## The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach

Optics Express, Vol. 15, Issue 14, pp. 8871-8883 (2007)

http://dx.doi.org/10.1364/OE.15.008871

Acrobat PDF (2733 KB)

### Abstract

We explain the origin of the electric and particular the magnetic polarizabiltiy of metamaterials employing a fully electromagnetic plasmonic picture. As example we study an U-shaped split-ring resonator based metamaterial at optical frequencies. The relevance of the split-ring resonator orientation relative to the illuminating field for obtaining a strong magnetic response is outlined. We reveal higher-order magnetic resonances and explain their origin on the basis of higher-order plasmonic eigenmodes caused by an appropriate current flow in the split-ring resonator. Finally, the conditions required for obtaining a negative index at optical frequencies in a metamaterial consisting of split-ring resonators and wires are investigated.

© 2007 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. D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A.-C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity,” J. Appl. Phys. **98**, 063505 (2005). [CrossRef]

*ϵ*) material are a typical example for a MM with strongly dispersive effective permittivity

*ϵ*

_{eff}and permeability

*μ*

_{eff}[3]. For this purpose one usually relies on polaritonic materials or semiconductors with strong ex-citonic resonances [4

4. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter **17**, 3717–3734 (2005). [CrossRef] [PubMed]

5. V. Yannopapas, “Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres,” Phys. Rev. B **75**, 035112 (2007). [CrossRef]

6. V. Yannopapas and N. V. Vitanov, “Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength,” Phys. Rev. B **74**, 193304 (2006). [CrossRef]

7. W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” IRE Trans. Antennas Propag. **10**, 82–95 (1962). [CrossRef]

9. J. P. 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]

10. D. R. Smith, W. J. Padilla, 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]

11. S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. **94**, 037402 (2005). [CrossRef] [PubMed]

12. N. Liu, H. Guo, L. Fu, H. Schweizer, S. Kaiser, and H. Giessen, “Electromagnetic resonances in single and double split-ring resonator metamaterials in the near infrared,” phys. stat. sol. (b) **224**, 1251–1255 (2007). [CrossRef]

13. M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and M. Soukoulis, “Left-handed metama-terials: detailed numerical studies of the transmission properties,” J. Opt. A: Pure Appl. Opt **7**, S12–S22 (2005). [CrossRef]

14. 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]

15. K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” New J. of Physics **7**, 168 (2005). [CrossRef]

16. V. V. Varadan and A. R. Tellakula,, “Effective properties of split-ring resonator metamaterials using measured scattering parameters: Effect of gap orientation,” J. Appl. Phys. **100**, 034910 (2006). [CrossRef]

17. P. Markoš and C. M. Soukoulis,, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E **65**, 036622 (2002). [CrossRef]

18. T.P. Meyrath, T. Zentgraf, and H. Giessen, “Lorentz Model for Metamaterials: Optical Frequency Resonance Circuits,” Phys. Rev. B **75**, 205102 (2007). [CrossRef]

19. U.K. Chettiar, A.V. Kildishev, T.A. Klar, and V.M. Shalaev, “Negative index metamaterial combining magnetic resonators with metal films,” Opt. Express **14**, 7872–7877 (2006). [CrossRef] [PubMed]

## 2. Resonances at normal incidence

20. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metama-terials at 100 Terahertz,” Science **306**, 1351–1353 (2004). [CrossRef] [PubMed]

14. 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*=

_{II}*l*

_{⊥}= 300 nm. The in-plane width of the wires forming the SRR is

*w*= 40 nm and their height is

*h*= 15 nm. The simulation of the spectral response was performed by using the Fourier Modal Method (FMM) [22

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

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

## 3. Resonances for parallel incidence

*h*to its width

*w*. Similar resonances appear for single wires with the same size and aspect ratio. These resonances are of no importance in the present context and will be neglected in the further discussion.

23. 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]

**D**

_{1}symmetry axis for these modes, whereas the illuminating plane wave provides only a symmetric field.

24. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomo-geneous metamaterials,” Phys. Rev. E **71**, 036617 (2005). [CrossRef]

24. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomo-geneous metamaterials,” Phys. Rev. E **71**, 036617 (2005). [CrossRef]

*n*

_{eff},

*ϵ*

_{eff}and

*μ*

_{eff}) are shown in Fig. 2(b–d), and the asymmetry of the structure was fully taken into account. For retrieving the effective material parameters we have applied the procedure as described in [24

24. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomo-geneous metamaterials,” Phys. Rev. E **71**, 036617 (2005). [CrossRef]

*ϵ*

_{eff}, a well pronounced Lorentz-type resonance can be observed for both SRR orientations at

_{1}= 3100 cm

^{-1}. The field of the eigenmodes in the two side arms of the SRR has no particular influence on the effective properties. The response is dominated by an electric dipole radiating from the bottom of the U-shaped SRR parallel to the gap. The eigenmode excitation is identical for both cases shown on top of Fig. 2. Hence, due to the electron oscillation in the bottom wire of the SRR the electric dipole alters mainly the electric polarizability of the medium irrespective of the SRR orientation. Essentially the same consideration holds for the higher-order resonance at

_{3}= 8000 cm

^{-1}. Although it is not being as well defined, the increase of the imaginary part of

*ϵ*

_{eff}and the general shape of the real part are well in accordance with a Lorentz resonance.

2. D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A.-C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity,” J. Appl. Phys. **98**, 063505 (2005). [CrossRef]

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

27. A. Farjadpour, David Roundy, Alejandro Rodriguez, M. Ibanescu, Peter Bermel, J. D. Joannopoulos, Steven G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. **31**, 2972–2974 (2006). [CrossRef] [PubMed]

21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

23. 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]

20. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metama-terials at 100 Terahertz,” Science **306**, 1351–1353 (2004). [CrossRef] [PubMed]

*conditio sine qua non*for a magnetic resonance to appear is a mode profile that induces currents having opposite directions in both arms. In Fig. 4 the direction of the currents is indicated by arrows. The direction of all currents were deduced from FDTD simulations. The first-order resonance at

_{1}= 3900 cm

^{-1}(Fig. 4-left) and the third-order resonance at

_{3}= 8100 cm

^{-1}(Fig. 4-right) meet this condition. It is evident that the base length affects the frequencies of the plasmonic eigenmodes and thus the spectral position of the effective material resonance [14

14. 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]

*h*of the SRR as will be shown later. The effective permittivity is only slightly affected in the spectral domain of the permeability resonances.

_{2}= 6400 cm

^{-1}the currents in the two side arms of the SRR point in the same direction as indicated in Fig. 4-center. This eigenmode corresponds to the lowest-order resonance, which can be excited at normal incidence with the electric field polarization perpendicular to the gap. The scattered field is mainly generated by an electrical dipole rather than a quadrupole. Hence, only the effective permittivity is altered at that frequency. Anti-resonances are observed in the effective permeability, characterized by a negative imaginary part and an inverted line shape for the real part. Their presence can be attributed to the excitation of higher Bloch eigenmodes with non-negligible amplitude [28

28. T. Koschny, P. Markosš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E **68**, 065602 (2003). [CrossRef]

2. D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A.-C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity,” J. Appl. Phys. **98**, 063505 (2005). [CrossRef]

## 4. How to obtain a negative index?

*l*

_{∥}= 300 nm of the SRR. The SRR height

*h*was assumed to be 40 nm to ensure a strong magnetic resonance with slightly negative values within a narrow spectral domain. We concentrate here on the first-order eigenmode of the SRR. The metallic wires are infinitely extended into the direction parallel to the polarization of the incident electric field. They have a width of 100 nm in the opposite transversal direction and a height of 80 nm in the longitudinal direction. They are placed such that they have both equal distances to adjacent SRRs and to both arms of a SSR. The period of the structure remained 400 nm. This new unit cell hosts two elements hosts, the SRR and the metallic wire. To counteract the decrease of the plasma frequency by ‘dilution’ the wires are quite massive and resemble a metallic plate.

_{1}= 4100 cm

^{-1}. For smaller frequencies, the effective permittivity of the medium, which comprises both elements, is dominated by the wire, whereas for frequencies larger than the plasma frequency the effective permittivity is dominated by the resonance of the second-order (dipolar) plasmonic eigenmode at

_{2}= 7900 cm

^{-1}. This holds for both the real and the imaginary part. Between these two frequency domains a smooth transition takes place [see Fig. 5(c,d)].

30. K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, “Experimental observation of true left-handed transmission peaks in metamaterials,” Opt. Lett. **29**, 2623–2625 (2004). [CrossRef] [PubMed]

*h*. We consider in the figure only the transmittance and the real part of the effective material parameters. Due to the stronger ‘dilution’ of the metal, the effective plasma frequency experiences a larger downshift. Above the frequency of the second-order plasmonic eigenmode, the permittivity is dominated by the effective permittivity provided by the SRRs, hence it is rather independent of the wire height. The effective permeability is likewise independent of the wire height. The increase of the permittivity in the spectral domain of the magnetic resonance for thinner wires causes a less negative refractive index. In the present example a negative refractive index requires a wire height of at least 40 nm. A detailed investigation of the strength of the resonance and the figure-of-merit for the material as a function of the detailed geometrical parameters (SRR and wire size and shape, as well as their relative position in the unit cell) is beyond the scope of this paper and will appear elsewhere. This figure-of-merit has been defined as the ratio of real to imaginary part of the refractive index [31

31. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative index metamaterial at 780 nm wavelength”, Opt. Lett. **32**, 53–55 (2007). [CrossRef]

*h*= 80 nm. First, the structure is placed on a substrate with a refractive index of

*n*= 1.5. Such a substrate has only a minor impact on the dispersion of the refractive index. A more realistic structure will be potentially embedded in a dielectric medium. Figure 7 shows likewise the real part of the effective refractive index if the structure is embedded in a dielectric host medium with a refractive index of 1.5 and finally when this structure is placed on a substrate with the same refractive index. Again, the presence of the substrate does alter the dispersion, but the surrounding media strongly shifts all resonances towards smaller wavenumbers. This is in complete agreement with the shift of the spectral position of the plasmon resonances in small metallic nanoparticles if they are embedded in a dielectric host media, except this shift of the dispersion remains qualitatively the same.

## 5. Conclusion and outlook

*π*out of phase. The corresponding currents which can be excited at the frequencies of the SRR plasmonic eigenmodes are the origin of this field. Because the currents are driven by the electric field component of the incident wave, the orientation of the electric field vector relative to the SRR is an important parameter.

## Acknowledgments

## References and links

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

2. | D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A.-C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity,” J. Appl. Phys. |

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

4. | V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter |

5. | V. Yannopapas, “Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres,” Phys. Rev. B |

6. | V. Yannopapas and N. V. Vitanov, “Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength,” Phys. Rev. B |

7. | W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” IRE Trans. Antennas Propag. |

8. | S. A. Schelkunoff and H. T. Friis, ”Antennas: theory and practice”, New York. John Wiley & Son (1952). |

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

10. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

11. | S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. |

12. | N. Liu, H. Guo, L. Fu, H. Schweizer, S. Kaiser, and H. Giessen, “Electromagnetic resonances in single and double split-ring resonator metamaterials in the near infrared,” phys. stat. sol. (b) |

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

14. | 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 |

15. | K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” New J. of Physics |

16. | V. V. Varadan and A. R. Tellakula,, “Effective properties of split-ring resonator metamaterials using measured scattering parameters: Effect of gap orientation,” J. Appl. Phys. |

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

18. | T.P. Meyrath, T. Zentgraf, and H. Giessen, “Lorentz Model for Metamaterials: Optical Frequency Resonance Circuits,” Phys. Rev. B |

19. | U.K. Chettiar, A.V. Kildishev, T.A. Klar, and V.M. Shalaev, “Negative index metamaterial combining magnetic resonators with metal films,” Opt. Express |

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

21. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

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

23. | 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 |

24. | D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomo-geneous metamaterials,” Phys. Rev. E |

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

26. | F. Garwe, C. Rockstuhl, C. Etrich, U. Hübner, U. Bauerschäfer, F. Setzpfandt, M. Augustin, T. Pertsch, A. Tünnermann, and F. Lederer, “Evaluation of gold nanowire pairs as a potential negative index material,” Appl. Phys. B |

27. | A. Farjadpour, David Roundy, Alejandro Rodriguez, M. Ibanescu, Peter Bermel, J. D. Joannopoulos, Steven G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. |

28. | T. Koschny, P. Markosš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

29. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W.J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter |

30. | K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, “Experimental observation of true left-handed transmission peaks in metamaterials,” Opt. Lett. |

31. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative index metamaterial at 780 nm wavelength”, Opt. 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: April 30, 2007

Revised Manuscript: June 21, 2007

Manuscript Accepted: June 21, 2007

Published: July 3, 2007

**Citation**

Carsten Rockstuhl, Thomas Zentgraf, Ekaterina Pshenay-Severin, Jörg Petschulat, Arkadi Chipouline, Jürgen Kuhl, Thomas Pertsch, Harald Giessen, and Falk Lederer, "The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach," Opt. Express **15**, 8871-8883 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8871

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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]
- D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A.-C. Tarot, "Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity," J. Appl. Phys. 98, 063505 (2005). [CrossRef]
- L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Elec. Eng. 94, 65-68 (1947).
- V. Yannopapas and A. Moroz, "Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges," J. Phys. Condens. Matter 17, 3717-3734 (2005). [CrossRef] [PubMed]
- V. Yannopapas, "Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres," Phys. Rev. B 75, 035112 (2007). [CrossRef]
- V. Yannopapas and N. V. Vitanov, "Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength," Phys. Rev. B 74, 193304 (2006). [CrossRef]
- W. Rotman, "Plasma simulation by artificial dielectrics and parallel-plate media," IRE Trans. Antennas Propag. 10, 82-95 (1962). [CrossRef]
- S. A. Schelkunoff and H. T. Friis, "Antennas: theory and practice", (New York, John Wiley & Son, 1952).
- J. P. 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]
- D. R. Smith, W. J. Padilla, 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. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, "Midinfrared resonant magnetic nanostructures exhibiting a negative permeability," Phys. Rev. Lett. 94, 037402 (2005). [CrossRef] [PubMed]
- N. Liu, H. Guo, L. Fu, H. Schweizer, S. Kaiser, and H. Giessen, "Electromagnetic resonances in single and double split-ring resonator metamaterials in the near infrared," Phys. Status Solidi B 224, 1251-1255 (2007). [CrossRef]
- M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt 7, S12-S22 (2005). [CrossRef]
- 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]
- K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, "Investigation of magnetic resonances for different split-ring resonator parameters and designs," New J. Phys. 7, 168 (2005). [CrossRef]
- V. V. Varadan and A. R. Tellakula, "Effective properties of split-ring resonator metamaterials using measured scattering parameters: Effect of gap orientation," J. Appl. Phys. 100, 034910 (2006). [CrossRef]
- P. Markoš and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002). [CrossRef]
- T. P. Meyrath, T. Zentgraf, and H. Giessen, "Lorentz model for Metamaterials: Optical frequency resonance circuits," Phys. Rev. B 75, 205102 (2007). [CrossRef]
- U. K. Chettiar, A. V. Kildishev, T. A. Klar, and V. M. Shalaev, "Negative index metamaterial combining magnetic resonators with metal films," Opt. Express 14, 7872-7877 (2006). [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]
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [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]
- 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]
- D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005). [CrossRef]
- 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]
- F. Garwe, C. Rockstuhl, C. Etrich, U. Hübner, U. Bauerschäfer, F. Setzpfandt, M. Augustin, T. Pertsch, A. Tünnermann, and F. Lederer, "Evaluation of gold nanowire pairs as a potential negative index material," Appl. Phys. B 84, 139-148 (2006). [CrossRef]
- A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006). [CrossRef] [PubMed]
- T. Koschny, P. Markoš, D. R. Smith and C. M. Soukoulis, "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 68, 065602 (2003). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin wire structures," J. Phys. Condens. Matter 10, 4785-4809 (1997). [CrossRef]
- K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, "Experimental observation of true left-handed transmission peaks in metamaterials," Opt. Lett. 29, 2623-2625 (2004). [CrossRef] [PubMed]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007). [CrossRef]

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