Abstract

In negative-index materials (NIMs), the self-steepening (SS) effect is proven to be anomalous in two aspects: First, it can be either positive or negative, with the zero SS point determined by the size of split-ring resonator circuit elements. Second, the negative SS parameter can have a very large value compared to an ordinary positive-index material. We present a theoretical investigation on modulation instability (MI) to identify the role of the anomalous SS effect in NIM. We find that the first anomaly of SS doesn’t influence MI, yet the controllable zero SS point can be used to manipulate MI, and thus manipulate the generation of solitons. The second anomaly, however, leads to significant changes in the MI condition and property, compared with the case of an ordinary positive-index material. Numerical simulations confirm the theoretical results and show that negative SS moves the center of generated pulse toward the leading side, and shifts a part of energy of the generated pulse toward the red side, opposite to the case of positive SS.

© 2006 Optical Society of America

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  1. G. P. Agrawal. Nonlinear Fiber Optics, 3nd edn. (San Diego, Academic, 2001).
  2. V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509 (1987).
    [CrossRef]
  3. M. J. Potosek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921 (1987).
    [CrossRef]
  4. A. Höök and M. Karlsson, “Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion,” Opt. Lett. 18, 1388 (1993).
    [CrossRef] [PubMed]
  5. F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
    [CrossRef]
  6. F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27 (1997).
    [CrossRef]
  7. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eand μ,” Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  8. J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  9. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with zimultane-ously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
    [CrossRef] [PubMed]
  10. R. A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of a negative refractive index of refraction,” Science 292, 77 (2001).
    [CrossRef] [PubMed]
  11. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
    [CrossRef]
  12. E. Schonbrun, M. Tinker, W. Park, and J. -B. Lee, “Negative refraction in a Si-polymer photonic Crystal membrane,” IEEE Photonics Technol. Lett. 17, 1196 (2005).
    [CrossRef]
  13. 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 (2005).
    [CrossRef]
  14. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  15. S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
    [CrossRef]
  16. M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
    [CrossRef]
  17. 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).
    [CrossRef]
  18. N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schröinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
    [CrossRef]
  19. I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
    [CrossRef]
  20. 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] [PubMed]
  21. 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 (2005).
    [CrossRef] [PubMed]
  22. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282 (1997).
    [CrossRef]
  23. M. Marklund, P. K. Shukla, and L. Stenflo, “Ultra-short solitons and kinetic effects in nonlinear metamaterials,” arXiv: nlin./060162.

2005 (6)

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

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

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] [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 (2005).
[CrossRef] [PubMed]

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 (2005).
[CrossRef]

2004 (3)

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (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).
[CrossRef]

2003 (2)

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (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] [PubMed]

2001 (1)

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

2000 (2)

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

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

1997 (2)

1994 (1)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
[CrossRef]

1993 (1)

1987 (2)

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509 (1987).
[CrossRef]

M. J. Potosek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921 (1987).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eand μ,” Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27 (1997).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
[CrossRef]

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).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal. Nonlinear Fiber Optics, 3nd edn. (San Diego, Academic, 2001).

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] [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 (2005).
[CrossRef] [PubMed]

Anand, S.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

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).
[CrossRef]

Berrier, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

Bischoff, S.

Bloemer, M. J.

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 (2005).
[CrossRef] [PubMed]

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] [PubMed]

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Bsichoff, S.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
[CrossRef]

Cai, W.

Chettiar, U. K.

Christiansen, P. L.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
[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] [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 (2005).
[CrossRef] [PubMed]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27 (1997).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
[CrossRef]

Drachev, V. P.

Gorkunov, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Höök, A.

Karlsson, M.

Kildishev, A. V.

Kivshar, Y. S.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Kourakis, I.

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

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Lapine, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Lazarides, N.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schröinger 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 Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

Marklund, M.

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultra-short solitons and kinetic effects in nonlinear metamaterials,” arXiv: nlin./060162.

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] [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 (2005).
[CrossRef] [PubMed]

McPeake, D.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Mulot, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

Nemat-Nasser, S. C.

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

O’Brien, S.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with zimultane-ously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (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 Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

Pendry, J. B.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Pendry, J.B.

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (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 Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Potosek, M. J.

Qiu, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

Ramakrishna, S. A.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Sarychev, A. K.

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] [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 (2005).
[CrossRef] [PubMed]

Schonbrun, E.

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

Schultz, S.

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

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

Shadrivov, I. V.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Shalaev, V. M.

Shelby, R. A.

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

Shen, Y. R.

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).
[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]

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultra-short solitons and kinetic effects in nonlinear metamaterials,” arXiv: nlin./060162.

Smith, D. R.

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

Smith, D.R.

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

Sørensen, M. P.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27 (1997).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
[CrossRef]

Stenflo, L.

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultra-short solitons and kinetic effects in nonlinear metamaterials,” arXiv: nlin./060162.

Sukhotskova, N. A.

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509 (1987).
[CrossRef]

Swillo, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[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] [PubMed]

Talneau, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

Thylén, L.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative Refraction at Infrared-Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 93, 73902 (2004).
[CrossRef]

Tinker, M.

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

Tsironis, G. P.

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

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eand μ,” Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Vier, D. C.

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

Vysloukh, V. A.

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509 (1987).
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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).
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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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IEEE Photonics Technol. Lett. (1)

E. Schonbrun, M. Tinker, W. Park, and J. -B. Lee, “Negative refraction in a Si-polymer photonic Crystal membrane,” IEEE Photonics Technol. Lett. 17, 1196 (2005).
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F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60 (1994).
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S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
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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).
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N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schröinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
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M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
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T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282 (1997).
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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] [PubMed]

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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Sov. J. Quantum Electron. (1)

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509 (1987).
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Figures (4)

Fig. 1.
Fig. 1.

Refractionindex n (green lines), SS parameter Cs (blue lines), and GVD parameter β 2 (red lines), versus ω/ωpe for ωpm /ωpe = 0.6 (a), 0.8 (b) and 1.0 (c). β 2 is calculatedin units of 1/(2πcωpe ).

Fig. 2
Fig. 2

(a) MI gain spectrum with different SS parameter S. (b) Variations of the critical frequency and the fastest frequency with self-steepening parameter S.

Fig. 3.
Fig. 3.

Temporal (left column) and spectral (right column) distributions of the field intensity at different propagation distances for different SS parameters: bottom row: S = 0, medium row: S = 0.2, and top row: S = -0.2. The blue, red and green lines are for Z = 0, 2 and 4, respectively.

Fig. 4.
Fig. 4.

Evolution of the 2nd-order soliton in (a) ordinary positive-index material for S = 0.2 and (b) NIM for S = -0.2.

Equations (19)

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2 E ( x , t ) z 2 = μ ε 2 E ( z , t ) t 2 + μ 2 P nl ( z , t ) t 2 .
ε ( ω ) = ε 0 ( 1 ω pe 2 ω ( ω + i γ e ) ) , μ ( ω ) = μ 0 ( 1 ω pm 2 ω ( ω + i γ m ) ) ,
ω ε ( ω ) = m = 0 { m [ ω ε ( ω ) ] m ! ω m ω = ω 0 ( ω ω 0 ) m } m = 0 Θ m ( ω ω 0 ) m ,
ω μ ( ω ) = m = 0 { m [ ω μ ( ω ) ] m ! ω m ω = ω 0 ( ω ω 0 ) m } m = 0 Ξ m ( ω ω 0 ) m .
2 E ˜ ( z , ω ) z 2 = m = 0 [ Θ m ( ω ω 0 ) m ] l = 0 [ Ξ l ( ω ω 0 ) l ] E ˜ ( z , ω )
ω l = 0 [ Ξ l ( ω ω 0 ) i ] P ˜ nl ( z , ω ) .
( i 2 k 0 2 z 2 + z + 1 V t ) A = i m = 2 i m δ m m ! m A t m + i m = 0 i m ϒ m m ! ( 1 + i ω 0 t ) m t m ( A 2 A ) ,
A ξ = i β 2 2 2 A τ 2 + m = 3 i m + 1 δ m m ! m A τ m + m = 0 i m + 1 ϒ m m ! ( 1 + i ω 0 τ ) m τ m ( A 2 A )
+ i 2 k 0 ( 2 A ξ 2 2 V 2 A τ ξ ) ,
β 2 = δ 2 1 ( k 0 V 2 ) ,
A ξ = i β 2 2 2 A τ 2 + i C nl ( 1 + i C s ω 0 τ ) ( A 2 A ) ,
β 2 = 1 2 cn ω 0 ( 1 + 3 ω pe 2 ω pm 2 ω 0 4 ) ( 1 ω pe 2 ω pm 2 ω 0 4 ) 2 cn 3 ω 0 ,
C nl = χ ( 3 ) ω 0 2 nc ( 1 ω pm 2 ω 0 2 ) ,
C s = 1 + ω pm 2 ω pe 2 ω 0 4 n 2 ω 0 4 ω 0 2 + ω pm 2 ω pm 2 ω 0 2 .
ω ω pe = [ 1 ( 1 ω pm 2 ω pe 2 ) 1 2 ] 1 2 .
U Z = i δ 2 2 U T 2 + iϑN ( 1 + iS T ) ( U 2 U ) ,
g ( ϖ ) = ϖ 2 2 ( 2 δ ϑ b + ϖ 2 2 ) S 2 b 2 ϖ 2 .
ϖ c = 4 b 4 S 2 b 2 , ϖ max = 2 b 2 S 2 b 2 .
U ( T , 0 ) = U 0 [ 1 + u 0 cos ( ϖ T ) ] ,

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