## Numerical study of a new negative index material in mid-infrared spectrum |

Optics Express, Vol. 20, Issue 23, pp. 25744-25751 (2012)

http://dx.doi.org/10.1364/OE.20.025744

Acrobat PDF (1133 KB)

### Abstract

We explored numerically a new negative index material in mid-infrared spectrum, based on a thin wire net pairs and a square ring pairs array, which exhibits simultaneously hyper-transmission, polarization independence and small period. The mechanism implementing the negative refractive index was analyzed using retrieved optical constants as well as a valid analytical expression of the effective permeability that was deduced by virtue of a simple equivalent circuit model and applied to account for the magnetic property of this metamaterial.

© 2012 OSA

## 1. Introduction

1. S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express **13**(13), 4922–4930 (2005). [CrossRef] [PubMed]

2. 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**(13), 137404 (2005). [CrossRef] [PubMed]

1. S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express **13**(13), 4922–4930 (2005). [CrossRef] [PubMed]

3. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. **31**(12), 1800–1802 (2006). [CrossRef] [PubMed]

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

5. Z. Ku and S. R. Brueck, “Comparison of negative refractive index materials with circular, elliptical and rectangular holes,” Opt. Express **15**(8), 4515–4522 (2007). [CrossRef] [PubMed]

6. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express **15**(18), 11536–11541 (2007). [CrossRef] [PubMed]

7. X. Wang, Y. H. Ye, C. Zheng, Y. Qin, and T. J. Cui, “Tunable figure of merit for a negative-index metamaterial with a sandwich configuration,” Opt. Lett. **34**(22), 3568–3570 (2009). [CrossRef] [PubMed]

8. C. Helgert, C. Menzel, C. Rockstuhl, E. Pshenay-Severin, E. B. Kley, A. Chipouline, A. Tünnermann, F. Lederer, and T. Pertsch, “Polarization-independent negative-index metamaterial in the near infrared,” Opt. Lett. **34**(5), 704–706 (2009). [CrossRef] [PubMed]

9. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. **101**(10), 103902 (2008). [CrossRef] [PubMed]

9. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. **101**(10), 103902 (2008). [CrossRef] [PubMed]

6. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express **15**(18), 11536–11541 (2007). [CrossRef] [PubMed]

10. C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. **106**(6), 067402 (2011). [CrossRef] [PubMed]

*λ/a*~2) [6

6. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express **15**(18), 11536–11541 (2007). [CrossRef] [PubMed]

**15**(18), 11536–11541 (2007). [CrossRef] [PubMed]

10. C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. **106**(6), 067402 (2011). [CrossRef] [PubMed]

9. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. **101**(10), 103902 (2008). [CrossRef] [PubMed]

## 2. Design and simulation

*a*and line width

*w*

_{n}as indicated in the unit cell in Fig. 1(b). The SRP array was expected to supply negative permeability originating from strong magnetic resonances. The permeability of the SRP can be tuned by scaling geometrical parameters independently or synergistically, which includes the period

*a*, side length

*l*, strip width

*w*, metalization thickness

*t*and dielectric spacer thickness

*d*etc. The separate electric and magnetic elements allow us to overlap easily the negative permittivity and negative permeability in the present spectrum.

_{11}and transmission parameters S

_{21}. With regard to the constituent material, a loss-free zinc sulfide (ZnS) that is transparent throughout the spectrum from ~1 μm to ~10 μm was chosen as dielectric spacer whose refractive index is nearly 2.24 in the region [12]. The metal element was defined as silver whose dielectric behavior is described by Drude dispersion model, i.e.,

*ω*= 2π × 21.75 × 10

_{ep}^{14}rad/s and the collision frequency

*ν*= 2π × 4.35 × 10

_{c}^{12}rad/s [13

13. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. **22**(7), 1099–20 (1983). [CrossRef] [PubMed]

## 3. Results and discussion

*w*= 0.34 μm,

_{n}*l*= 1.60 μm

*w*= 0.47 μm,

*t*= 0.06 μm and

*d*= 0.22 μm through the electromagnetic calculation. The complex scattering parameters are displayed in Figs. 2(a) and 2(b). These scattering parameters both the magnitude and phase have the same features as that in the negative index region presented in the Ref [1

1. S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express **13**(13), 4922–4930 (2005). [CrossRef] [PubMed]

14. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **71**(33 Pt 2B), 036617 (2005). [CrossRef] [PubMed]

_{21}indicates the presence of a negative index band. The extracted complex refractive index using a classical retrieval procedure within Ref [14

14. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **71**(33 Pt 2B), 036617 (2005). [CrossRef] [PubMed]

*n(λ)*is real part of the refractive index and

*κ(λ)*is extinction coefficient. The figure of merit as an absorptive loss indicter was also calculated by

*(λ)*is 6.9 around an operating wavelength of 8.1 μm where

*n(λ) =*−1. The corresponding transmission given by |S

_{21}|

^{2}is up to 90.2%. Furthermore, a ratio of 3.0 of operating wavelength to period is acquired. It should be mentioned that there is a bad transmission about 60% around 8.8 μm wavelength because of the large extinction coefficient although another

*n(λ) =*−1 is obtained here, which is also the case in literatures.

### 3.1. Mechanism of implementing the NIM

*ε*, and the permeability, i.e.,

_{eff}(λ) = ε_{1}(λ) + iε_{2}(λ)*μ*, in order to explain the fulfillment of the NIM. Figure 2(e) explicitly demonstrates that the NIM is double negative in the range of 7.5 μm - 8.7 μm and is single negative between 8.7 μm and 9.3 μm. And now, we analyze the contribution of the WNP and the SRP array to the permittivity and permeability. Figure 2(f) reveals the retrieved real part permittivity and permeability of the isolated WNP and the isolated SRP array (they are denoted as subscripts of “1n” and “1r”, respectively). Clearly, the WNP displays dispersive permittivity ε

_{eff}(λ) = μ_{1}(λ) + iμ_{2}(λ)_{1n}(λ) with a plasma frequency of 3.0 μm, which is mainly dominated by the cutoff wavelength of the hole waveguide in the WNP [9

**101**(10), 103902 (2008). [CrossRef] [PubMed]

_{1r}(λ) with a electric resonance of 3.25 μm rather than infinity, which stems from the additional depolarization effect due to its finite-long nature [15

15. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. **93**(10), 107402 (2004). [CrossRef] [PubMed]

_{1}(λ) of the combined system and lead to a new plasma frequency of 7.5 μm and a new resonance frequency around 3.1 μm in ε

_{1}(λ) (see Fig. 2(e)). Actually, the ε

_{1}(λ) is to first approximation equal to the sum of ε

_{1n}(λ) and ε

_{1r}(λ) [15

15. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. **93**(10), 107402 (2004). [CrossRef] [PubMed]

_{1n}(λ) of + 1.0 implies that the WNP is nonmagnetic in the present spectrum. Figures 2(e) and 2(f) illustrate that the permeability of the SRP array is almost same as the one of the combined system. This indicates that the magnetic property of the SRP array mostly determines that of the combined system and is not distinctly affected by the WNP. This is because the WNP is nonmagnetic and can also be explained by a numerical model of the permeability deduced from RLC equivalent circuit model shown in next section.

### 3.2. Numerical model of permeability

16. 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**(24), 3356–3358 (2005). [CrossRef] [PubMed]

17. J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. **31**(24), 3620–3622 (2006). [CrossRef] [PubMed]

*z*component of E-field distribution in between the transversal metallic strips of the square rings and shown in Fig. 3(b). At the same time, the E-field closed the two branches of antiparallel currents sustained by USSP and then a virtual current loop between the metallic layers on a perpendicular plane to the incoming magnetic field formed. Furthermore, due to the accumulated opposite charges in the transversal strips, electric field is also expected to be confined within the separation gap between two nearest square rings. The E-field profiles in the spacer confirmed this picture, as is depicted in Fig. 3(c).

17. J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. **31**(24), 3620–3622 (2006). [CrossRef] [PubMed]

18. J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B **73**(4), 041101 (2006). [CrossRef]

*L*can be expressed as

_{m}_{0}is the permeability of vacuum,

*l*is the effective length of the vertical strip of the square rings under which the most of all magnetic energy was trapped, as depicted in Fig. 3(d). We regard roughly it as the average length of the square ring, i.e.,

_{eff}*l*. Since there are the larger charge distribution areas (the transversal strip) at the ends of USSP compared with cut-wire pairs, the plate capacitance

_{eff}= l-w*C*has a form of

_{m}_{0}is the permittivity of vacuum, ε

_{r}is the relative permittivity of the dielectric spacer, and

*f*is a fitting factor less than 1.0, which is used to adjust the effective charge distribution area of the transversal strip when the strip width vary, and 0.8 was chosen here. The gap capacitor approximated by two parallel wires of length 0.5

*l*and diameter

*t*is described as

*g = a-l*is separation distance between two nearest neighboring rings. Since one major source of the line broadening of optical constants originates from the damping of the constituent metal, dispersive resistances of the effective vertical strips are taken into account and have an expression of

19. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. **95**(22), 223902 (2005). [CrossRef] [PubMed]

*IS*, where

*S*=

*l*is the area of the current loop. This leads the magnetization

_{eff}d*N*is the number of magnetic atoms and

*V*is the volume of the array. One can easily derive

*I*and

*H*. Applying the mentioned above equivalent circuit model illustrated in Fig. 4(a) and Kirchhoff’s voltage and current law (KVL and KCL), the circuit equations can be written as andwhere

*U*is the voltage drop over the parallel branch, and

_{p}*I*is the branch current in the parallel branch of

_{R}*R*and

*L*. Solve the Eqs. (2) and (3) above and yieldsWe denote

_{m}*U*into Eq. (1), the expression of

_{p}*I*versus

*H*is immediately acquired as followingAs a result, the numerical model of analytical permeability is derived asFor the combined system consisting of a WNP and a SRP array, in fact, we substitute the equivalent circuit model in Fig. 4(a) for 4(b). The resistance and capacitance introduced by WNP have expressions of

*w*of the WNP decrease to 0. Additionally, we find C

_{n}_{g}(≈7.6 aF) ~C

_{s}(≈6.1 aF),

*irC*~0 and

_{s}ω*R ~6r*, these explain why the WNP has few influence on the magnetic property of the SRP array.

### 3.3. Achievement of hyper-transmission

*ε*and

_{1}(λ) = μ_{1}(λ)*ε*. As to the real part, we adjust the “magnetic plasma frequency” to equate the electric plasma frequency so that

_{2}(λ) = μ_{2}(λ)*μ*as close to

_{1}(λ)*ε*as possible (see Fig. 2(e), Fig. 5(b) also). Although the minus imaginary permittivity

_{1}(λ)*ε*originating from the disturbance of magnetic resonance is much weaker than imaginary permeability

_{2}(λ)*μ*around the magnetic resonance frequency, this does not prevent the accordance of the two imaginary part values at those frequencies far from the center frequency of the magnetic resonance. In Fig. 5(b), the devised NIM with a strip width of 0.47 μm exhibits better impedance match relative to the other two samples, thus present the best transmission performance in double negative regime marked by dashed box, as is shown in Fig. 5(c). It is worth mentioning that the magnetic plasma frequency is more sensitive than the electric plasma frequency to the changing of strip width, as is illustrated in Fig. 5(b). This implies that we can control relatively independently the permittivity and permeability, which allows us to match easily the impedance.

_{2}(λ)## 4. Summary

## Acknowledgments

## References and links

1. | S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express |

2. | 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. |

3. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. |

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

5. | Z. Ku and S. R. Brueck, “Comparison of negative refractive index materials with circular, elliptical and rectangular holes,” Opt. Express |

6. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express |

7. | X. Wang, Y. H. Ye, C. Zheng, Y. Qin, and T. J. Cui, “Tunable figure of merit for a negative-index metamaterial with a sandwich configuration,” Opt. Lett. |

8. | C. Helgert, C. Menzel, C. Rockstuhl, E. Pshenay-Severin, E. B. Kley, A. Chipouline, A. Tünnermann, F. Lederer, and T. Pertsch, “Polarization-independent negative-index metamaterial in the near infrared,” Opt. Lett. |

9. | A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. |

10. | C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. |

11. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

12. | E. D. Palik, |

13. | M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. |

14. | D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

15. | T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. |

16. | 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. |

17. | J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. |

18. | J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B |

19. | J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. |

**OCIS Codes**

(260.3060) Physical optics : Infrared

(310.6860) Thin films : Thin films, optical properties

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: August 8, 2012

Revised Manuscript: October 13, 2012

Manuscript Accepted: October 18, 2012

Published: October 30, 2012

**Citation**

Dengmu Cheng, Jianliang Xie, Peiheng Zhou, Huibin Zhang, Nan Zhang, and Longjiang Deng, "Numerical study of a new negative index material in mid-infrared spectrum," Opt. Express **20**, 25744-25751 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25744

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### References

- S. Zhang, W. Fan, K. J. Malloy, S. R. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13(13), 4922–4930 (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(13), 137404 (2005). [CrossRef] [PubMed]
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31(12), 1800–1802 (2006). [CrossRef] [PubMed]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- Z. Ku and S. R. Brueck, “Comparison of negative refractive index materials with circular, elliptical and rectangular holes,” Opt. Express 15(8), 4515–4522 (2007). [CrossRef] [PubMed]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express 15(18), 11536–11541 (2007). [CrossRef] [PubMed]
- X. Wang, Y. H. Ye, C. Zheng, Y. Qin, and T. J. Cui, “Tunable figure of merit for a negative-index metamaterial with a sandwich configuration,” Opt. Lett. 34(22), 3568–3570 (2009). [CrossRef] [PubMed]
- C. Helgert, C. Menzel, C. Rockstuhl, E. Pshenay-Severin, E. B. Kley, A. Chipouline, A. Tünnermann, F. Lederer, and T. Pertsch, “Polarization-independent negative-index metamaterial in the near infrared,” Opt. Lett. 34(5), 704–706 (2009). [CrossRef] [PubMed]
- A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. 101(10), 103902 (2008). [CrossRef] [PubMed]
- C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106(6), 067402 (2011). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, 1995).
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1985).
- M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–1120 (1983). [CrossRef] [PubMed]
- D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005). [CrossRef] [PubMed]
- T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93(10), 107402 (2004). [CrossRef] [PubMed]
- 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(24), 3356–3358 (2005). [CrossRef] [PubMed]
- J. Zhou, E. N. Economon, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31(24), 3620–3622 (2006). [CrossRef] [PubMed]
- J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73(4), 041101 (2006). [CrossRef]
- J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). [CrossRef] [PubMed]

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