## Nonlinear properties of split-ring resonators

Optics Express, Vol. 16, Issue 20, pp. 16058-16063 (2008)

http://dx.doi.org/10.1364/OE.16.016058

Acrobat PDF (359 KB)

### Abstract

In this letter, the properties of split-ring resonators (SRRs) loaded with high-Q capacitors and nonlinear varactors are theoretically analyzed and experimentally measured. We demonstrate that the resonance frequency *f _{m}
* of the nonlinear SRRs can be tuned by increasing the incident power.

*f*moves to lower and higher frequencies for the SRR loaded with one varactor and two back-to-back varactors, respectively. For high incident powers, we observe bistable tunable metamaterials and hysteresis effects. Moreover, the coupling between two nonlinear SRRs is also discussed.

_{m}© 2008 Optical Society of America

## 1. Introduction

1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science **305**, 788–792 (2004) [CrossRef] [PubMed]

2. C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science **315**, 47–49 (2007) [CrossRef] [PubMed]

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave. Theory Technol. **47**, 2075–2084 (1999) [CrossRef]

4. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and *µ*,” Sov. Phys. Usp. **10**, 509–514 (1968) [CrossRef]

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

7. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**, 77–79 (2002) [CrossRef]

8. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-handed Materials,” Phys. Rev. Lett. **91**, 037401–4 (2003) [CrossRef] [PubMed]

12. D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. **92**, 264104 (2008) [CrossRef]

14. A. Degiron, J. J. Mock, and D. R. Smith, “Modulating and tuning the response of metamaterials at the unit cell level,” Opt. Express **15**, 1115–1127 (2007) [CrossRef] [PubMed]

15. L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonators and ferrite rods,” Opt. Express **16**, 8825–8834 (2008) [CrossRef] [PubMed]

9. I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express **14**, 9344–9349 (2006) [CrossRef] [PubMed]

11. I. Gil, J. Bonache, J. Garcia-Garcia, and F. Martin, “Tunable metamaterial transmission lines based on varactor loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech **54**, 2665–2674 (2007) [CrossRef]

16. D. Wang, L. Ran, H. Chen, M. Mu, J. A. Kong, and B. Wu, “Active left-handed material collaborated with microwave varactors,” Appl. Phys. Lett. **91**, 164101 (2007) [CrossRef]

10. D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, and M. V. Gorkunov, “Self-tuning mechanisms of nonlinear split-ring resonators,” Appl. Phys. Lett. **91**, 144107 (2007) [CrossRef]

17. J. Carbonell, V. E. Boria, and D. Lippens, “Nonlinear effects in split-ring resonators loaded with heterostructure barrier varactors,” Microwave Opt. Technol. Lett. **50**, 474–479 (2008) [CrossRef]

11. I. Gil, J. Bonache, J. Garcia-Garcia, and F. Martin, “Tunable metamaterial transmission lines based on varactor loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech **54**, 2665–2674 (2007) [CrossRef]

12. D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. **92**, 264104 (2008) [CrossRef]

16. D. Wang, L. Ran, H. Chen, M. Mu, J. A. Kong, and B. Wu, “Active left-handed material collaborated with microwave varactors,” Appl. Phys. Lett. **91**, 164101 (2007) [CrossRef]

10. D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, and M. V. Gorkunov, “Self-tuning mechanisms of nonlinear split-ring resonators,” Appl. Phys. Lett. **91**, 144107 (2007) [CrossRef]

12. D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. **92**, 264104 (2008) [CrossRef]

## 2. Nonlinear properties of a single nonlinear SRR

18. K. Aydin and E. Ozbay, “Capacitor-loaded split ring resonators as tunable metamaterial components,” Journal of Applied Physics, **101**, 024911 (2007) [CrossRef]

*C*(

*V*

*) (*

_{D}*V*is the voltage across the diode) is described as the following, provided by the manufacturer SPICE model.

_{D}*C*(

*V*)=

_{D}*C*

_{0}(1-

*V*/

_{D}*V*)

_{p}_{-M}, where

*C*

_{0}=2.2 pF is the DC rest capacitance,

*V*=1.5 V is the intrinsic potential and

_{p}*M*=0.8. The dissipative current is given by

*V*<

_{D}*V*, then the voltage across the diode can be expressed by the charge,

_{p}*q*=Q

_{D}/

*C*

_{0}.

*I*is the current in the resonator,

*L*is the inductance of the resonator determined by the ring geometry,

*R*

*is the resistance, and*

_{S}*ε*is the driven term provided by the loop antenna in the experiment. For small excitations,

*I*

*can be neglected so the current can be estimated by*

_{D}*I*≈

*dQ*/

_{D}*dt*. The equation of motion is now

*ω*

^{2}

_{0}=1/(

*LC*

_{0}) and

*γ*=

*ω*

^{2}

_{0}

*R*

_{S}*C*

_{0}. Expand the restoring term

*V*by the Taylor series for small oscillations (the oscillation amplitude satisfies (1-

_{D}*M*)|

*q*|<

*V*) and omit the higher order terms,

_{p}*ε*(

*t*)=

*f*

*cos*(

*ωt*), where

*f*is the excitation amplitude and

*ω*is the excitation frequency, the equation of motion is further estimated by

*ω*=

*ω*

_{0}+

*δ*. When

*δ*is small, the driven frequency is close to the resonance frequency. Without the

*q*

^{2}and

*q*

^{3}term, the oscillator is linear and the amplitude of oscillation,

*b*, is given by

*b*

^{2}(

*δ*

^{2}+

*γ*

^{2}/4)=

*ω*

^{2}

_{0}

*f*

^{2}/4. The nonlinear

*q*

^{2}and

*q*

^{3}terms make the eigen-frequency amplitude dependent,

*ω*

_{0}→

*ω*

_{0}+

*κb*

^{2}, where

*δ*→

*δ*-

*κb*

^{2}, and the oscillation amplitude satisfies the equation

*q*

^{2}and the real roots give the amplitude of oscillations. When the external excitation is small, the oscillation amplitude is also small and the higher orders of

*b*may be neglected and the oscillation can be considered to be linear. When the excitation power is larger, the curve is distorted and the resonance shifts to a lower frequency, since, in our case,

*κ*is negative. When the excitation power is large enough, there are three real roots of

*b*

^{2}and the curve is folded over, see Fig. 1(b). The branch in the middle is unstable and the oscillation tends to go to the other two branches. In experiment, the oscillation follows the lower branch until it jumps to the higher branch for forward sweep, and follows the higher branch until it jumps to the lower branch. So, the hysteresis effect is observed in our experimental measurements. Note, when the voltage on the varactor is larger than 0.5 V, the nonlinear DC dissipative current

*V*=

_{T}*k*/

_{B}T*e*is the thermal voltage,

*k*is the Boltzmann constant,

_{B}*T*is the temperature, and

*e*is the electron charge) sets in and this increases the loss on the SRR and the reflection dip measured is not as strong as the small oscillation case. See Fig. 1(a), the reflection minimum increases from around -40 dB to -3 dB when the input power is increased from -15 dBm to 9 dBm. This is not covered in the simplified model of the nonlinear oscillator model.

*C*(

*V*) of the two varactors is now symmetric. This configuration has the same effect of one heterostructure barrier varator (HBV) diode [17

17. J. Carbonell, V. E. Boria, and D. Lippens, “Nonlinear effects in split-ring resonators loaded with heterostructure barrier varactors,” Microwave Opt. Technol. Lett. **50**, 474–479 (2008) [CrossRef]

## 3. Mutual coupling between two nonlinear SRRs

## 4. Conclusion

## Acknowledgments

## References and links

1. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

2. | C. M. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science |

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave. Theory Technol. |

4. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and |

5. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

6. | P. Markos and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. |

7. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

8. | A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-handed Materials,” Phys. Rev. Lett. |

9. | I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express |

10. | D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, and M. V. Gorkunov, “Self-tuning mechanisms of nonlinear split-ring resonators,” Appl. Phys. Lett. |

11. | I. Gil, J. Bonache, J. Garcia-Garcia, and F. Martin, “Tunable metamaterial transmission lines based on varactor loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech |

12. | D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. |

13. | E. Ozbay, K. Aydin, S. Butun, K. Kolodziejak, and D. Pawlak, “Ferroelectric based tuneable SRR based metamaterial for microwave applications,” in |

14. | A. Degiron, J. J. Mock, and D. R. Smith, “Modulating and tuning the response of metamaterials at the unit cell level,” Opt. Express |

15. | L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonators and ferrite rods,” Opt. Express |

16. | D. Wang, L. Ran, H. Chen, M. Mu, J. A. Kong, and B. Wu, “Active left-handed material collaborated with microwave varactors,” Appl. Phys. Lett. |

17. | J. Carbonell, V. E. Boria, and D. Lippens, “Nonlinear effects in split-ring resonators loaded with heterostructure barrier varactors,” Microwave Opt. Technol. Lett. |

18. | K. Aydin and E. Ozbay, “Capacitor-loaded split ring resonators as tunable metamaterial components,” Journal of Applied Physics, |

19. | L. D. Landau and E. M. Lifshitz, |

**OCIS Codes**

(350.4010) Other areas of optics : Microwaves

**ToC Category:**

Metamaterials

**History**

Original Manuscript: September 4, 2008

Revised Manuscript: September 19, 2008

Manuscript Accepted: September 19, 2008

Published: September 24, 2008

**Citation**

Bingnan Wang, Jiangfeng Zhou, Thomas Koschny, and Costas M. Soukoulis, "Nonlinear properties of split-ring resonators," Opt. Express **16**, 16058-16063 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16058

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### References

- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- C. M. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-49 (2007). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave.Theory Technol. 47, 2075-2084 (1999). [CrossRef]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968) [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622-8 (2002).
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2002). [CrossRef]
- A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-handed Materials," Phys. Rev. Lett. 91, 037401-4 (2003). [CrossRef] [PubMed]
- I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, "Tunable split-ring resonators for nonlinear negative-index metamaterials," Opt. Express 14, 9344-9349 (2006). [CrossRef] [PubMed]
- D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, and M. V. Gorkunov, "Self-tuning mechanisms of nonlinear split-ring resonators," Appl. Phys. Lett. 91, 144107 (2007). [CrossRef]
- I. Gil, J. Bonache, J. Garcia-Garcia, and F. Martin, "Tunable metamaterial transmission lines based on varactor loaded split-ring resonators," IEEE Trans. Microwave Theory Tech 54, 2665-2674 (2007). [CrossRef]
- D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104 (2008). [CrossRef]
- E. Ozbay, K. Aydin, S. Butun, K. Kolodziejak, and D. Pawlak, "Ferroelectric based tuneable SRR based metamaterial for microwave applications," in Proceedings of the 37th European Microwave Conference, 497-9 (2007).
- A. Degiron, J. J. Mock, and D. R. Smith, "Modulating and tuning the response of metamaterials at the unit cell level," Opt. Express 15, 1115-1127 (2007). [CrossRef] [PubMed]
- L. Kang, Q. Zhao, H. Zhao, and J. Zhou, "Magnetically tunable negative permeability metamaterial composed by split ring resonators and ferrite rods," Opt. Express 16, 8825-8834 (2008). [CrossRef] [PubMed]
- D. Wang, L. Ran, H. Chen, M. Mu, J. A. Kong, and B. Wu, "Active left-handed material collaborated with microwave varactors," Appl. Phys. Lett. 91, 164101 (2007). [CrossRef]
- J. Carbonell, V. E. Boria, and D. Lippens, "Nonlinear effects in split-ring resonators loaded with heterostructure barrier varactors," Microwave Opt. Technol. Lett. 50, 474-479 (2008). [CrossRef]
- K. Aydin and E. Ozbay, "Capacitor-loaded split ring resonators as tunable metamaterial components," J. Appl. Phys. 101, 024911 (2007). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Mechanics, 3rd Edition (Course of Theoretical Physics Vol. 1).

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