## Discrete dissipative localized modes in nonlinear magnetic metamaterials |

Optics Express, Vol. 19, Issue 27, pp. 26500-26506 (2011)

http://dx.doi.org/10.1364/OE.19.026500

Acrobat PDF (1049 KB)

### Abstract

We analyze the existence, stability, and propagation of dissipative discrete localized modes in one- and two-dimensional nonlinear lattices composed of weakly coupled split-ring resonators (SRRs) excited by an external electromagnetic field. We employ the near-field interaction approach for describing quasi-static electric and magnetic interaction between the resonators, and demonstrate the crucial importance of the electric coupling, which can completely reverse the sign of the overall interaction between the resonators. We derive the effective nonlinear model and analyze the properties of nonlinear localized modes excited in one-and two-dimensional lattices. In particular, we study nonlinear magnetic domain walls (the so-called switching waves) separating two different states of nonlinear magnetization, and reveal the bistable dependence of the domain wall velocity on the external field. Then, we study two-dimensional localized modes in nonlinear lattices of SRRs and demonstrate that larger domains may experience modulational instability and splitting.

© 2011 OSA

## 1. Introduction

2. D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B **82**, 155128 (2010). [CrossRef]

3. M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B **28**, 263–269 (2002). [CrossRef]

4. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B **84**, 045424 (2011). [CrossRef]

5. C. Denz, S. Flach, and Yu. S. Kivshar, eds. *Nonlinearities in Periodic Structures and Metamaterials* (Springer-Verlag, Heidelberg, 2009). [CrossRef]

6. I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. **4**, 69–74 (2006). [CrossRef]

7. J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. **44**, 2099–2109 (1996). [CrossRef]

9. M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. **88**, 083503 (2006). [CrossRef]

6. I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. **4**, 69–74 (2006). [CrossRef]

6. I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. **4**, 69–74 (2006). [CrossRef]

10. N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. **97**, 157406 (2006). [CrossRef] [PubMed]

14. W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E **81**, 016604 (2010). [CrossRef]

2. D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B **82**, 155128 (2010). [CrossRef]

15. D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B **83**, 235420 (2011). [CrossRef]

*switching waves*) separating the regions of different values of the magnetization, and study the motion of such domain walls under the action of an external field, revealing the hysteresis in the dependence of the velocity on the applied field. We also analyze the existence and stability of nonlinear localized modes in two-dimensional magnetic lattices of SRRs including the decay of magnetic domains via modulational instability.

## 2. Near-field interaction between split-ring resonators

*I*in a split-ring resonator with indices (n,q,m), corresponding to the axes (x,y,z), has the form [2

_{n,q,m}2. D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B **82**, 155128 (2010). [CrossRef]

*L,R,C*are inductance, resistance and capacitance of the resonator,

*Q*is the charge in the resonator,

_{n,q,m}**j**describes indices of the nearest neighboring resonators,

*M*and

*P*are electric and magnetic interaction coefficients, respectively, ℰ

*is an external electromotive force.*

_{n,q,m}**4**, 69–74 (2006). [CrossRef]

*U*across the gap of the resonator:

*C*=

_{NL}*C*

_{0}+ Δ

*C*(|

_{NL}*U*)|

_{n,q,m}^{2}), where the nonlinear correction to the capacitance is small. Using the slowly varying approximation [6

**4**, 69–74 (2006). [CrossRef]

*τ*=

*ω*

_{0}

*t*,

*ω*–

*ω*

_{0})/

*ω*

_{0}, and Ψ

*= ℐ*

_{n,q,m}*/ℐ*

_{n,q,m}*, where ℐ*

_{c}*=*

_{c}*ω*

_{0}

*C*

_{0}

*U*is the characteristic nonlinear current,

_{c}*U*=

_{c}*E*×

_{c}*d*is the characteristic nonlinear voltage,

_{g}*γ*=

*R/Lω*

_{0}is the damping coefficient, and

*is the amplitude of the harmonic current*

_{n,q,m}*I*. The effective interaction coefficients

_{n,q,m}*K*

**=**

_{j}*κ*

_{j,H}

*ω*/

*ω*

_{0}–

*κ*

_{j,E}

*ω*

_{0}/

*ω*, where normalized electric and magnetic interaction coefficients are

*κ*

_{j,H}=

*M/L*and

**82**, 155128 (2010). [CrossRef]

7. J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. **44**, 2099–2109 (1996). [CrossRef]

16. F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B **244**, 1170–1175 (2007). [CrossRef]

**82**, 155128 (2010). [CrossRef]

*r*

_{0}=2.25 mm, track width of 0.5 mm, metal thickness of 0.03 mm, gap width of 1 mm. For such resonators, the interaction coefficients were calculated using our approach presented in Ref. [2

**82**, 155128 (2010). [CrossRef]

17. M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. **12**, 083062 (2010). [CrossRef]

*quantitatively*describe near-field interaction of the resonators. Importantly, the effective interaction coefficient

*K*

**in Eq. (2) depends on the difference between electric and magnetic coefficients, which for experimentally realistic parameters may become comparable. In contrast to the previous works, which assumed only approximate magnetic interaction between the resonators, when the interaction of in-plane resonators is identical in x- and y-directions, in our case we see from the Figs. 1 (a,b) that the interaction is strongly different in x- and y-direction. Remarkably, the corresponding effective interaction coefficients**

_{j}*K*can have different signs.

## 3. Discrete dissipative localized modes

### 3.1. Domain walls in SRR arrays

18. N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. **93**, 743–746 (2011). [CrossRef]

*c/r*

_{0}= 2.4. From Fig. 1 we find that the corresponding effective interaction coefficient

*K*= −0.02. In the bistable regime, when two stable uniform solutions exist in the chain, we observe the possibility of formation of switching waves, which represent a transition from one uniform distribution to another with a change in resonator number n. Such waves are initially excited by non-uniform distribution of the external field, and then they are supported by a uniform external excitation. The profiles of such waves are close to a step function, with typical domain wall structure shown in Fig. 3(a).

_{z}*v*is obtained by averaging over sufficient time. The velocity sign is defined positive if the motion of the switching wave leads to the expansion of the region occupied by the upper branch of bistability; otherwise, the velocity is negative. Figure 3(b,c) shows the velocity of the switching waves as a function of the external excitation. We find that for the selected parameters there exist a narrow range of external excitations

*S*for which we observe bistable behavior, see Fig. 3(c). Interestingly, one of the branches in bistable regime corresponds to the stationary domain wall, while another branch - to the one moving in negative direction.

### 3.2. Dissipative solitons in two-dimensional lattices

*a/r*

_{0}= 2.4,

*c/r*

_{0}= 1. From the results presented in Fig. 1 we find the effective interaction coefficients for this regime as

*K*= 0.06,

_{x}*K*= −0.02, and use these values in our further simulations of Eqs. (2). Our two-dimensional structure supports a family of nonlinear localized solutions – discrete dissipative solitons. In bistable regime, when the interaction between resonators is sufficiently weak, such solutions of Eqs. 2 can be found using perturbation theory, with the smallness parameter defined by the interaction coefficients. However for realistic interaction constants, shown in Fig. 1, the existence and stability of dissipative solitons can be found only numerically. Fig. 4 shows two possible stable soliton solutions, consisting of just one strongly excited resonator (Fig. 4(a)) or five excited resonators (Fig. 4(b)). For such narrow solitons, as well as for narrow solitons in continuous media [19], stability of the upper branches of the bistability curves is not generally required for the stability of the solitons, since the modulational instability is calculated for the homogeneously excited structure. However, for wider excitations, the modulational instability starts to manifest itself, with the larger structures showing instability and splitting into several smaller solitons. For example, if we initially excite the structure with wide area corresponding to the upper branch of the bistability curve (see Fig. 5(a)), then the excitation becomes unstable in z-direction, and we observe formation of two stable narrow solitons, which are shown in Fig. 5(b). Demonstrated decay of the wide soliton occurs within 100 dimensionless time units

_{z}*τ*.

## 4. Conclusions

20. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. **99**, 153901 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | N. Engheta and R. W. Ziolkowski, eds. |

2. | D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B |

3. | M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B |

4. | A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B |

5. | C. Denz, S. Flach, and Yu. S. Kivshar, eds. |

6. | I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. |

7. | J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. |

8. | E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. |

9. | M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. |

10. | N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. |

11. | M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E |

12. | N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E |

13. | M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E |

14. | W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E |

15. | D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B |

16. | F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B |

17. | M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. |

18. | N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. |

19. | N. N. Rosanov, |

20. | Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

**ToC Category:**

Metamaterials

**History**

Original Manuscript: November 9, 2011

Revised Manuscript: November 30, 2011

Manuscript Accepted: November 30, 2011

Published: December 13, 2011

**Citation**

Nikolay N. Rosanov, Nina V. Vysotina, Anatoly N. Shatsev, Ilya V. Shadrivov, David A. Powell, and Yuri S. Kivshar, "Discrete dissipative localized modes in nonlinear magnetic metamaterials," Opt. Express **19**, 26500-26506 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26500

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### References

- N. Engheta and R. W. Ziolkowski, eds. Electromagnetic Metamaterials: Physics and Engineering Explorations (Wiley-IEEE Press, 2006), 440 pp.
- D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B82, 155128 (2010). [CrossRef]
- M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B28, 263–269 (2002). [CrossRef]
- A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011). [CrossRef]
- C. Denz, S. Flach, and Yu. S. Kivshar, eds. Nonlinearities in Periodic Structures and Metamaterials (Springer-Verlag, Heidelberg, 2009). [CrossRef]
- I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl.4, 69–74 (2006). [CrossRef]
- J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech.44, 2099–2109 (1996). [CrossRef]
- E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett.38, 371–373 (2002). [CrossRef]
- M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett.88, 083503 (2006). [CrossRef]
- N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett.97, 157406 (2006). [CrossRef] [PubMed]
- M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E77, 036608 (2008). [CrossRef]
- N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E77, 065601(R) (2008). [CrossRef]
- M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E80, 046605 (2009). [CrossRef]
- W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E81, 016604 (2010). [CrossRef]
- D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83, 235420 (2011). [CrossRef]
- F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B244, 1170–1175 (2007). [CrossRef]
- M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys.12, 083062 (2010). [CrossRef]
- N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett.93, 743–746 (2011). [CrossRef]
- N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, Berlin, 2002).
- Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett.99, 153901 (2007). [CrossRef] [PubMed]

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