Abstract

We have investigated the modulation instability (MI) in a negative index media (NIM) or metamaterial (MM) using a new generalized model describing the pulse propagation in a NIM embedded into a Kerr medium. We have found that one could control the gain of MI in an MM by tuning the initial electric or magnetic field amplitudes. Our model successfully recovers previously proposed models to describe pulse propagation in MMs exhibiting Kerr nonlinearity. Moreover, it contains a few additional terms connecting both the electric and magnetic field envelopes in an MM.

© 2011 Optical Society of America

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  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] [PubMed]
  3. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications(Wiley, 2006).
  4. A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
    [CrossRef]
  5. V. G. Veselago, “The electrodynamics of substance with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514(1968).
    [CrossRef]
  6. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
    [CrossRef] [PubMed]
  7. E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (2005).
    [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. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
    [CrossRef] [PubMed]
  10. A. Chowdhury and J. A. Tataronis, “Long wave—short wave resonance in nonlinear negative refractive index media,” Phys. Rev. Lett. 100, 153905 (2008).
    [CrossRef] [PubMed]
  11. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press, 2009).
  12. Y. M. Liu and G. Bartal, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
    [CrossRef] [PubMed]
  13. G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
    [CrossRef] [PubMed]
  14. M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
    [CrossRef]
  15. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
    [CrossRef] [PubMed]
  16. S. Wen, Y. Wang, W. Su, Y. Xiang, and X. Fu, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
    [CrossRef]
  17. 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]
  18. 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]
  19. 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]
  20. I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
    [CrossRef]
  21. P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
    [CrossRef]
  22. P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A 81, 023808 (2010).
    [CrossRef]
  23. G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
    [CrossRef] [PubMed]
  24. M. Stockman, “Criteria for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  25. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
    [CrossRef] [PubMed]
  26. X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulation instability of copropagating light beams in nonlinear metamaterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
    [CrossRef]
  27. A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
    [CrossRef]
  28. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

2010 (3)

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A 81, 023808 (2010).
[CrossRef]

A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
[CrossRef]

2009 (2)

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulation instability of copropagating light beams in nonlinear metamaterials,” J. Opt. Soc. Am. B 26, 564–571 (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]

2008 (2)

S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

A. Chowdhury and J. A. Tataronis, “Long wave—short wave resonance in nonlinear negative refractive index media,” Phys. Rev. Lett. 100, 153905 (2008).
[CrossRef] [PubMed]

2007 (5)

Y. M. Liu and G. Bartal, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
[CrossRef]

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (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]

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

2006 (2)

2005 (6)

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (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]

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

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

2004 (2)

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
[CrossRef] [PubMed]

2000 (2)

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

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] [PubMed]

1968 (1)

V. G. Veselago, “The electrodynamics of substance with simultaneously negative value 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, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

Anand, S.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

Bartal, G.

Y. M. Liu and G. Bartal, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Berrier, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

Bloemer, M. J.

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
[CrossRef] [PubMed]

Cai, W.

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications(Wiley, 2006).

Centini, M.

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[CrossRef]

Chettiar, U. K.

Chowdhury, A.

A. Chowdhury and J. A. Tataronis, “Long wave—short wave resonance in nonlinear negative refractive index media,” Phys. Rev. Lett. 100, 153905 (2008).
[CrossRef] [PubMed]

D’Aguanno, G.

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
[CrossRef] [PubMed]

D’Ceglia, D.

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[CrossRef]

Dai, X.

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulation instability of copropagating light beams in nonlinear metamaterials,” J. Opt. Soc. Am. B 26, 564–571 (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]

Drachev, V. P.

Fan, D.

Feng, S.

S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

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.

Grzegorczyk, T. M.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press, 2009).

Halterman, K.

S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

Hu, Y.

Itoh, T.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications(Wiley, 2006).

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]

Kildishev, A. V.

Kinsler, P.

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A 81, 023808 (2010).
[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]

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]

Lee, J.-B.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (2005).
[CrossRef]

Liu, Y. M.

Y. M. Liu and G. Bartal, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Mattiucci, N.

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
[CrossRef] [PubMed]

Mulot, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

Park, W.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (2005).
[CrossRef]

Pendry, J. B.

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

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Qiu, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press, 2009).

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]

Sarma, A. K.

A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
[CrossRef]

Sarychev, A. K.

Scalora, M.

M. Scalora, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
[CrossRef] [PubMed]

Schonbrun, E.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (2005).
[CrossRef]

Schultz, S.

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

Shalaev, V. M.

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]

Sihvola, A.

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
[CrossRef]

Smith, D. R.

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] [PubMed]

Stockman, M.

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

Su, W.

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. Wang, W. Su, Y. Xiang, and X. Fu, “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 in modulation instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef] [PubMed]

Swillo, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Talneau, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

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]

Tataronis, J. A.

A. Chowdhury and J. A. Tataronis, “Long wave—short wave resonance in nonlinear negative refractive index media,” Phys. Rev. Lett. 100, 153905 (2008).
[CrossRef] [PubMed]

Thylen, L.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

Tinker, M.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (2005).
[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 substance with simultaneously negative value 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] [PubMed]

Wang, Y.

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

Wen, S.

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulation instability of copropagating light beams in nonlinear metamaterials,” J. Opt. Soc. Am. B 26, 564–571 (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. Wang, W. Su, Y. Xiang, and X. Fu, “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 in modulation instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef] [PubMed]

Xiang, Y.

X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulation instability of copropagating light beams in nonlinear metamaterials,” J. Opt. Soc. Am. B 26, 564–571 (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. Wang, W. Su, Y. Xiang, and X. Fu, “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 in modulation instability in negative-index material,” Opt. Express 14, 1568–1575(2006).
[CrossRef] [PubMed]

Yuan, H.

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000(2005).
[CrossRef] [PubMed]

Europhys. Lett. (1)

A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, “Negative refraction in a Si-polymer photonic crystal membrane,” IEEE Photon. Technol. Lett. 17, 1196–1198 (2005).
[CrossRef]

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

Metamaterials (1)

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (3)

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]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A 81, 023808 (2010).
[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]

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

S. Wen, Y. Wang, W. Su, Y. Xiang, and X. Fu, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[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, D. D’Ceglia, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. Centini, and M. J. Bloemer, “Gap solitons in a nonlinear quadratic negative-index cavity,” Phys. Rev. E 75, 066606 (2007).
[CrossRef]

Phys. Rev. Lett. (9)

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

Y. M. Liu and G. Bartal, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902–213905 (2004).
[CrossRef] [PubMed]

S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef] [PubMed]

A. Chowdhury and J. A. Tataronis, “Long wave—short wave resonance in nonlinear negative refractive index media,” Phys. Rev. Lett. 100, 153905 (2008).
[CrossRef] [PubMed]

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

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] [PubMed]

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

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared-wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substance with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514(1968).
[CrossRef]

Other (3)

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications(Wiley, 2006).

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press, 2009).

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

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Figures (3)

Fig. 1
Fig. 1

(a) Variation of refractive index n, electric nonlinear coefficient P n l , electric self-steepening parameter P s , and electric coupling coefficient P s e with the normalized frequency ω 0 / ω p e . ω p m / ω p e = 0.8 . P n l is calculated in units of ω p e χ E ( 3 ) / c , while P s and P s e are calculated in the units of 1 / ω p e . γ ˜ e = γ ˜ m = 0.01 . (b) Variation of refractive index n, magnetic nonlinear coefficient Q n l , magnetic self-steepening parameter Q s , and magnetic coupling coefficient Q s h with the normalized frequency ω 0 / ω p e . ω p m / ω p e = 0.8 . Q n l is calculated in units of ω p e χ M ( 3 ) / c , while Q s and Q s h are calculated in the units of 1 / ω p e . γ ˜ e = γ ˜ m = 0.01 .

Fig. 2
Fig. 2

MI gain as a function of normalized perturbation frequency for four different values of ω 0 / ω p e ( = q ) with a 0 = 1 and T 0 = 10 fs .

Fig. 3
Fig. 3

MI gain as a function of normalized perturbation frequency for four different values of reduced self-steepening parameter s with ω 0 / ω p e = 0.5 and T 0 = 10 fs .

Equations (13)

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ε ( ω ˜ ) = ε 0 [ 1 1 ω ˜ ( ω ˜ + i γ ˜ e ) ] ; μ ( ω ˜ ) = μ 0 [ 1 ( ω p m / ω p e ) 2 ω ˜ ( ω ˜ + i γ ˜ m ) ] ,
( 2 z 2 + 2 ) E ( r , t ) ( . E ( r , t ) ) = 2 t 2 [ μ * ε * E ( r , t ) ] + 2 t 2 [ μ * P NL ( r , t ) ] + t ( × M NL ( r , t ) ) ( 2 z 2 + 2 ) H ( r , t ) ( . H ( r , t ) ) = 2 t 2 [ μ * ε * H ( r , t ) ] + 2 t 2 [ ε * M NL ( r , t ) ] t ( × P NL ( r , t ) )
t × M NL ( r , t ) μ NL t ( × H ) and t × P NL ( r , t ) ε NL t ( × E ) .
A ξ = i 2 k 0 2 A i β 2 2 2 A τ 2 + i P n l ( 1 + i P s τ ) | A | 2 A + i Q n l | B | 2 ( A + i P s e A τ ) B ξ = i 2 k 0 2 B i β 2 2 2 B τ 2 + i Q n l ( 1 + i Q s τ ) | B | 2 B + i P n l | A | 2 ( B + i Q s h B τ ) ,
P n l = ω 0 2 μ ( ω 0 ) ε 0 χ E ( 3 ) 2 k 0 , P s = [ 1 ω 0 ( 1 + γ μ ( ω 0 ) ) 1 k 0 V ] , P s e = 1 ω 0 ( 1 + α ε ( ω 0 ) ) , Q n l = ω 0 2 ε ( ω 0 ) μ 0 χ M ( 3 ) 2 k 0 , Q s = [ 1 ω 0 ( 1 + α ε ( ω 0 ) ) 1 k 0 V ] , Q s h = 1 ω 0 ( 1 + γ μ ( ω 0 ) ) ,
β 2 = [ { α γ + ω 0 μ ( ω 0 ) α / / 2 + ω 0 ε ( ω 0 ) γ / / 2 1 / V 2 } / k 0 ] , γ = [ ω μ ( ω ) ] / ω | ω = ω 0 , γ / = 2 [ ω μ ( ω ) ] / 2 ω | ω = ω 0 , α = [ ω ε ( ω ) ] / ω | ω = ω 0 , α / = 2 [ ω ε ( ω ) ] / 2 ω | ω = ω 0 , V = 2 k 0 / [ ω 0 ε ( ω 0 ) γ + ω 0 μ ( ω 0 ) α ] .
Z = ξ L D , T = τ T 0 ; U = A A 0 , V = B B 0 ; X = x L , Y = y L , u = N E U , v = N H V ,
u Z = i sgn ( k 0 ) 2 T 2 u i sgn ( β 2 ) 2 2 u T 2 + i ( 1 + i S E T ) | u | 2 u + i | v | 2 u C E | v | 2 u T v Z = i sgn ( k 0 ) 2 T 2 v i sgn ( β 2 ) 2 2 v T 2 + i ( 1 + i S H T ) | v | 2 v + i | u | 2 v C H | u | 2 v T ,
u Z = i sgn ( β 2 ) 2 2 u T 2 + i ( 1 + i S E T ) | u | 2 u + i | v | 2 u , v Z = i sgn ( β 2 ) 2 2 v T 2 + i ( 1 + i S H T ) | v | 2 v + i | u | 2 v ,
u ( Z , T ) = [ a 0 + a ( Z , T ) ] exp ( i Ω 0 a Z ) and v ( Z , T ) = [ b 0 + b ( Z , T ) ] exp ( i Ω 0 b Z ) ,
a Z = i sgn ( β 2 ) 2 2 a T 2 + i a 0 2 ( a + a * ) + i a 0 b 0 ( b + b * ) S E a 0 2 ( 2 a τ + a * τ ) , b Z = i sgn ( β 2 ) 2 2 b T 2 + i b 0 2 ( b + b * ) + i a 0 b 0 ( a + a * ) S H b 0 2 ( 2 b τ + b * τ ) .
K = 1 2 [ 4 s Ω ± ( 4 s 2 Ω 2 + Ω 4 4 + 4 δ ( a 0 2 + b 0 2 ) Ω 2 ) 1 2 ] ,
g ( Ω ) = 2 Im ( K ) = [ 4 ( a 0 2 + b 0 2 ) Ω 2 4 s 2 Ω 2 Ω 4 / 4 ] 1 2 .

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