Abstract

We show that modulation instability may occur even in the normal group-velocity dispersion regime, or in the case of no group-velocity dispersion in metamaterials with a nonlinear electric polarization. The physical origin of the modulation instability is the additional second-order nonlinear dispersion effect resulted from the combination of the linear dispersive magnetic permeability with the nonlinear electric polarization. Based on the Drude model, a numerical simulation is performed to confirm the theoretical predictions, and a detailed discussion on the role of the second-order nonlinear dispersion effect in modulation instability in both negative-index and positive-index regions of metamaterial is presented.

© 2008 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
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    [CrossRef]
  6. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. St. J. Russell, "Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber," Opt. Lett. 28, 2225-2227 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780nm wavelength," Opt. Lett. 32, 53-55 (2007).
    [CrossRef]
  13. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  14. 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]
  15. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
    [CrossRef]
  16. 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]
  17. A. K. Popov and V. M. Shalaev, "Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification," Appl. Phys. B 84, 131-137 (2006).
    [CrossRef]
  18. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, "Second-harmonic generation from magnetic metamaterials," Science 313, 502-504 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  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. 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]
  23. I. Kourakis and P. K. Shukla, "Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials," Phys. Rev. E 72, 016626 (2005).
    [CrossRef]
  24. 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]
  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. M. J. Potosek, "Modulation instability in an extended nonlinear Schrödinger equation," Opt. Lett. 12, 921-923 (1987).
    [CrossRef]
  27. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  28. 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]
  29. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
    [CrossRef] [PubMed]
  30. A. K. Popov and V. M. Shalaev, "Compensating losses in negative-index metamaterials by optical parametric amplification," Opt. Lett. 31, 2169-2171 (2006).
    [CrossRef] [PubMed]

2007 (4)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (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]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, "Experiments on second- and third-harmonic generation from magnetic metamaterials," Opt. Express 15, 5238-5247 (2007).
[CrossRef] [PubMed]

2006 (6)

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]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

A. K. Popov and V. M. Shalaev, "Compensating losses in negative-index metamaterials by optical parametric amplification," Opt. Lett. 31, 2169-2171 (2006).
[CrossRef] [PubMed]

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

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, "Second-harmonic generation from magnetic metamaterials," Science 313, 502-504 (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 (5)

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]

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]

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]

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 (3)

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]

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

T. Tanemura, Y. Ozeki, and K. Kikuchi, "Modulational instability and parametric amplification induced by loss dispersion in optical fibers," Phys. Rev. Lett. 93, 163902 (2004).
[CrossRef] [PubMed]

2003 (3)

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]

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. St. J. Russell, "Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber," Opt. Lett. 28, 2225-2227 (2003).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

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

1997 (1)

S. Coen and M. Haelterman, "Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber," Phys. Rev. Lett. 79, 4139-4142 (1997).
[CrossRef]

1994 (1)

F. Kh. 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 (1)

1987 (2)

M. J. Potosek, "Modulation instability in an extended nonlinear Schrödinger equation," Opt. Lett. 12, 921-923 (1987).
[CrossRef]

G. P. Agrawal, "Modulation instability induced by cross-phase modulation," Phys. Rev. Lett. 59, 880-883 (1987).
[CrossRef] [PubMed]

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]

Appl. Phys. B (1)

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

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

Nat. Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Commun. (1)

F. Kh. 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 (2)

Opt. Lett. (8)

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. B (1)

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]

Phys. Rev. E (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]

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]

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]

Phys. Rev. Lett. (7)

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

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

T. Tanemura, Y. Ozeki, and K. Kikuchi, "Modulational instability and parametric amplification induced by loss dispersion in optical fibers," Phys. Rev. Lett. 93, 163902 (2004).
[CrossRef] [PubMed]

G. P. Agrawal, "Modulation instability induced by cross-phase modulation," Phys. Rev. Lett. 59, 880-883 (1987).
[CrossRef] [PubMed]

S. Coen and M. Haelterman, "Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber," Phys. Rev. Lett. 79, 4139-4142 (1997).
[CrossRef]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[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]

Phys. Today (1)

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

Science (1)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, "Second-harmonic generation from magnetic metamaterials," Science 313, 502-504 (2006).
[CrossRef] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (5)

Fig. 1
Fig. 1

Refraction index n, SS parameter s 1 , second-order nonlinear dispersion parameter, GVD β 2 , and the fourth-order dispersion β 4 , versus ω ω pe for ω pm ω pe = 0.8 . β 2 is calculated in units of 1 ( c ω pe ) , β 4 is calculated in units of 10 3 ( c ω pe 3 ) , s = 0.2 .

Fig. 2
Fig. 2

MI gain spectra in the negative-index region of MM. (a) Normal dispersion regime for s 1 = 0.46 , s 2 = 0.14 , and I 0 = 25 . (b) Zero GVD point for s 1 = 0.3122 , s 2 = 0.1424 , and I 0 = 10 .

Fig. 3
Fig. 3

Temporal distributions of the field intensity at different propagation distance for different parameters at the zero GVD point for (a) s 1 = 0.3122 , s 2 = 0.1424 , and (b) s 1 = 0 , s 2 = 0.1424 .

Fig. 4
Fig. 4

The MI gain spectra for different second-order nonlinear dispersion in the normal dispersion regime of MM for I 0 = 10 .

Fig. 5
Fig. 5

Joint influence of SS and the second-order nonlinear dispersion on MI in the anomalous dispersion regime of (a) the negative-index region and (b) the positive-index region for I 0 = 1 .

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

A ξ = i β 2 2 2 A τ 2 + m = 3 i m + 1 β m m ! m A τ m + i Υ 0 [ A 2 A + i S 1 τ ( A 2 A ) S 2 2 τ 2 ( A 2 A ) ] ,
S 1 = 1 ω 0 ( k 0 V ) 1 + Υ 1 Υ 0 ,
S 2 = Υ 1 ( ω 0 Υ 0 ) β 2 ( 4 k 0 ) + Υ 2 ( 2 Υ 0 ) ,
u Z = i σ 2 2 u T 2 + b 3 6 m u T m + i b 4 24 m u T m + i N [ u 2 u + i s 1 T ( u 2 u ) s 2 2 T 2 ( u 2 u ) ] ,
a Z = i ( σ 2 + 2 N s 2 I 0 ) 2 a T 2 + b 3 6 3 a T 3 + i b 4 24 4 a T 4 + i N I 0 ( a + a * ) N s 1 I 0 ( 2 a T + a * T ) i N s 2 I 0 2 a * T 2 ,
a ( Z , T ) = a 1 exp [ i ( K Z Ω T ) ] + a 2 exp [ i ( K Z Ω T ) ] ,
K = 2 N I 0 s 1 Ω + b 3 6 Ω 3 ± Ω 2 4 σ N I 0 + Ω 2 + Q 1 ( Ω ) + Q 2 ( Ω ) + 4 N 2 I 0 2 s 1 2 ,
Q 1 ( Ω ) = b 4 6 Ω 2 ( 2 N I 0 + σ Ω 2 + b 4 24 Ω 4 ) ,
Q 2 ( Ω ) = 4 s 2 N I 0 ( 2 N I 0 + 2 σ Ω 2 + b 4 6 Ω 4 + 3 N I 0 s 2 Ω 2 ) .
g ( Ω ) = Ω 4 σ N I 0 Ω 2 Q 1 ( Ω ) Q 2 ( Ω ) 4 N 2 I 0 2 s 1 2 .
4 σ N I 0 + Ω 2 + Q 1 ( Ω ) + Q 2 ( Ω ) + 4 N 2 I 0 2 s 1 2 < 0 .
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 ) ) ,
β 2 = 1 c ω 0 n ( 1 + 3 ω pe 2 ω pm 2 ω 0 4 ) 1 c ω 0 n 3 ( 1 ω pe 2 ω pm 2 ω 0 4 ) 2 ,
β 4 = 60 ε 0 μ 0 ω pe 2 ω pm 2 k 0 ω 0 6 3 β 2 2 k 0 ,
s 1 = s ( 1 + ω pm 2 ω pe 2 ω 0 4 n 2 ω 0 4 ω pm 2 + ω 0 2 ω pm 2 ω 0 2 ) ,
s 2 = s 2 [ ω 0 2 ω 0 2 ω pm 2 1 4 n 2 ( 1 + 3 ω pe 2 ω pm 2 ω 0 4 ) + 1 4 n 4 ( 1 ω pe 2 ω pm 2 ω 0 4 ) 2 ] .

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