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] [PubMed]
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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]
  7. 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]
  8. G. P. Agrawal, "Modulation instability induced by cross-phase modulation," Phys. Rev. Lett. 59, 880-883 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
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  11. 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]
  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]
  19. 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]
  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. 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)

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

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

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

2006 (6)

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]

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]

2005 (5)

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]

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]

2004 (3)

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 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]

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)

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]

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]

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)

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

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

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)

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

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. 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]

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]

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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