## Role of the anomalous self-steepening effect in modulation instability in negative-index material

Optics Express, Vol. 14, Issue 4, pp. 1568-1575 (2006)

http://dx.doi.org/10.1364/OE.14.001568

Acrobat PDF (370 KB)

### 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

## 1. Introduction

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]

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

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]

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

## 2. Modelling the pulse propagation in NIM

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]

*E*propagates along the

*z*direction. Both

*E*and the nonlinear polarization

*P*

_{nl}are assumed to be polarized parallel to the

*x*axis. The dielectric susceptibility

*ε*and magnetic permeability

*μ*are dispersive in a NIM, otherwise the energy density could be negative [7

07. 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]

*ω*

_{pe}and

*ω*

_{pm}are the respective electric and magnetic plasma frequencies,

*γ*

_{e}and

*γ*

_{m}are the respective electric and magnetic loss terms, which are very small and are neglected in the following analysis for simplicity, and

*ε*

_{0}and

*μ*

_{0}are the respective vacuum susceptibility and magnetic permeability.

*ε*(

*ω*) and

*μ*(

*ω*) in powers of

*ω*, thus enabling us to treat the material parameters as a power series whichwe can truncate to an approriate order Howeverfor simplicity it is better to expand

*ωε*(

*ω*) and

*ωμ*(

*ω*) abouta suitable

*ω*

_{0}instead,

*E*(

*z*,

*t*) = (1/2)

*A*(

*z*,

*t*)exp(

*ik*

_{0}

*z*-

*iω*

_{0}

*t*) +

*c*.

*c*, where

*k*

_{0}=

*n*(

*ω*

_{0})

*ω*

_{0}/

*c*is wave number in medium at the carrier frequency

*ω*

_{0},

*n*(

*ω*

_{0}) is the refractive index of medium at

*ω*

_{0}, and assume that,

*P*

_{nl}(

*z*,

*t*) =

*ε*

_{0}χ

^{(3)}∣

*E*(

*z*,

*t*)∣

^{2}

*E*(

*z*,

*t*), which characterizesaKerrnonlnearity. Withtheseenvelope-carrier substitutions, taking the inverse Fourier transform of the obtained equation, we have

*V*= 2

*k*

_{0}/(Θ

_{0}Ξ

_{1}+ Θ

_{1}Ξ

_{0}) is the group velocity of the pulse,

_{m}=

*m*!

*ω*

_{0}

*ε*

_{0}χ

^{(3)}Ξ

_{m}/(2

*k*

_{0}). Introducing co-moving variables, τ =

*t*-

*z*/

*V*,

*ξ*=

*z*, Eq. (6) is transformed to the following generalized NLSE

*β*

_{2}=

*∂*

^{2}

*k*/

*∂*

*ω*

^{2}∣

_{ω0}.

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

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]

*∂*

^{2}

*A*/

*∂*

*ξ*

^{2}=

*i*ϒ

_{0}

*∂*(∣

*A*∣

^{2}

*A*)/

*∂ξ*,

*∂*

^{2}

*A*/

*∂τ∂ξ*=

*i*ϒ

_{0}

*∂*(∣

*A*∣

^{2}

*A*)/

*∂τ*. Thus we obtainthe following NLSE

*C*

_{nl}= ϒ

_{0}and

*C*

_{s}= 1 +

*ω*

_{0}ϒ

_{1}/ϒ

_{0}-

*ω*

_{0}(

*k*

_{0}

*V*) are the nonlinear and SS coefficients, respectively. For lossless Drude model, the expressions for

*β*

_{2},

*C*

_{nl}and

*C*

_{s}are

*C*

_{s}. Inits peer in ordnary positi ve-index materials,

*C*

_{s}= 1. To disclose the anomalous properties of the SS parameter, we plot the variation of

*n*,

*C*

_{s}, and

*β*

_{2}with

*ω*/

*ω*

_{pe}for different values ofthe ratio

*ω*

_{pm}/

*ω*

_{pe}for

*γ*

_{e}=

*γ*

_{m}= 0 in Fig. 1. We see that, first, like GVD [10

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]

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]

*ω*

_{pm}/

*ω*

_{pe}< 1, with the zero SS point located at

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]

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]

## 3. Role of the anomalous self-steepening in MI in NIM

*T*=

*τ*/

*τ*

_{p},

*Z*=

*ξ*/

*l*

_{d}, and

*U*=

*A*/

*A*

_{0}, where

*τ*

_{p}is the width ofthe input pulse,

*l*

_{d}=

*β*

_{2}∣ the dispersion length,

*l*

_{nl}= 1/(∣

*C*

_{nl}∣

*A*

_{0}the amplitude ofthe input field,to transform Eq. (9) into the normalized form:

*N*=

*l*

_{d}/

*l*

_{nl},

*S*=

*C*

_{s}/(

*τ*

_{p}

*ω*

_{0}) is the normalized self-steepening coefficient,

*δ*= ±1 stands for normal and anomalous GVD respectively, and ϑ = ±1 for focusing and defocusing nonlin-earity respectively.

**12**, 921 (1987). [CrossRef]

*ϖ*is frequency for perturbation,

*b*=

*U*

_{0}is the normalized initial amplitude ofthe wave. Eq. (15) shows that MI in NIM appears for focusing nonlinearity (ϑ = 1) and anomalous dispersion (ϑ = -1), or defocusing nonlnearity (

*δ*= -1) and normaldispersion (

*δ*= 1), as in ordinary materials [1, 3

**12**, 921 (1987). [CrossRef]

*k*

_{0}, it doesn’t alter the sign of nonlnearity coefficient determined by the combination of signs of

*μ*, nand χ

^{(3)}, as Eq.(11) showns. For NIM,

*μ*and

*n*are simultaneously negative, and thus the sign of nonlinearity coefficient is only determined by χ

^{(3)}, as in ordinary materials From Eq.(15), we can obtain the critical frequency for MI to occur, and the fastest growing frequency for occurring MI

*ϖ*and SS parameter; and Fig.2(b) is the variation of the critical frequency and the fastest growing frequency with SS parameter. We see that the SS effect leads to the reduction of the gain and movement of the positions of the critical frequency and the fastest growing frequency to lower frequencies. As the SS parameter increases further to exceed a critical value, MI can’t appear. As stated before, SS effect can be engineered, thus the MI can be manipulated. These results illustrate not only the unusual nonlinear effects that can be seen in NIMs, but also the new ways of manipulating solitons.

## 4. Numerical simulations

*ω*

_{pm}/

*ω*

_{pe}= 0.8 in the case of focusing nonlinearity and anomalous dispersion. In this case, the SS parameter can vary from 0.867 to -1.5, with the zero SS point located at

*ω*/

*ω*

_{pe}= 0.632, as Fig. 1(b) shows. The initial field distribution is a cosinoidally modulated plane wave,

*U*

_{0}is the initial amplitude of background wave which is set to be 1 in the following simulations,

*u*

_{0}= 0.05 is the initial amplitude of modulation wave. In a laboratory experiment, noise in the system provides the seed from which instability frequencies begin to develop. However, for the numerical simulations, it is necessary to explicitly provide an initial seed at the frequency

*ϖ*to stimulate a response (induced MI). In the following simulations,

*ϖ*is chosen such that it is the fastest growing frequency in the case of

*S*= 0.

## 5. Conclusion

## Acknowledgements

## References and links

1. | G. P. Agrawal. |

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. |

3. | M. J. Potosek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. |

4. | A. Höök and M. Karlsson, “Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion,” Opt. Lett. |

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. |

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 |

07. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eand μ,” Sov. Phys. Usp. |

8. | J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

09. | 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. |

10. | R. A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of a negative refractive index of refraction,” Science |

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. |

12. | E. Schonbrun, M. Tinker, W. Park, and J. -B. Lee, “Negative refraction in a Si-polymer photonic Crystal membrane,” IEEE Photonics Technol. Lett. |

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. |

14. | A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. |

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 |

16. | M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E |

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 |

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

19. | I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E |

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. |

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. |

22. | T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. |

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

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 11, 2006

Revised Manuscript: February 11, 2006

Manuscript Accepted: February 13, 2006

Published: February 20, 2006

**Citation**

Shuangchun Wen, Yuanjiang Xiang, Wenhua Su, Yonghua Hu, Xiquan Fu, and Dianyuan Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express **14**, 1568-1575 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1568

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### References

- G. P. Agrawal. Nonlinear Fiber Optics, 3nd edn. (San Diego, Academic, 2001).
- V. A. Vysloukh, 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 Schr¨odinger equation," Opt. Lett. 12, 921 (1987). [CrossRef]
- A. H¨o¨ok, M. Karlsson, "Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion," Opt. Lett. 18, 1388 (1993). [CrossRef] [PubMed]
- F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, M. P. Sørensen, "Modulational instability in optical fibers near the zero dispersion point," Opt. Commun. 108, 60 (1994). [CrossRef]
- 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]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- 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 simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative refractive index of refraction," Science 292, 77 (2001). [CrossRef] [PubMed]
- 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]
- 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]
- V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356 (2005). [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]
- 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]
- 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]
- 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]
- 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 Schr¨odinger 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, A. M. Zheltikov, "Dispersion-free pulse propagation in a negative-index material," Opt. Lett. 30, 1998 (2005). [CrossRef] [PubMed]
- T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282 (1997). [CrossRef]
- M. Marklund, P. K. Shukla, and L. Stenflo, "Ultra-short solitons and kinetic effects in nonlinear metamaterials," arXiv: nlin./060162.

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