## Coupling strength of complex plasmonic structures in the multiple dipole approximation |

Optics Express, Vol. 19, Issue 22, pp. 22156-22166 (2011)

http://dx.doi.org/10.1364/OE.19.022156

Acrobat PDF (1676 KB)

### Abstract

We present a simple model to calculate the spatial dependence of the interaction strength between two plasmonic objects. Our approach is based on a multiple dipole approximation and utilizes the current distributions at the resonances in single objects. To obtain the interaction strength, we compute the potential energy of discrete weighted dipoles associated with the current distributions of the plasmonic modes in the scattered fields of their mutual partners. We investigate in detail coupled stacked plasmonic wires, stereometamaterials and plasmon-induced transparency materials. Our calculation scheme includes retardation and can be carried out in seconds on a standard PC.

© 2011 OSA

## 1. Introduction

1. C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. **7**, 318–322 (2007). [CrossRef] [PubMed]

4. S.-C. Yang, H. Kobori, C.-L. He, M.-H. Lin, H.-Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett. **10**, 632–637 (2010). [CrossRef] [PubMed]

5. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science **306**, 1351–1353 (2004). [CrossRef] [PubMed]

6. N. Liu and H. Giessen “Coupling effects in optical metamaterials,” Angew. Chem., Int. Ed. **49**, 9838–9852 (2010). [CrossRef]

8. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics **1**, 41–48 (2007). [CrossRef]

9. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. **7**, 31–37 (2008). [CrossRef]

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

12. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**, 758–762 (2009). [CrossRef] [PubMed]

13. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. **101**, 047401 (2008). [CrossRef] [PubMed]

14. N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. **9**, 1663–1667 (2009). [CrossRef] [PubMed]

15. M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. **10**, 2721–2726 (2010). [CrossRef] [PubMed]

17. J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tunability,” Nano Lett. **10**, 3184–3189 (2010). [CrossRef] [PubMed]

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

18. H. Liu, J. X. Cao, and S. N. Zhu, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B **81**, 241403 (2010). [CrossRef]

19. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. **9**, 707–715 (2010). [CrossRef]

20. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. **10**, 2342–2348 (2010). [CrossRef] [PubMed]

21. E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. **120**, 5444–5454 (2004). [CrossRef] [PubMed]

22. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science **302**, 419–422 (2003). [CrossRef] [PubMed]

23. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. **4**, 899–903 (2004). [CrossRef]

24. T. J. Davis, D. E. Gomez, and K. C. Vernon, “Simple model for the hybridization of surface plasmon resonances in metallic nanoparticles,” Nano Lett. **10**, 2618–2625 (2010). [CrossRef] [PubMed]

25. I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magneto-electric point scattering theory for metamaterial scatterers,” Phys. Rev. B **83**, 245102 (2011). [CrossRef]

### 1.1. Modes in plasmonic systems

26. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express **14**, 8827–8836 (2006). [CrossRef] [PubMed]

27. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. **10**, 3596–3603 (2010). [CrossRef] [PubMed]

28. J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. **9**, 2372–2377 (2009). [CrossRef] [PubMed]

*A*of a standing wave (Fig.1, right side) [29

_{i}29. T. Meyrath, T. Zentgraf, and H. Giessen, “Lorentz model for metamaterials: optical frequency resonance circuits,” Phys. Rev. B **75**, 205102 (2007). [CrossRef]

*p⃗*with the radiation pattern expressed in Eq. (1): Summing up the electric fields of all discrete dipoles of the object (Eq. (2)) gives us a good approximation for the scattered field of the nanoparticle at the resonance: In the static case, the work of a current in an external field is given by

*U*= –

*E⃗*·

*j⃗*. For the time-harmonic case the cycle-averaged potential energy

*U*for a point-current dipole

*p⃗*

_{2}in an external field

*E⃗*can be written as in Eq. (3) For two extended objects, we also discretize the current distribution in the second object into a set of oscillating dipole moments

*p⃗*

_{2,}

*and then calculate the scalar product of the scattered field of object one,*

_{j}*E⃗*

_{scat,1}, with the discretized currents of object two (Eq. (4)): The calculated potential energy

*U*is then proportional to the coupling strength of the system at one specific frequency

*ω*. Due to the efficient calculation of

*U*, the number of dipoles can easily be increased until convergence is reached.

### 1.2. Relative phase between two coupled particle plasmons

*φ*of

*E⃗*

_{scat,1}and

*p⃗*

_{2}the complex scalar

*φ*, Eq. (4) yields an extremal interaction energy

*U*.

## 2. Structures

### 2.1. Perpendicular cut wire pair

*n*= 1). One wire is aligned along the x axis and the other is aligned along the y-axis and hence perpendicular to the first one (see Fig. 2a).

*d*= 300 nm the splitting becomes smaller than half the linewidth of the resonance and hence could not be extracted anymore from the spectrum.

_{z}26. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express **14**, 8827–8836 (2006). [CrossRef] [PubMed]

*N*dipoles as described above and calculated the interaction energy at the resonance at

*ν*= 200 THz using our model. The coupling results for

*N*= 1,3,5,9 and 15 discrete dipoles with their amplitude set to the appropriate value which we obtained from the respective fundamental plasmon mode are plotted in Fig. 3a as solid line. Already for a discretization level of

*N*= 9 dipoles (blue curve) the agreement of our model with the full Maxwell simulations is excellent. Figure 3b shows the phase shift between the modes in the top and bottom wires, which is more or less flat at

*ϕ*≈ 0 degrees for these wire separations.

### 2.2. Dipole-quadrupole structure

*n*= 1.55). Two of them are in the lower plane and have a length of 315 nm. They are laterally separated by a distance (center to center) of 300 nm. Above them is a single gold nanowire of length 355 nm which is perpendicular to the bottom wires. All three wires have a width of 80 nm and height of 40 nm, the stacking distance in z-direction is 70 nm. The system resembles a plasmonic analog of electromagnetically induced transparency (EIT) [12

12. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**, 758–762 (2009). [CrossRef] [PubMed]

*λ*= 1.66

_{vac}*μ*m that is polarized in the x-direction (the direction of the long axis of the top nanowire), the dipole is excited. By near-field coupling, the dark quadrupolar mode of the lower wires (which is not excited by the incoming light) is excited. A peculiarity of this EIT system is the fact that the excitation of the quadrupole takes only place if the dipole is laterally shifted away from the symmetry axis of the quadrupole by a shift

*s*(see Fig. 5a). The

*experimental*transmittance spectra from reference [12

12. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**, 758–762 (2009). [CrossRef] [PubMed]

*s*are shown in Fig. 5c. In order to extract the coupling strength between the dipole and the quadrupole in dependence on this parameter

*s*, a fit of the experimental spectra to a classical EIT model [31

31. C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. **70**, 37 (2002). [CrossRef]

*s*, we discretized the currents in the three wires in a similar fashion as above. Again, sinusoidal modes in each single wire have been assumed. The solid line in Fig. 6a shows the resulting coupling strength for different numbers of discretized dipoles. The coupling strength behaves more or less linearly with lateral shift for small values of

*s*, and saturates when the position of the dipole bar approaches the edges of the quadrupole bars underneath. A single dipole per wire can already describe the initial linear behavior. We determine that a number of

*N*= 30 (green curve) gives a quite good agreement between our simple model and the experiment for larger lateral shifts as well, and convergence has been achieved. The extremal condition described above in section 1.2 leads to the phase relation between the upper and lower dipole and quadrupole as plotted in Fig. 6b. The phase is more or less flat at

*ϕ*≈ 0 degrees for the stacking distance of 70 nm, regardless of the shift parameter

*s*.

### 2.3. Coupled split ring resonators

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

*θ*from 0 to 180 degrees.

26. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express **14**, 8827–8836 (2006). [CrossRef] [PubMed]

*λ*= 1.5

_{vac}*μ*m. Hence, a particle plasmon along the total wire with a more or less sinusoidal current distribution is excited (see Fig. 7). We treat the split ring resonator element as a wire which was bent twice [26

**14**, 8827–8836 (2006). [CrossRef] [PubMed]

18. H. Liu, J. X. Cao, and S. N. Zhu, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B **81**, 241403 (2010). [CrossRef]

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

*N*= 9 up to

*N*= 100. It is clear that for such a complex system, a single straight dipole cannot account for the coupling behavior upon twisting at all. We find that

*N*= 50 gives already satisfactory agreement between the FITD simulation (which actually agrees quite well with experimental observations, see Liu et al. [10

**3**, 157–162 (2009). [CrossRef]

*N*= 100, our model converges well. For a more complex structure we need more elementary dipoles to account for the associated modes. Our model reproduces the characteristic features of the system with a modest splitting at

*θ*= 0°, a minimum at a critical angle of

*θ*≈ 60°, and a stronger splitting at

_{c}*θ*= 180°.

*π*/5 at

*θ*= 0° twist angle to slightly above

*π*for

*θ*= 180° twist angle. The latter result can be easily understood, as the twist angle of 180° results in a

*π*phase shift between the two excitations and because the distance gives only a very small retardation. This small retardation might be responsible for the small deviation of the phase to a value of

*π*. One should mention that the interplay of external excitation and internal coupling in such a stereometamaterial is very intricate and far from being understood in a simple model system.

## 3. Summary/outlook

*N*= 100 discrete dipoles with the strength of the current. Summing then over all simple interactions between mutual dipoles and calculating the interaction energy, we were able to derive the total splitting energy of our complex systems within a computing time of less than a minute on a standard PC. We obtained very good overall agreement with much more complex FITD simulations and were even able to calculate coupling strengths for distances where extraction from FITD data was no longer possible due to the small energy splitting. We were also able to determine the phase behavior between the individual oscillators. Retardation was taken into account in all cases. Our model with our quite simple assumptions should be able to predict the behavior of even more complex nanostructures in the near-field as well in as in the far field with very small effort. Our method can therefore facilitate parameter studies for the design of novel nanoplasmonic devices which are based on coupled structures, such as novel polarizers [32

32. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science **325**, 1513–1515 (2009). [CrossRef] [PubMed]

20. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. **10**, 2342–2348 (2010). [CrossRef] [PubMed]

33. N. Liu, M. Hentschel, Th. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science **332**, 1407–1410 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. |

2. | N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmon hybridization in stacked cut-wire metamaterials,” Adv. Mater. |

3. | B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal nanoparticle gratings: influence of dipolar particle interaction on the plasmon resonance,” Phys. Rev. Lett. |

4. | S.-C. Yang, H. Kobori, C.-L. He, M.-H. Lin, H.-Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett. |

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

6. | N. Liu and H. Giessen “Coupling effects in optical metamaterials,” Angew. Chem., Int. Ed. |

7. | M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A |

8. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

9. | N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. |

10. | N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics |

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

12. | N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. |

13. | S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. |

14. | N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. |

15. | M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. |

16. | J. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science |

17. | J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tunability,” Nano Lett. |

18. | H. Liu, J. X. Cao, and S. N. Zhu, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B |

19. | B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. |

20. | N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. |

21. | E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. |

22. | E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science |

23. | P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. |

24. | T. J. Davis, D. E. Gomez, and K. C. Vernon, “Simple model for the hybridization of surface plasmon resonances in metallic nanoparticles,” Nano Lett. |

25. | I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magneto-electric point scattering theory for metamaterial scatterers,” Phys. Rev. B |

26. | C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express |

27. | J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. |

28. | J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. |

29. | T. Meyrath, T. Zentgraf, and H. Giessen, “Lorentz model for metamaterials: optical frequency resonance circuits,” Phys. Rev. B |

30. | CST. CST Microwave Studio (2009). |

31. | C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. |

32. | J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science |

33. | N. Liu, M. Hentschel, Th. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Plasmonics

**History**

Original Manuscript: June 7, 2011

Revised Manuscript: July 9, 2011

Manuscript Accepted: July 11, 2011

Published: October 24, 2011

**Virtual Issues**

Collective Phenomena (2011) *Optics Express*

**Citation**

Lutz Langguth and Harald Giessen, "Coupling strength of complex plasmonic structures in the multiple dipole approximation," Opt. Express **19**, 22156-22166 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-22156

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### References

- C. Dahmen, B. Schmidt, and G. von Plessen, “Radiation damping in metal nanoparticle pairs,” Nano Lett. 7, 318–322 (2007). [CrossRef] [PubMed]
- N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmon hybridization in stacked cut-wire metamaterials,” Adv. Mater. 19, 3628–3632 (2007). [CrossRef]
- B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal nanoparticle gratings: influence of dipolar particle interaction on the plasmon resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000). [CrossRef] [PubMed]
- S.-C. Yang, H. Kobori, C.-L. He, M.-H. Lin, H.-Y. Chen, C. Li, M. Kanehara, T. Teranishi, and S. Gwo, “Plasmon hybridization in individual gold nanocrystal dimers: direct observation of bright and dark modes,” Nano Lett. 10, 632–637 (2010). [CrossRef] [PubMed]
- S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004). [CrossRef] [PubMed]
- N. Liu and H. Giessen “Coupling effects in optical metamaterials,” Angew. Chem., Int. Ed. 49, 9838–9852 (2010). [CrossRef]
- M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A 7, S12–S22 (2005).
- V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007). [CrossRef]
- N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31–37 (2008). [CrossRef]
- N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3, 157–162 (2009). [CrossRef]
- D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011). [CrossRef]
- N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009). [CrossRef] [PubMed]
- S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008). [CrossRef] [PubMed]
- N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9, 1663–1667 (2009). [CrossRef] [PubMed]
- M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10, 2721–2726 (2010). [CrossRef] [PubMed]
- J. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010). [CrossRef] [PubMed]
- J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tunability,” Nano Lett. 10, 3184–3189 (2010). [CrossRef] [PubMed]
- H. Liu, J. X. Cao, and S. N. Zhu, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403 (2010). [CrossRef]
- B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010). [CrossRef]
- N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010). [CrossRef] [PubMed]
- E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120, 5444–5454 (2004). [CrossRef] [PubMed]
- E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003). [CrossRef] [PubMed]
- P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4, 899–903 (2004). [CrossRef]
- T. J. Davis, D. E. Gomez, and K. C. Vernon, “Simple model for the hybridization of surface plasmon resonances in metallic nanoparticles,” Nano Lett. 10, 2618–2625 (2010). [CrossRef] [PubMed]
- I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magneto-electric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83, 245102 (2011). [CrossRef]
- C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14, 8827–8836 (2006). [CrossRef] [PubMed]
- J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010). [CrossRef] [PubMed]
- J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009). [CrossRef] [PubMed]
- T. Meyrath, T. Zentgraf, and H. Giessen, “Lorentz model for metamaterials: optical frequency resonance circuits,” Phys. Rev. B 75, 205102 (2007). [CrossRef]
- CST. CST Microwave Studio (2009).
- C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37 (2002). [CrossRef]
- J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009). [CrossRef] [PubMed]
- N. Liu, M. Hentschel, Th. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011). [CrossRef] [PubMed]

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