66, 155411 (2002).
[CrossRef]

2001 (2)

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

2000 (2)

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, 4184–4187 (2000).
[CrossRef]

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

1999 (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]

1985 (1)

A. I. Maimistov and Y. M. Sklyarov, “Coherent interaction of light pulses with a three-level medium,” Opt. Spectrosc. 59, 459–461 (1985).

1981 (1)

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567–2583 (1981).
[CrossRef]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1967 (1)

V. G. Veselago, “Properties of materials having simultaneously negative values of dielectric (ξ) and magnetic (μ) susceptibilities,” Sov. Phys. Solid State 8, 2854 (1967).

1945 (1)

L. I. Mandelshtam, “Group velocity in crystalline arrays,” Zh. Eksp. Teor. Fiz. 15, 475–478 (1945).

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).

Akozbek, N.

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).

Bilotti, F.

F. Bilotti and L. Sevgi, “Metamaterials: definitions, properties, applications, and FDTD-based modeling and simulation (invited paper),” Int. J. RF Microwave Computer-Aided Eng. 22, 422–438 (2012).
[CrossRef]

Bloemer, M.

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Boltasseva, A.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Nat. Acad. Sci. 109, 8834–8838 (2012).

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Byrne, J. A.

J. A. Byrne, I. R. Gabitov, and G. Kovačič, “Polarization switching of light interacting with a degenerate two-level optical medium,” Physica D 186, 69–92 (2003).
[CrossRef]

Cai, W.

Chen, M.

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]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977).

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

D’Aguanno, G.

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1977).

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]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Eberly, J. H.

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567–2583 (1981).
[CrossRef]

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Gabitov, I.

A. Maimistov, I. Gabitov, and N. Litchinitser, “Solitary waves in nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

Gabitov, I. R.

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16, 401–409 (2010).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Effect of an optical negative index thin film on optical bistability,” Opt. Lett. 32, 151–153 (2007).
[CrossRef]

A. I. Maimistov, I. R. Gabitov, and E. V. Kazantseva, “Quadratic solitons in negative refractive index medium,” Opt. Spectrosc. 102, 90–97 (2007).
[CrossRef]

I. R. Gabitov, A. I. Maimistov, A. Korotkevich, and J. B. McMahon, “Solitary waves in plasmonic Bragg gratings,” Appl. Phys. A 89, 277–281 (2007).
[CrossRef]

A. I. Maimistov and I. R. Gabitov, “Nonlinear optical effects in artificial materials,” Eur. Phys. J. Spec. Top. 147, 265–286 (2007).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

I. R. Gabitov, R. Indik, N. M. Litchinitser, A. I. Maimistov, V. M. Shalaev, and J. Soneson, “Double-resonant optical materials with embedded metal nanostructures,” J. Opt. Soc. Am. B 23, 535–542 (2006).
[CrossRef]

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Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
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G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Nat. Acad. Sci. 109, 8834–8838 (2012).

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).
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W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
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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).
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V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
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N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
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A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
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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).
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J. A. Byrne, I. R. Gabitov, and G. Kovačič, “Polarization switching of light interacting with a degenerate two-level optical medium,” Physica D 186, 69–92 (2003).
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I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16, 401–409 (2010).
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N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Effect of an optical negative index thin film on optical bistability,” Opt. Lett. 32, 151–153 (2007).
[CrossRef]

I. R. Gabitov, A. I. Maimistov, A. Korotkevich, and J. B. McMahon, “Solitary waves in plasmonic Bragg gratings,” Appl. Phys. A 89, 277–281 (2007).
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A. I. Maimistov, I. R. Gabitov, and E. V. Kazantseva, “Quadratic solitons in negative refractive index medium,” Opt. Spectrosc. 102, 90–97 (2007).
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A. I. Maimistov and I. R. Gabitov, “Nonlinear optical effects in artificial materials,” Eur. Phys. J. Spec. Top. 147, 265–286 (2007).
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N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
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I. R. Gabitov, R. Indik, N. M. Litchinitser, A. I. Maimistov, V. M. Shalaev, and J. Soneson, “Double-resonant optical materials with embedded metal nanostructures,” J. Opt. Soc. Am. B 23, 535–542 (2006).
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L. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys. Rev. B 66, 155411 (2002).
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S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
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M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
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I. R. Gabitov, A. I. Maimistov, A. Korotkevich, and J. B. McMahon, “Solitary waves in plasmonic Bragg gratings,” Appl. Phys. A 89, 277–281 (2007).
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D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
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G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Nat. Acad. Sci. 109, 8834–8838 (2012).

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V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
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V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
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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, 4184–4187 (2000).
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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).
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S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
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Pacheco, J.

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, 4184–4187 (2000).
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L. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys. Rev. B 66, 155411 (2002).
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S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
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J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
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J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett 85, 3966–3969 (2000).
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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]

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V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
[CrossRef]

Poliakov, E.

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Popov, A. K.

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

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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).
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Scalora, M.

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
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R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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, 4184–4187 (2000).
[CrossRef]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

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F. Bilotti and L. Sevgi, “Metamaterials: definitions, properties, applications, and FDTD-based modeling and simulation (invited paper),” Int. J. RF Microwave Computer-Aided Eng. 22, 422–438 (2012).
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I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Shalaev, V. M.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Nat. Acad. Sci. 109, 8834–8838 (2012).

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, 41–47 (2007).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Effect of an optical negative index thin film on optical bistability,” Opt. Lett. 32, 151–153 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

I. R. Gabitov, R. Indik, N. M. Litchinitser, A. I. Maimistov, V. M. Shalaev, and J. Soneson, “Double-resonant optical materials with embedded metal nanostructures,” J. Opt. Soc. Am. B 23, 535–542 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (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]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

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R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

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V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).

Sklyarov, Y. M.

A. I. Maimistov and Y. M. Sklyarov, “Coherent interaction of light pulses with a three-level medium,” Opt. Spectrosc. 59, 459–461 (1985).

Smith, D. R.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. 23, 391–403 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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, 4184–4187 (2000).
[CrossRef]

Soneson, J.

Soukoulis, C. M.

P. Markoš and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622 (2002).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

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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]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

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M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

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N. Lazarides, and G. P. Tsironis, “Coupled nonlinear Schrodinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
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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, 4184–4187 (2000).
[CrossRef]

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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Wu, B.-I.

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]

Yuan, H.

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]

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Demonstration of near-infrared negative-index materials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Zhang, Y.

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Zheltikov, A.

M. Scalora, M. Syrchin, N. Akozbek, E. Poliakov, G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Zheludev, N. I.

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[CrossRef]

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R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Appl. Phys. A (1)

I. R. Gabitov, A. I. Maimistov, A. Korotkevich, and J. B. McMahon, “Solitary waves in plasmonic Bragg gratings,” Appl. Phys. A 89, 277–281 (2007).
[CrossRef]

Appl. Phys. B (1)

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

Eur. Phys. J. Spec. Top. (1)

A. I. Maimistov and I. R. Gabitov, “Nonlinear optical effects in artificial materials,” Eur. Phys. J. Spec. Top. 147, 265–286 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16, 401–409 (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]

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]

Int. J. RF Microwave Computer-Aided Eng. (1)

F. Bilotti and L. Sevgi, “Metamaterials: definitions, properties, applications, and FDTD-based modeling and simulation (invited paper),” Int. J. RF Microwave Computer-Aided Eng. 22, 422–438 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. 23, 391–403 (2006).
[CrossRef]

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

Nat. Mater. (1)

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
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Nat. Photonics (2)

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Figures (6)

Fig. 1.
Fig. 1.

Λ-configuration system with transition frequencies ω+ and ω. The transition with the frequency ω+ corresponds to the response of the metamaterial in a regime with a positive index of refraction, and the transition with the frequency ω corresponds to the response of the metamaterial in a regime with a negative index of refraction. The medium is initially prepared so that a nonzero coherence value, ρ0, between the two lower levels is present.

Fig. 2.
Fig. 2.

Due to the resonant nature of the negative-refractive-index phenomenon, the sign of the refractive index depends on the frequency. The intermediate (forbidden) range of frequencies, in which ϵ and μ have (real parts with) opposite signs and there is no wave propagation, is depicted by the green dot.

Fig. 3.
Fig. 3.

Schematic of the physical setup investigated numerically: in this case red light, the pump, is injected into the Λ-configuration metamaterial. This red light is partly converted into blue light, the signal, which travels in the opposite direction.

Fig. 4.
Fig. 4.

Self-trapped, copropagating pulse pair. Nonvanishing initial conditions: ρ0=0.3, n+=0.8, n=0.2. Amplitude of the injected Gaussian pulse E+0(t): A=2.

Fig. 5.
Fig. 5.

Self-trapped, copropagating pulse pair and counterpropagating radiation. Nonvanishing initial conditions: ρ0=0.2, n+=0.3, n=0.7. Amplitude of the injected Gaussian pulse E+0(t): A=1.4.

Fig. 6.
Fig. 6.

Hot spot. Nonvanishing initial conditions: ρ0=0.2, n+=n=0.5. Amplitude of the injected Gaussian pulse E+0(t): A=2.

Equations (22)

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×E=1cBt,×H=1cDt,
·D=0,·B=0,
B=H+4πM,D=E+4πP+4πQ,
2Pt2+ωP2P=ωP2γ4πE,
2Mt2+ωM2M=β4π2Ht2.
Q=Nljl,j=13al*aj·rlj,rlj=rjl*,rll=0,
itam=ωmamn=13E·rmnan,
r23=r32=0.
ωc=2πNd2ω0,
E(z,t)=E+(z,t)eiθ++E(z,t)eiθ+c.c.,
θ±=ω0ωc(k±zω±t).
k2(ω)=ω2μ(ω)ϵ(ω),
ϵ(ω)=1+γΩP2ΩP2ω2,μ(ω)=1+βω2ΩM2ω2,
Q=i[ρ+(z,t,ν)α+*eiθ++ρ(z,t,ν)α*eiθc.c.]g(ν)dν,
E±z+1v±E±t=2ω±μ±ϵ±|α±|2ρ±(ν)g(ν)dν,
ρ+t=iνρ++12[E+(Nn+)Eρ*],
ρt=iνρ+12[E(Nn)E+ρ],
ρt=12[E+*ρ+Eρ+*],
Nt=12[E+ρ+*+E+*ρ++Eρ*+E*ρ],
n±t=12[E±ρ±*+E±*ρ±],
1v±=ω±2(1+γΩP2ΩP2ω±2)(1+βω2ΩM2ω±2)×[2ω±+γΩP2(ΩP2(1+γ)ω±2)(ΩP2ω±2)+βΩM2(ΩM2+ω±2(β1))(ΩM2ω±2)].
E+0(t)=Ae(ttd)2/2σ2,

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