Abstract

We analyze modulation instability (MI) in metamaterials with saturable and focusing nonlinearity according to the propagation model, including the self-steepening (SS) parameter and the second-order nonlinear dispersion (SOND). A detailed discussion on the common role of saturable nonlinearity, SS, and SOND terms on the MI is presented. It is found that MI is irrespective of the sign of the SS parameter and dependent on the sign of SOND. Generally, SS and saturable nonlinearity suppress the MI generation, and SOND promotes the MI in the anomalous group-velocity dispersion (GVD) region. Moreover, linear stability analysis predicts that MI may occur in both the abnormal GVD and normal GVD regime, even at the zero-GVD point, which is to say that some interesting MI phenomena appear involving the additional excited SOND term induced by the dispersive magnetic permeability. In particular, the saturable nonlinearity changes the generation condition of MI seriously. Finally, the numerical simulation is performed to confirm the theoretical predictions.

© 2011 Optical Society of America

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (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, 1509–1511 (1987).
    [CrossRef]
  3. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138(1986).
    [CrossRef] [PubMed]
  4. F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(1994).
    [CrossRef]
  5. A. Hook and M. Karlsson, “Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion,” Opt. Lett. 18, 1388–1390 (1993).
    [CrossRef] [PubMed]
  6. J. M. Soto-Crespo and E. M. Wright, “Generation of pulse trains in the normal dispersion regime of a dielectric medium,” Appl. Phys. Lett. 59, 2489–2491 (1991).
    [CrossRef]
  7. G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
    [CrossRef] [PubMed]
  8. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [CrossRef] [PubMed]
  9. J. Miguel Hickmann, S. B. Cavalcanti, N. M. Borges, E. A. Gouveia, and A. S. Gouveia-Neto, “Modulational instability in semiconductor-doped glass fibers with saturable nonlinearity,” Opt. Lett. 18, 182–184 (1993).
    [CrossRef] [PubMed]
  10. X. Q. Zhong and A. P. Xiang, “Cross-phase modulation induced modulation instability in single-mode optical fibers with saturable nonlinearity,” Opt. Fiber Technol. 13, 271–279 (2007).
    [CrossRef]
  11. P. T. Dinda and K. Porsezian, “Impact of fourth-order dispersion in the modulational instability spectra of wave propagation in glass fibers with saturable nonlinearity,” J. Opt. Soc. Am. B 27, 1143–1152 (2010).
    [CrossRef]
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    [CrossRef]
  13. J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57 (6), 37–43 (2004).
    [CrossRef]
  14. V. M. Shalaev, W. Cai, U. K. Chettiar, H. -K. 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]
  15. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  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. 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]
  18. 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]
  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. 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. 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]
  22. S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
    [CrossRef]
  23. 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]
  24. Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
    [CrossRef]
  25. 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]
  26. A. Maluckov, L. Hadžievski, N. Lazarides, G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
    [CrossRef]
  27. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]

2010 (1)

2009 (2)

2008 (1)

A. Maluckov, L. Hadžievski, N. Lazarides, G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

2007 (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]

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

X. Q. Zhong and A. P. Xiang, “Cross-phase modulation induced modulation instability in single-mode optical fibers with saturable nonlinearity,” Opt. Fiber Technol. 13, 271–279 (2007).
[CrossRef]

2006 (2)

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]

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

2005 (4)

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]

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]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. -K. 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 (2)

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57 (6), 37–43 (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)

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

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]

2000 (1)

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

1994 (1)

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(1994).
[CrossRef]

1993 (2)

1991 (1)

J. M. Soto-Crespo and E. M. Wright, “Generation of pulse trains in the normal dispersion regime of a dielectric medium,” Appl. Phys. Lett. 59, 2489–2491 (1991).
[CrossRef]

1990 (1)

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

1989 (1)

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef] [PubMed]

1987 (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, 1509–1511 (1987).
[CrossRef]

1986 (1)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138(1986).
[CrossRef] [PubMed]

Abdullaev, F. K.

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(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, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (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]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef] [PubMed]

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef] [PubMed]

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]

Bischoff, S.

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(1994).
[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] [PubMed]

Borges, N. M.

Cai, W.

Cavalcanti, S. B.

Chettiar, U. K.

Christiansen, P. L.

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(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]

da Silva, G. L.

Dai, X.

Darmanyan, S. A.

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(1994).
[CrossRef]

Dinda, P. T.

Drachev, V. P.

Fan, D.

Fu, X.

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]

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

Gleria, I.

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]

Gouveia, E. A.

Gouveia-Neto, A. S.

Hadžievski, L.

A. Maluckov, L. Hadžievski, N. Lazarides, G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138(1986).
[CrossRef] [PubMed]

Hickmann, J. Miguel

Hook, A.

Hu, Y.

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]

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.

A. Maluckov, L. Hadžievski, N. Lazarides, G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[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]

Lyra, M. L.

Maluckov, A.

A. Maluckov, L. Hadžievski, N. Lazarides, G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[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] [PubMed]

Pendry, J. B.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57 (6), 37–43 (2004).
[CrossRef]

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

Porsezian, K.

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]

Rothenberg, J. E.

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

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]

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.

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]

Smith, D. R.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57 (6), 37–43 (2004).
[CrossRef]

Sombra, A. S. B.

Sørensen, M. P.

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(1994).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo and E. M. Wright, “Generation of pulse trains in the normal dispersion regime of a dielectric medium,” Appl. Phys. Lett. 59, 2489–2491 (1991).
[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]

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

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

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, 1509–1511 (1987).
[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]

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138(1986).
[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]

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

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138(1986).
[CrossRef] [PubMed]

Tsironis, G. P.

A. Maluckov, L. Hadžievski, N. Lazarides, G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[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]

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, 1509–1511 (1987).
[CrossRef]

Wang, Y.

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

Wen, S.

Wright, E. M.

J. M. Soto-Crespo and E. M. Wright, “Generation of pulse trains in the normal dispersion regime of a dielectric medium,” Appl. Phys. Lett. 59, 2489–2491 (1991).
[CrossRef]

Xiang, A. P.

X. Q. Zhong and A. P. Xiang, “Cross-phase modulation induced modulation instability in single-mode optical fibers with saturable nonlinearity,” Opt. Fiber Technol. 13, 271–279 (2007).
[CrossRef]

Xiang, Y.

Yuan, H. -K.

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).
[CrossRef]

Zharov, A. A.

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

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] [PubMed]

Zhong, X. Q.

X. Q. Zhong and A. P. Xiang, “Cross-phase modulation induced modulation instability in single-mode optical fibers with saturable nonlinearity,” Opt. Fiber Technol. 13, 271–279 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

J. M. Soto-Crespo and E. M. Wright, “Generation of pulse trains in the normal dispersion regime of a dielectric medium,” Appl. Phys. Lett. 59, 2489–2491 (1991).
[CrossRef]

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

Opt. Commun. (1)

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64(1994).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. (1)

X. Q. Zhong and A. P. Xiang, “Cross-phase modulation induced modulation instability in single-mode optical fibers with saturable nonlinearity,” Opt. Fiber Technol. 13, 271–279 (2007).
[CrossRef]

Opt. Lett. (3)

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Phys. Rev. B (1)

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Figures (9)

Fig. 1
Fig. 1

Refraction index n, SS parameter s 1 , SOND s 2 , and GVD parameter β 2 versus ω / ω pe for ω pm / ω pe = 0.8 . β 2 is calculated in units of 1 / ( 2 π c ω pe ) , s = 0.2 .

Fig. 2
Fig. 2

(a) Dependences of the OMF Ω opt and (b) peak gain g max on SS parameter | s 1 | at different saturation parameter κ, and (c) the OMF Ω opt and (d) peak gain g max as the function of the input power I 0 at different SS parameter s 1 and saturation parameter κ.

Fig. 3
Fig. 3

Contour plots of the gain spectrum on SS parameter s 1 at different saturation parameter κ: (a)  κ = 0.01 , (b)  κ = 0.20 , and (c)  κ = 1.0 , where I 0 = 1 , N = 1 .

Fig. 4
Fig. 4

Dependences of the OMF Ω opt , in the (a) anomalous GVD and (c) normal GVD, and the peak gain g max in the (b) anomalous GVD and (d) normal GVD, on the SOND parameter s 2 at different saturation parameter κ, where I 0 = 1 and N = 1 .

Fig. 5
Fig. 5

Gain spectrum on SOND parameter s 2 and gain frequency Ω at different GVD regions: (a) anomalous GVD and (b) normal GVD, where I 0 = 1 , N = 1 , and κ = 1.0 .

Fig. 6
Fig. 6

Contour plots of the gain spectrum on normalized frequency ω / ω pe in (a) the anomalous GVD region of the NIMs, (b) the normal GVD region of the negative-index region, and (c) the anomalous GVD region of the positive-index MMs, where κ = 0.1 and N = 1 .

Fig. 7
Fig. 7

Dependence of the MI gain spectrum on the saturation parameter κ at the zero GVD point, where s 1 = 0.3122 , s 2 = 0.1424 , N = 1 . The input powers are (a)  I 0 = 1 and (b)  I 0 = 2 .

Fig. 8
Fig. 8

Temporal distributions of the normalized field intensity at different propagation distance in the anomalous GVD region for (a)  s 1 = 0.2 s 2 = 0 ; (b)  s 1 = 0.2 , s 2 = 0 , where κ = 0.2 , ϖ = 1.0 , and U 0 = 1 .

Fig. 9
Fig. 9

Temporal distributions of the normalized field intensity at different propagation distance at the zero-GVD point for (a)  s 1 = 0 and s 2 = 0.1424 ; (b)  s 1 = 0.3122 , s 2 = 0.1424 , where κ = 0.1 , ϖ = 1.0 , and U 0 = 4 . The black arrow denotes the direction of movement of the peak intensity.

Equations (18)

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A ξ = i β 2 2 2 A T 2 + m = 3 i m + 1 δ m m ! m A T m + i γ 0 [ | A | 2 1 + κ | A | 2 A + i S 1 T ( | A | 2 1 + κ | A | 2 A ) S 2 2 T 2 ( | A | 2 1 + κ | A | 2 A ) ] .
S 1 = 1 / ω 0 ( k 0 V ) 1 + γ 1 / γ 0 ,
S 2 = γ 1 / ( ω 0 γ 0 ) β 2 / ( 4 k 0 ) + γ 2 / ( 2 γ 0 ) .
A ξ = i sgn ( β 2 ) 2 2 A τ 2 + i N [ | A | 2 1 + κ | A | 2 A + i s 1 T ( | A | 2 1 + κ | A | 2 A ) s 2 2 T 2 ( | A | 2 1 + κ | A | 2 A ) ] ,
u ¯ = I 0 exp ( i N I 0 Z / ( 1 + κ I 0 ) ) ,
K = N I 0 S Ω ( 1 + κ I 0 ) 2 + N I 0 S Ω 1 + κ I 0 ± 1 2 Ω { 4 N 2 I 0 2 S 2 ( 1 + κ I 0 ) 4 + 4 N I 0 sgn ( β 2 ) ( 1 + κ I 0 ) 2 + Ω 2 + Π ( s 2 ) } 1 / 2 ,
Π ( s 2 ) = [ 8 N 2 I 0 2 s 2 2 ( 1 + κ I 0 ) 3 + 4 N 2 I 0 2 s 2 2 ( 1 + κ I 0 ) 2 + 4 N I 0 s 2 sgn ( β 2 ) ( 1 + κ I 0 ) 2 + 4 N I 0 s 2 sgn ( β 2 ) 1 + κ I 0 ] Ω 2 + 8 N 2 I 0 2 s 2 ( 1 + κ I 0 ) 3 .
g = | Ω | { 4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 4 N I 0 sgn ( β 2 ) ( 1 + κ I 0 ) 2 Ω 2 Π ( s 2 ) } 1 / 2 .
4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 + 4 N I 0 sgn ( β 2 ) ( 1 + κ I 0 ) 2 + Ω 2 + Π ( s 2 ) < 0 .
Ω c 2 = [ 4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 8 N 2 I 0 2 s 2 ( 1 + κ I 0 ) 3 4 N I 0 sgn ( β 2 ) ( 1 + κ I 0 ) 2 ] / [ 8 N 2 I 0 2 s 2 2 ( 1 + κ I 0 ) 3 + 4 N 2 I 0 2 s 2 2 ( 1 + κ I 0 ) 2 + 4 N I 0 s 2 sgn ( β 2 ) ( 1 + κ I 0 ) 2 + 4 N I 0 s 2 sgn ( β 2 ) 1 + κ I 0 + 1 ] .
ε ( ω ) = ε 0 ( 1 ω pe 2 / ω 2 ) , μ ( ω ) = μ 0 ( 1 ω pm 2 / ω 2 ) ,
s 1 = s ( ω pm 2 ω pe 2 ω 4 n 2 ω 4 ω pm 2 + ω 2 ω pm 2 ω 2 ) , s 2 = s 2 [ ω 2 ( ω 2 ω pm 2 ) 1 4 n 2 ( 1 + 3 ω pe 2 ω 2 ω pm 2 ω 2 ) + 1 4 n 4 ( 1 ω pe 2 ω 2 ω pm 2 ω 2 ) 2 ] ,
g = | Ω | { 4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 4 N I 0 sgn ( β 2 ) ( 1 + κ I 0 ) 2 Ω 2 } 1 / 2 .
Ω c = 4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 4 N I 0 sgn ( β 2 ) ( 1 + κ I 0 ) 2 .
s 1 2 < sgn ( β 2 ) ( 1 + κ I 0 ) 2 N I 0 .
g = | Ω | { 8 N 2 I 0 2 s 2 ( 1 + κ I 0 ) 3 [ 4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 + 8 N 2 I 0 2 s 2 2 Ω 2 ( 1 + κ I 0 ) 3 + 4 N 2 I 0 2 s 2 2 Ω 2 ( 1 + κ I 0 ) 2 + Ω 2 ] } 1 / 2 .
Ω c 2 = [ 8 N 2 I 0 2 s 2 ( 1 + κ I 0 ) 3 4 N 2 I 0 2 s 1 2 ( 1 + κ I 0 ) 4 ] / [ 8 N 2 I 0 2 s 2 2 ( 1 + κ I 0 ) 3 + 4 N 2 I 0 2 s 2 2 ( 1 + κ I 0 ) 2 + 1 ] .
s 2 < s 1 2 2 ( 1 + κ I 0 ) .

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