## On the reinterpretation of resonances in split-ring-resonators at normal incidence

Optics Express, Vol. 14, Issue 19, pp. 8827-8836 (2006)

http://dx.doi.org/10.1364/OE.14.008827

Acrobat PDF (578 KB)

### Abstract

We numerically study the spectral response of ‘U’-shaped split-ring-resonators at normal incidence with respect to the resonator plane. Based on the evaluation of the near-field patterns of the resonances and their geometry-dependent spectral positions, we obtain a comprehensive and consistent picture of their origin. We conclude that all resonances can be understood as plasmonic resonances of increasing order of the entire structure. In particular, for an electrical field polarized parallel to the gap the so-called LC-resonance corresponds to the fundamental plasmonic mode and, contrary to earlier interpretations, the electrical resonance is a second order plasmon mode of the entire structure. The presence of further higher order modes is discussed.

© 2006 Optical Society of America

## 1. Introduction

2. D. R. Smith, S. Schultz, P. Markoš, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

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

*ε*

_{1}

*µ*

_{2}+

*ε*

_{2}

*µ*

_{1}<0 where the subscripts 1 and 2 denote the real and imaginary part of permittivity and permeability, respectively. In such a medium a multitude of counterintuitive physical effects may take place and very appealing applications can be viewed, with the perfect lens being the most prominent example [5

5. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

6. W. Rotman, “Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media,” IRE Trans. Ant. Propagat. **10**, 82–95 (1962). [CrossRef]

8. J. P. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors, and Enhanced Non-Linear Phenomena,” IEEE Trans. Microwave Theory Tech. **47**, 2075–2084 (1999). [CrossRef]

10. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

11. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**, 3198–3200 (2005). [CrossRef] [PubMed]

12. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. **95**, 137404 (2005). [CrossRef] [PubMed]

13. D. R. Smith, W. J. Padila, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

13. D. R. Smith, W. J. Padila, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

14. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science **306**, 1361–1353 (2004). [CrossRef]

15. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C.M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. **84**, 2943–2945 (2004). [CrossRef]

16. N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Magnetic response of split-ring resonators in the far-infrared frequency regime,” Opt. Lett. **30**, 1348–1350 (2005). [CrossRef] [PubMed]

14. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science **306**, 1361–1353 (2004). [CrossRef]

17. C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B **84**, 219–227 (2006). [CrossRef]

## 2. Analyzed structures and numerical tools

*n*

_{S}=1.5. The height of the structure is

*h*=20 nm, the period is Λ=500 nm, the thickness of the wire is

*w*=60 nm and the length of the SRR parallel to the gap is

*l*

_{‖}=400 nm. One parameter of the SRRs that was changed in the simulations is the length of the SRRs perpendicular to the gap

*l*

_{⊥}. For simplifying the description of the SRRs in the text, we use throughout the manuscript the phrase “a bottom of the ‘U’-structure” for the base wire piece of the SRR, and “legs of the ‘U’-structure” for the two wire pieces of the SRR perpendicular to the gap.

18. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

19. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

20. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

## 3. Results

### 3.1. Near-field distributions of the plasmon modes

*ν̄*=5,000 cm

^{-1}is broad and rather strong. In addition, two higher-order resonances are observed at

*ν̄*=9,000 cm

^{-1}and

*ν̄*=11,400 cm

^{-1}. Also the same residual spectral features appear at higher frequencies. The Wood anomaly in reflection is visible once more at

*ν̄*=13,000 cm

^{-1}. The plasmonic peak (fifth resonance) at around

*ν̄*=17,500 cm

^{-1}is associated with a charge density oscillation perpendicular to the bottom of the ‘U’-structure (oscillation in

*y*-direction).

*π*occurs along each node of the magnitude of the electric field. The phases are rather constant in the spatial domain between the nodes as expected for a resonance. The fields are referenced to the total field at

*z*=20 nm above the structure in reflection. The illuminating field is a linearly polarized plane wave at normal incidence (positive

*z*-direction). Please note that the amplitude of the field component in the polarization direction is a superposition of the incident and the reflected field, whereas the other two components are due to reflected fields only.

*all*excited resonances can be attributed to

*plasmonic*modes of the entire SRR. With increasing order of the mode the resonance frequency becomes larger. By labeling the appearing modes according to the number of their nodes in the magnitude of the

*E*

_{z}-component, we can recognize modes up to an order of six. The character of the various plasmon modes is best explained looking at the

*E*

_{z}-component. Modes with an odd number of nodes are excited if the incident field is polarized parallel to the gap, whereas modes with an even number of nodes are excited for polarization perpendicular to the gap. This is a result of the D

_{1}symmetry group for the SRR and the illuminating light field. For the polarization perpendicular to the gap (Fig. 3), the internal field has to be in phase at both legs of the SRR. The SRR has mirror symmetry with respect to this polarization. Hence, for preserving this symmetry, the reflected fields have to have an equal phase along the legs of the ‘U’-structure. As a zero order mode was not found (e.g., a mode with a constant phase across the SRR), all modes must have a non-vanishing even number of nodes (2,4,6, ..) to meet the required symmetry. The strength of the highest order mode at

*ν̄*=11,364 cm

^{-1}is low but its symmetry can be inferred from the figure (e.g., there are seven maxima in the

*E*

_{z}-component and six nodes).

*π*. Consequently, only plasmon modes with an odd number of nodes (1,3,5, ..) can be excited (see Fig. 4).

23. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver Nanowires as Surface Plasmon Resonators,” Phys. Rev. Lett. **95**, 258403 (2005). [CrossRef]

*E*

_{z}-component of the field, the same line of argumentation on the exact mode character applies to all the other field components. With an increasing resonance frequency the nodes in the amplitude in the respective modes increases.

## 3.2. Spectral response as a function of the geometry

*l*

_{⊥}=60 nm=

*w*in Fig. 5 corresponds to a single nanowire. For the polarization of the electric field parallel to the gap as shown in Fig. 5a), the strength of the first resonance remains nearly constant whereas the second and third resonance become significantly weaker with decreasing

*l*

_{⊥}. The strength of these higher-order resonances becomes weaker with a decreasing geometrical size of the structure. Particularly the third-order resonance is no longer visible for the structure with

*l*

_{⊥}=273 nm, but remains visible for structures with a slightly larger leg length. The second resonance appears with negligible strength at a frequency of

*ν̄*=12,500 cm

^{-1}for the structure with

*l*

_{⊥}=103 nm. In the limit of a vanishing leg length of the SRR where the length

*l*

_{⊥}is equal to the width

*w*, only a dipolar mode can be excited.

15. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C.M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. **84**, 2943–2945 (2004). [CrossRef]

*l*

_{⊥}can be understood by unfolding conceptually the ‘U’ into an extended single wire piece that supports the electron oscillations. Its resonance frequency depends dominantly on the ratio of the entire ‘U’- structure length to its height. An increasing ratio results in a stronger resonance shift towards lower frequencies. Therefore, reducing the length of the unfolded ‘U’-structure as in the present simulation shifts the resonance to higher frequencies. The third geometrical parameter, namely the width, has minor influence.

*ν̄*=13,000 cm

^{-1}independently of the exact SRR geometry in Fig. 5. The fifth resonance at

*ν̄*=17,500 cm

^{-1}appears nearly at the same frequency but is decreased in strength for a reduced length of

*l*

_{⊥}. This can be easily understood, as the resonance is associated with a charge density oscillation perpendicular to the wire pieces that forms the legs of the ‘U’-structure. If these particular piece of wires vanishes, the plasmon is no longer excitable.

*l*

_{⊥}, the resonances associated with the plasmon excitation in the entire SRR shift simultaneously to higher frequencies. The extracted resonance positions for the first two modes are also shown in Fig. 6. The strength of the first resonance gets significantly weaker than for the second resonance. The third resonance is weakly seen for the structure with a length of

*l*

_{⊥}=273 nm but vanishes at a further reduction of

*l*

_{⊥}.

*l*

_{⊥}is changed. The part of the ‘U’ that causes the resonance, an oscillation of the charge density perpendicular to the bottom of the SRR along the width of this wire piece, is not affected. This observation is similar to the response with the polarization parallel to the gap.

## 4. Conclusion

## Acknowledgment

## References and links

1. | John D. Joannopoulos |

2. | D. R. Smith, S. Schultz, P. Markoš, and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

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

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

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

6. | W. Rotman, “Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media,” IRE Trans. Ant. Propagat. |

7. | S. A. Schelkunoff and H. T. Friis |

8. | J. P. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors, and Enhanced Non-Linear Phenomena,” IEEE Trans. Microwave Theory Tech. |

9. | L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Elec. Eng., Part 3 , |

10. | V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. |

11. | G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. |

12. | S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. |

13. | D. R. Smith, W. J. Padila, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. |

14. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science |

15. | N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C.M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. |

16. | N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Magnetic response of split-ring resonators in the far-infrared frequency regime,” Opt. Lett. |

17. | C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B |

18. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

19. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

20. | P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

21. | A. Taflove and S. C. Hagness |

22. | R. W. Wood, “Anomalous Diffraction Grating,” Phys. Rev. |

23. | H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver Nanowires as Surface Plasmon Resonators,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(260.5740) Physical optics : Resonance

**ToC Category:**

Metamaterials

**History**

Original Manuscript: June 30, 2006

Manuscript Accepted: August 2, 2006

Published: September 18, 2006

**Citation**

Carsten Rockstuhl, Falk Lederer, Christoph Etrich, Thomas Zentgraf, Jürgen Kuhl, and Harald Giessen, "On the reinterpretation of resonances in split-ring-resonators at normal incidence," Opt. Express **14**, 8827-8836 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8827

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### References

- J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
- D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, New York, 1982).
- 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]
- W. Rotman, "Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media," IRE Trans. Ant. Propag. 10, 82-95 (1962). [CrossRef]
- S. A. Schelkunoff and H. T. FriisAntennas, Theory and Practice (New York, John Wiley & Sons, 1952).
- J. P. Pendry, A. J. Holden, D. J. Robbins and W. J. Stewart, "Magnetism from Conductors, and Enhanced Non-Linear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
- L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Electr. Eng., 94, 65-68 (1947).
- V. M. Shalaev,W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005). [CrossRef]
- G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, "Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials," Opt. Lett. 30, 3198-3200 (2005). [CrossRef] [PubMed]
- S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]
- D. R. Smith, W. J. Padila, D. C. Vier, S. C. Nemat-Nasser and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [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, 1361-1353 (2004). [CrossRef]
- N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou and C. M. Soukoulis, "Electric coupling to the magnetic resonance of split ring resonators," Appl. Phys. Lett. 84, 2943-2945 (2004). [CrossRef]
- N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu,M. Kafesaki, E. N. Economou, T. Koschny and C. M. Soukoulis, "Magnetic response of split-ring resonators in the far-infrared frequency regime," Opt. Lett. 30, 1348-1350 (2005). [CrossRef] [PubMed]
- C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, "Resonances of split-ring resonator metamaterials in the near infrared," Appl. Phys. B 84, 219-227 (2006). [CrossRef]
- L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
- L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996). [CrossRef]
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).
- R. W. Wood, "Anomalous diffraction grating," Phys. Rev. 48, 928-936 (1935). [CrossRef]
- H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, "Silver Nanowires as Surface Plasmon Resonators," Phys. Rev. Lett. 95, 258403 (2005). [CrossRef]

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