## Structural-configurated magnetic plasmon bands in connected ring chains

Optics Express, Vol. 17, Issue 14, pp. 11486-11494 (2009)

http://dx.doi.org/10.1364/OE.17.011486

Acrobat PDF (494 KB)

### Abstract

Magnetic resonance coupling between connected split ring resonators (SRRs) and magnetic plasmon (MP) excitations in the connected SRR chains were theoretically studied. By changing the connection configuration, two different coupling behaviors were observed, and therefore two kinds of MP bands were formed in the connected ring chains accordingly. From the extracted dispersion properties of MPs, forward and backward characteristics of the guided waves are well exhibited corresponding to the homo- and hetero-connected chains. Notably, thanks to the conductive coupling the revealed MP waves both have wide bandwidth even starting from the zero frequency. These results are suggested to provide instructions to build new kinds of subwavelength waveguides.

© 2009 Optical Society of America

## 1. Introduction

1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled resonator optical waveguides: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999).
[CrossRef]

2. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**, R16356–R16359 (2000).
[CrossRef]

5. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004).
[CrossRef]

6. W. N. Hardy and L. A. Whitehead, “Split-ring resonator for use in magnetic resonance from 200–2000 MHz,” Rev. SCI. Instr. **52**, 213–216 (1981).
[CrossRef]

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

8. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001).
[CrossRef] [PubMed]

12. V. M. Shalaev, “Optical negative-index metamaterial,” Nat. Photon. **1**, 41–48 (2007).
[CrossRef]

13. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. **92**, 6252–6261 (2002).
[CrossRef]

16. O. Sydoruk, A. Radkovskaya, O. Zhuromskyy, E. Shamonina, M. Shamonin, C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, “Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell,” Phys. Rev. B **73**, 224406 (2006).
[CrossRef]

17. A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. B **73**, 036609 (2006).
[CrossRef]

20. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Selective switch made from a graded anosandwich chain,” Appl. Phys. Lett. **93**, 233102 (2008).
[CrossRef]

18. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. **97**, 243902 (2006).
[CrossRef]

21. M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, “Electroinductive waves in chains of complementary metamaterial elements,” Appl. Phys. Lett. **88**, 083503 (2006).
[CrossRef]

27. N. Liu and H. Giessen, “Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling,” Opt. Express **16**, 21233–21238 (2008).
[CrossRef] [PubMed]

28. N. Liu, H. Liu, S. N. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. **3**, 157–162 (2009).
[CrossRef]

25. 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. Stat. Sol. (b) **244**, No. 4, 1170–1175 (2007).
[CrossRef]

26. N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules”, Adv. Mater. **20**, 1–5 (2008).
[CrossRef]

## 2. Resonant coupling between connected SRR pairs

*ω*

_{0}=(

*LC*)

^{-1/2}, where

*L*and

*C*are effective inductance and capacitance respectively. Fig. 1(a) shows the square SRR model with the parameters marked on, side length

*a*=400 nm, thickness

*t*=60 nm, width

*w*=100 nm, and slit gap

*s*=20 nm. Its electromagnetic (EM) resonance property is numerically evaluated using a commercial software package (CST Microwave Studio), with which the EM response with respect to frequencies and field distributions can be conveniently simulated. Here, the metal is defined as gold, whose permittivity is defined by the Drude model with

*ω*

_{p}=1.37×10

^{16}s

^{-1}and

*γ*=4.08×10

^{13}s

^{-1}[22

22. T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, “Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission,” Opt. Express **14**, 11155–11163 (2006).
[CrossRef] [PubMed]

*ω*

_{0}=77.1 THz. For the coupled cases, two split resonances are clearly observed (red curves) for both homo-connected and hetero-connected structures. Although both cases exhibit the split modes due to the coupling effect, the mode shifts are quite different. The split energy of hetero-connected case is apparently larger than the other one. From simulations, we get these coupled eigen frequencies as

*ω*

_{homo+}=77.6 THz,

*ω*

_{homo-}=70.7 THz,

*ω*

_{hetero+}=82.3 THz and

*ω*

_{hetero-}=67.0THz, where subscript “+” and “-” correspond to the higher and lower energy levels, respectively. To get a detailed recognition of these coupled modes, magnetic field (perpendicular to the paper) distributions at these frequencies are depicted out as insets in Fig. 2(a) and (b). As expected, these two modes exhibit symmetric and antisymmetric resonance features as well as the most coupling cases. But what interest us most is that these two coupled structures reverse their eigen modes between higher and lower ones. It is clearly demonstrated that

*ω*

_{+}corresponds to the antisymmetric mode and

*ω*

_{-}to the symmetric one for the homo-connected case, and vice versa for the hetero-connected case.

25. 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. Stat. Sol. (b) **244**, No. 4, 1170–1175 (2007).
[CrossRef]

*L*(

*Q*̇

^{2}

_{1}+

*Q*̇

^{2}

_{2})/2 and the electrostatic energy stored in the capacitors (

*Q*

^{2}

_{1}+

*Q*

^{2}

_{2})/2

*C*, the total energy of this system contains three coupling items. The one is the current exchanges [18

18. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. **97**, 243902 (2006).
[CrossRef]

26. N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules”, Adv. Mater. **20**, 1–5 (2008).
[CrossRef]

*ω*

_{0}=(

*LC*)

^{-1/2}. We can transfer the form of 1/

*C*as

*Lω*

^{2}

_{0}. So the Lagrangian of the coupled system is written as

*Q*=

_{i}*A*exp(

_{i}*iωt*) and the normalized coupling coefficients as

*κ*

_{e1}and

*κ*

_{e2}correspond to the electric coupling coefficients of the two different connection cases. Since we have already obtained the eigen frequencies of the two coupled cases from the simulations, coupling coefficients are easily to be calculated out as

*η*=0.072,

*κ*=0.007,

_{m}*κ*

_{e1}=-0.166 and

*κ*

_{e2}=0.131. These data, therefore, provides us a ruler to evaluate the coupling strength as we are stepping into the underlying physics of this coupled system. As we know that the energy of the dipole-dipole interaction (both electric and magnetic) has the form of

**d**(i=1, 2) represent either the magnetic dipoles (

_{i}**m**=

_{i}*Q*̇

*) or electric dipoles (*

_{i}^{S}**p**=

_{i}*Q*). Comparing the equivalent coupled LC-circuits and effective dipoles configurations of split modes (see Fig. 3), we can draw the conclusion that the coefficients of magnetic coupling of the two cases are both positive, while that of electric coupling should be negative for homo-connect SRR pair and positive for the hetero-connected one. Regarding the different distance and relative orientations of the electric dipoles (

_{i}^{l}**p**,

_{1}**p**), the absolute value of

_{2}*κ*

_{e1}should be evidently larger than

*κ*

_{e2}, which agrees well with retrieved results through the former equations.

*κ*) is the strongest one while the magneto induction (

_{e}*κ*) is much weaker. Thus, alternating the position of the slit changes the contribution of electroinductive coupling and therefore reverse split eigen modes, which appropriately explains the simulation results. Furthermore, it tells us that although the system manifests a feature of magnetic resonance, the major coupling components are from the electric dipole-dipole interaction and current exchange. Here, the current exchange contributing the conductive coupling is accommodated in the community inductance, which is expected playing an important role to build a wide MP band as we extend these SRR pairs to the SRR chains exhibited in the next section.

_{m}## 3. MP modes in the chains of connected SRRs

13. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. **92**, 6252–6261 (2002).
[CrossRef]

20. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Selective switch made from a graded anosandwich chain,” Appl. Phys. Lett. **93**, 233102 (2008).
[CrossRef]

19. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express **16**, 3560–3565 (2008).
[CrossRef] [PubMed]

*ω*-

*k*space can be obtained, which may give a clear picture of the dispersion property of the guided wave via the coupled resonances.

*ω*-

*k*space. Firstly for the homo-connected SRR chain shown in Fig 4(a) and (c), two symmetric dispersion curves both starting from zero frequency (although very weak) to an upper limit are clearly observed in the ±

*k*bands. Except for a tiny horizontal line around this upper limit frequency, all the curved dispersion lie below the free-space light line shown as the white line in the Fig. 4(c). One remarkable feature should be noted is that the MP band has very wide bandwidth, which cannot be achieved in the chain system with isolated elements via induction couplings [3

3. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, Ari A. G. Requicha, and H. A. Atwater, “Plasmonics - a route to nanoscale optical devices,” Adv. Mater. **13**, 1501 (2001).
[CrossRef]

5. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004).
[CrossRef]

13. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. **92**, 6252–6261 (2002).
[CrossRef]

17. A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. B **73**, 036609 (2006).
[CrossRef]

19. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express **16**, 3560–3565 (2008).
[CrossRef] [PubMed]

20. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Selective switch made from a graded anosandwich chain,” Appl. Phys. Lett. **93**, 233102 (2008).
[CrossRef]

18. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. **97**, 243902 (2006).
[CrossRef]

*v*g=

*∂ω*/

*∂k*) and very large density of state (DOS). This upper limit frequency, characterized as

*ω*

_{mp}, can undoubtedly be modulated by the structural parameters of SRR chain.

**97**, 243902 (2006).
[CrossRef]

*k*regions. Fig. 5 depicts the magnetic field distributions along the CROW chain for several specific frequencies for the homo-connected case. The coincidence between the field intensity of propagating MP wave and the color map of the dispersion curve is evidently exhibited. As for the tiny horizontal line at upper limit frequency, it is likely come from a strong localized mode that the field concentrates at the beginning of the chain, as shown in Fig. 5 (84 THz case). This localized mode may support variety

*k*vectors resulting in such a flat dispersion branch, though it is very weak. It also should be mentioned that the dispersion map are extracted directly from the field distributions for a specific excitation case that the source is placed at the left side. Therefore, the dispersion curve with positive

*k*is well anticipated as has been displayed in Fig. 4(c). However, the negative propagation MP wave with negative

*k*is also presented symmetrically to the positive branch. After a little analysis, we may be aware of that it is actually a reflection wave that comes from the propagating wave reflected by every SRR units, and it is reasonably weaker than the right-propagation wave.

*k*range, we noticed that the energy still flows from left to right due to the much stronger dispersion in the -

*k*region with positive group velocity (

*∂ω*/

*∂k*)>0. As well as the transverse modes demonstrated in previous papers [2

2. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**, R16356–R16359 (2000).
[CrossRef]

5. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004).
[CrossRef]

*ω*/

*k*)(

*∂ω*/

*∂k*)<0. As for the present system, the most significance of the work is that such a backward wave can be accommodated in such a wide MP band that even extending to zero frequency. Although in two dimensional systems (e.g. photonic crystal) abnormal dispersion in the band structure is commonly seen, it was seldom reported in the one-dimensional system. In addition, a cross point of the MP wave with free-space light line is clearly observed in positive

*k*region of the dispersion map [

*k*~0.23 (π/a) and

*ω*~84 THz], which can be called as “Magnetic Plasmon Polariton-MPP”. This MPP mode is regarded as a phase matching point as well. Since the light radiation from the dipole source only propagates from left to right [see the sketch Fig. 4(a) and (b)] with a positive

*k*, there is no MPP appear in the negative

*k*region.

**97**, 243902 (2006).
[CrossRef]

## 4. Conclusions

## Acknowledgments

## References and Links

1. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled resonator optical waveguides: a proposal and analysis,” Opt. Lett. |

2. | M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B |

3. | S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, Ari A. G. Requicha, and H. A. Atwater, “Plasmonics - a route to nanoscale optical devices,” Adv. Mater. |

4. | S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. |

5. | W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

6. | W. N. Hardy and L. A. Whitehead, “Split-ring resonator for use in magnetic resonance from 200–2000 MHz,” Rev. SCI. Instr. |

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

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

9. | T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science |

10. | S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science |

11. | C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science |

12. | V. M. Shalaev, “Optical negative-index metamaterial,” Nat. Photon. |

13. | E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. |

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

15. | O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. |

16. | O. Sydoruk, A. Radkovskaya, O. Zhuromskyy, E. Shamonina, M. Shamonin, C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, “Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell,” Phys. Rev. B |

17. | A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. B |

18. | H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. |

19. | S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express |

20. | S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Selective switch made from a graded anosandwich chain,” Appl. Phys. Lett. |

21. | M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, “Electroinductive waves in chains of complementary metamaterial elements,” Appl. Phys. Lett. |

22. | T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, “Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission,” Opt. Express |

23. | T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, “Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures”, Appl. Phys. Lett. |

24. | T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, “Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials,” J. Appl. Phys. |

25. | 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. Stat. Sol. (b) |

26. | N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules”, Adv. Mater. |

27. | N. Liu and H. Giessen, “Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling,” Opt. Express |

28. | N. Liu, H. Liu, S. N. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photon. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

(260.2030) Physical optics : Dispersion

(260.5740) Physical optics : Resonance

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 16, 2009

Revised Manuscript: May 6, 2009

Manuscript Accepted: May 31, 2009

Published: June 24, 2009

**Citation**

T. Li, R. X. Ye, C. Li, H. Liu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, "Structural-configurated magnetic plasmon bands in connected ring chains," Opt. Express **17**, 11486-11494 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11486

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### References

- A. Yariv, Y. Xu. R. K. Lee, and A. Scherer, "Coupled resonator optical waveguides: a proposal and analysis," Opt. Lett. 24,711-713 (1999). [CrossRef]
- M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, R16356-R16359 (2000). [CrossRef]
- S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001). [CrossRef]
- S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003). [CrossRef] [PubMed]
- W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70,125429 (2004). [CrossRef]
- W. N. Hardy and L. A. Whitehead, "Split-ring resonator for use in magnetic resonance from 200-2000 MHz," Rev. SCI. Instr. 52,213-216 (1981). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004). [CrossRef] [PubMed]
- S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004). [CrossRef] [PubMed]
- C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007). [CrossRef] [PubMed]
- V. M. Shalaev, "Optical negative-index metamaterial," Nat. Photon. 1, 41-48 (2007). [CrossRef]
- E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002). [CrossRef]
- E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002). [CrossRef]
- O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005). [CrossRef]
- O. Sydoruk, A. Radkovskaya O. Zhuromskyy, E. Shamonina, M. Shamonin,C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, "Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell," Phys. Rev. B 73,224406 (2006). [CrossRef]
- A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic plasmon resonance," Phys. Rev. B 73, 036609 (2006). [CrossRef]
- H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006). [CrossRef]
- S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008). [CrossRef] [PubMed]
- S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008). [CrossRef]
- M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006). [CrossRef]
- T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (2006). [CrossRef] [PubMed]
- T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007). [CrossRef]
- T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008). [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. Stat. Sol. (B) 244 (4), 1170-1175 (2007). [CrossRef]
- N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008). [CrossRef]
- N. Liu and H. Giessen, "Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling," Opt. Express 16, 21233-21238 (2008). [CrossRef] [PubMed]
- N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009). [CrossRef]

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