Abstract

We study the modulational instability (MI) of the continuous wave propagating through negative-index materials, based on the coupled cubic–quintic nonlinear Schrödinger equations. A comparison is made with the already available results on MI without including the quintic term. We find that this term helps in achieving MI, in otherwise impossible combinations of dispersion and nonlinearity. For example, spatial MI can occur in the focusing regime, and temporal MI becomes possible in the anomalous dispersion for defocusing nonlinearity, while in the case of the normal dispersion regime, it occurs for focusing nonlinearity, and spatiotemporal MI can occur in the focusing nonlinearity and normal dispersion. Moreover, this additional term provides extra freedom to control the amplitude of the MI gain profile.

© 2012 Optical Society of America

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  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, 4184–4187 (2000).
    [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef]
  3. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  4. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
    [CrossRef]
  5. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
    [CrossRef]
  6. N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
    [CrossRef]
  7. A. K. Sarma, “Solitary waves in a negative index material with dispersive permittivity and permeability,” Eur. Phys. J. D 62, 421–424 (2011).
    [CrossRef]
  8. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  9. A. Joseph, K. Porsezian, and P. T. Dinda, “Modulational instability scenario in negative index materials,” J. Mod. Opt. 57, 436–443 (2010).
    [CrossRef]
  10. Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulational instability in metameterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
    [CrossRef]
  11. Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
    [CrossRef]
  12. Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fan, “Modulational instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
    [CrossRef]
  13. W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
    [CrossRef]
  14. X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulational instability of copropagating light beams in nonlinear metameterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
    [CrossRef]
  15. A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
    [CrossRef]
  16. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
    [CrossRef]
  17. I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
    [CrossRef]
  18. A. K. Sarma and M. Saha, “Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium,” J. Opt. Soc. Am. B 28, 944–948 (2011).
    [CrossRef]
  19. N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
    [CrossRef]
  20. S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
    [CrossRef]

2012 (1)

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[CrossRef]

2011 (3)

2010 (2)

A. Joseph, K. Porsezian, and P. T. Dinda, “Modulational instability scenario in negative index materials,” J. Mod. Opt. 57, 436–443 (2010).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

2009 (3)

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulational instability of copropagating light beams in nonlinear metameterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
[CrossRef]

2007 (2)

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fan, “Modulational instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

2006 (2)

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

2005 (3)

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

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

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

2001 (1)

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

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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Akozbek, N.

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

Bloemer, M. J.

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

Cheng, X.

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

D’Aguanno, G.

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

Dai, X.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulational instability in metameterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulational instability of copropagating light beams in nonlinear metameterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
[CrossRef]

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fan, “Modulational instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Dinda, P. T.

A. Joseph, K. Porsezian, and P. T. Dinda, “Modulational instability scenario in negative index materials,” J. Mod. Opt. 57, 436–443 (2010).
[CrossRef]

Fan, D.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulational instability in metameterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulational instability of copropagating light beams in nonlinear metameterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
[CrossRef]

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fan, “Modulational instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Frantzeskakis, D. J.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Fu, X.

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

Guo, J.

Hu, Y.

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Joseph, A.

A. Joseph, K. Porsezian, and P. T. Dinda, “Modulational instability scenario in negative index materials,” J. Mod. Opt. 57, 436–443 (2010).
[CrossRef]

Kevrekidis, P. G.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Kourakis, I.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Kumar, P.

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[CrossRef]

Lazarides, N.

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

Mattiucci, N.

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

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, 4184–4187 (2000).
[CrossRef]

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).
[CrossRef]

Poliakov, E. Y.

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

Porsezian, K.

A. Joseph, K. Porsezian, and P. T. Dinda, “Modulational instability scenario in negative index materials,” J. Mod. Opt. 57, 436–443 (2010).
[CrossRef]

Rompotis, N.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Saha, M.

Sarma, A. K.

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[CrossRef]

A. K. Sarma, “Solitary waves in a negative index material with dispersive permittivity and permeability,” Eur. Phys. J. D 62, 421–424 (2011).
[CrossRef]

A. K. Sarma and M. Saha, “Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium,” J. Opt. Soc. Am. B 28, 944–948 (2011).
[CrossRef]

Scalora, M.

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

Schultz, S.

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]

Shelby, R. A.

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

Shukla, P. K.

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Smith, D. R.

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]

Su, W.

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Syrchin, M. S.

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

Tang, Z.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Tsironis, G. P.

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

Tsitsas, N. L.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[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.

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]

Wen, S.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulational instability in metameterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulational instability of copropagating light beams in nonlinear metameterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
[CrossRef]

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fan, “Modulational instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

Xiang, Y.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulational instability in metameterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulational instability of copropagating light beams in nonlinear metameterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
[CrossRef]

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fan, “Modulational instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect on modulational instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef]

Zheltikov, A. M.

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

Zhou, W.

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

Appl. Phys. B (1)

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[CrossRef]

Eur. Phys. J. D (1)

A. K. Sarma, “Solitary waves in a negative index material with dispersive permittivity and permeability,” Eur. Phys. J. D 62, 421–424 (2011).
[CrossRef]

J. Mod. Opt. (1)

A. Joseph, K. Porsezian, and P. T. Dinda, “Modulational instability scenario in negative index materials,” J. Mod. Opt. 57, 436–443 (2010).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

W. Zhou, W. Su, X. Cheng, Y. Xiang, X. Dai, and S. Wen, “Copropagation of two pulses of different frequencies and modulational instabilities induced by cross-phase modulation in metameterials,” Opt. Commun. 282, 1440–1447 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Phys. Rev. E (5)

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulational instability in nonlinear oppositely directed coupler with a negative-index metameterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

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

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (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, 4184–4187 (2000).
[CrossRef]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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 (1)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

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Figures (5)

Fig. 1.
Fig. 1.

μeff as a function of (|H|2+|H|4)/Ec2, for Ω=1.1 (solid), Ω=1.15 (long-dashed), Ω=1.20 (dotted line), Ω=1.25 (dash-dotted line), A=3, F=0.4, and α^>0.

Fig. 2.
Fig. 2.

Gain profile for the spatial MI. (a) For self-focusing medium taking f=+1; G=4,2.6,1.5; (b) for defocusing medium taking f=1; G=0.2,0.3,0.4 for curves A, B, C, and the values of other parameters are a0=0.93, b0=0.676, and p=1.

Fig. 3.
Fig. 3.

Gain profile of temporal MI in a medium with the following properties: (a1) anomalous dispersion and focusing is plotted taking values of G=0.5;0.4;0.1; (b1) normal dispersion and focusing is plotted with G=1,0.8,0.7 for A, B, C, respectively. Other values of parameters are fixed s=0.006; a0=0.93; b0=0.676 for all the three A, B, and C for both plots. In (a2) anomalous dispersion and focusing for G=1, and in (b2) normal dispersion and focusing with G=0.9 taking s=0.006,0.0143,0.022. The values of other parameters are taken a0=0.93 and b0=0.676,0.871,0.907 for curves A, B, and C.

Fig. 4.
Fig. 4.

Gain profile of temporal MI in a medium with the following properties: (c1) anomalous dispersion and defocusing is plotted for G=1,2,3; (d1) normal dispersion and defocusing is plotted for G=0.3,0.4,0.5 and other values of the parameter fixed to be s=0.006; a0=0.93; b0=0.676; (c2) anomalous dispersion and defocusing with G=1. (d2) normal dispersion and defocusing with G=0.3 plotted taking different s values given as s=0.006,0.0143,0.022. The values of other parameters are a0=0.93 and b0=0.676,0.871,0.907.

Fig. 5.
Fig. 5.

Gain profile of spatiotemporal MI (a) for focusing and normal dispersion with G=1, (b) for focusing and anomalous dispersion G=0.4, (c) defocusing and normal dispersion G=0.4, (d) defocusing and anomalous dispersion with G=5. The values of other parameters are a0=0.93, b0=0.676, and s=0.006.

Equations (23)

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εeff=ε0(εD(|E|2,|E|4)ωp2ω2),μeff=μ0(1Fω2ω2ω0NL2(|H|2,|H|4)),
Ω2X6(α|H|2+|H|4)=A2Ec2(1X2)(X2Ω2)2,
ψ1ξ=i2k02ψ1iβ222ψ1τ2+iCnl(1+iCsτ)|ψ1|2ψ1+iDnl|ψ2|2(ψ1+iCseψ1τ)+iCqnl|ψ1|4ψ1,ψ2ξ=i2k02ψ2iβ222ψ2τ2+iDnl(1+iDsτ)|ψ2|2ψ2+iCnl|ψ1|2(ψ2+iDshψ2τ)+iDqnl|ψ2|4ψ2.
Cnl=ω02μ(ω0)ε0χE(3)2k0,Cqnl=ω02μ(ω0)ε0χE(5)2k0,Cs=[1ω0{1+γμ(ω0)}1k0V],Cse=1ω0{1+αε(ω0)},Dnl=ω02ε(ω0)μ0χM(3)2k0,Dqnl=ω02ε(ω0)μ0χM(5)2k0,Ds=[1ω0{1+αε(ω0)}1k0V],Dsh=1ω0{1+γμ(ω0)}
β2=[{αγ+ω0μ(ω0)α/2+ω0ε(ω0)γ/21/V2}/k0],γ=[ωμ(ω)]/ω|ω=ω0,γ=2[ωμ(ω)]/ω2|ω=ω0,α=[ωε(ω)]/ω|ω=ω0,α=2[ωε(ω)]/ω2|ω=ω0,V=2k0/[ω0ε(ω0)γ+ω0μ(ω0)α].
Z=ξLD,T=τT0,U=ψ1ψ01,V=ψ2ψ02X=xL,Y=yL,u=NEU,v=NHV,
uZ=isgn(k0)22uisgn(β2)22uT2+if(1+iSET)|u|2u+i|v|2uCE|v|2uT+igE|u|4u,
vZ=isgn(k0)22visgn(β2)22vT2+if(1+iSHT)|v|2v+i|u|2vCH|u|2vT+igH|v|4v.
u(X,Y,Z,T)=[a0+a(X,Y,Z,T)]exp(iΩ0aZ),
v(X,Y,Z,T)=[b0+b(X,Y,Z,T)]exp(iΩ0bZ),
aZ=i2a2iδ22aT2+ifa02(a+a*)fSE(2a02aT+a02a*T)+ia0b0(b+b*)CEb02aT+igE(3a04a+a05+2a04a*),
bZ=i2b2iδ22bT2+ib02(b+b*)fSH(2b02bT+b02b*T)+ia0b0(a+a*)CHa02bT+igH(3b04b+b05+2b04b*).
sgn(β2)=δ,andsgn(χ(3))=f.
k=12[(2cΩ+4sΩ)±4s2Ω2+Ω4+q4+2δq2Ω2+4(δΩ2+q2)(f(a02+b02)+2G)].
SHb02=SEa02=s,CEb02=CHa02=c,gEa04=gHb04=G,
gMI=2Im(k)=4s2Ω2Ω4q42δq2Ω24(δΩ2+q2)(f(a02+b02)+2G).
ks=±12q4+4q2(f(a02+b02)+2G).
f(a02+b02)+2Gq24.
gs=2Im(ks)=q44q2(f(a02+b02)+2G).
kt=12[(2cΩ+4sΩ)±4s2Ω2+Ω4+4δΩ2(f(a02+b02)+2G)].
δ(f(a02+b02)+2G)(s2+Ω24).
gt=2Im(kt)=4Ω2s2Ω44δΩ2(f(a02+b02)+2G).
f(a02+b02)+2G<4s2Ω2Ω4q42δq2Ω24(δΩ2+q2).

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