Abstract

The influence of symmetry breaking in a planar metamaterial on transparency effect is numerically investigated. The planar metamaterial’s cell is formed by three parallel metal wires. From numerical simulation results, we can see that the transparency effect results from the asymmetric coupling between the cut wires. The excited mechanism of the transparency effect is further analyzed by using the hybridization concept. It is found that the coupling fields between the cut wires play key roles and lead to the spectral splitting of the resonance, i.e., the classical electromagnetically induced transparency effect. The metamaterial sensor based on the refractive index variation of the surrounding material is also numerically demonstrated and yields a sensitivity of 9.47mm/RIU and a figure of merit of 13.5. In addition, the spectral response of the metamaterial is quantitatively described via the “three-particle” model. The analytically calculated results of the model show a good agreement with the simulation results.

© 2012 Optical Society of America

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2012 (6)

2011 (8)

M. Rahmani, B. Lukiyanchuk, B. Ng, K. G. A. Tavakkoli, Y. F. Liew, and M. H. Hong, “Influence of symmetry breaking in pentamers on Fano resonance and near-field energy localization,” Opt. Mater. Express 1, 1409–1415 (2011).
[CrossRef]

F. Y. Meng, F. Zhang, K. Zhang, and Q. Wu, “Low-loss magnetic metamaterial based on analog of electromagnetically induced transparency,” IEEE Trans. Magn. 47, 3347–3350 (2011).
[CrossRef]

M. Rahmani, B. Lukiyanchuk, B. Ng, K. G. A. Tavakkoli, Y. F. Liew, and M. H. Hong, “Generation of pronounced Fano resonances and tuning of subwavelength spatial light distribution in plasmonic pentamers,” Opt. Express 19, 4949–4956 (2011).
[CrossRef]

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

X. R. Jin, J. Park, H. Y. Zheng, S. Lee, Y. Lee, J. Y. Rhee, K. W. Kim, H. S. Cheong, and W. H. Jang, “Highly-dispersive transparency at optical frequencies in planar metamaterials based on two-bright-mode coupling,” Opt. Express 19, 21652–21657 (2011).
[CrossRef]

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

A. Ourir, R. Abdeddaim, and J. d. Rosny, “Double-T metamaterial for parallel and normal transverse electric incident waves, ” Opt. Lett. 36, 1527–1529 (2011).
[CrossRef]

2010 (6)

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,” Nature 9, 707–715 (2010).
[CrossRef]

K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A 81, 195128 (2010).

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[CrossRef]

K. Aydin, I. M. Pryce, and H. A. Atwater, “Symmetry breaking and strong coupling in planar optical metamaterials,” Opt. Express 18, 13407–13417 (2010).
[CrossRef]

B. Lukiyanchuk, 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]

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

2009 (4)

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]

B. Kanté, S. N. Burokur, A. Sellier, A. de Lustrac, and J. M. Lourtioz, “Controlling plasmon hybridization for negative refraction metamaterials,” Phys. Rev. B 79, 075121 (2009).
[CrossRef]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
[CrossRef]

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

2008 (1)

A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171–2175 (2008).
[CrossRef]

2007 (3)

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

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]

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

2006 (1)

2004 (1)

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

2002 (1)

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37–41 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

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

A. Tavakkoli, K. G.

Abdeddaim, R.

Al-Naib, I. A. I.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Alzar, C. L. G.

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37–41 (2002).
[CrossRef]

Atwater, H. A.

Aydin, K.

Bai, Q.

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Bettiol, A. A.

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

Burokur, S. N.

B. Kanté, S. N. Burokur, A. Sellier, A. de Lustrac, and J. M. Lourtioz, “Controlling plasmon hybridization for negative refraction metamaterials,” Phys. Rev. B 79, 075121 (2009).
[CrossRef]

Cao, W.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Chen, J.

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Cheng, H. F.

Cheng, Y.

Cheong, H. S.

Chiam, S. Y.

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

Choi, J. J.

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Chong, C. T.

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,” Nature 9, 707–715 (2010).
[CrossRef]

B. Lukiyanchuk, 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]

Chowdhury, D. R.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Christ, A.

A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171–2175 (2008).
[CrossRef]

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

Cong, J.

Cook, J. J. H.

K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A 81, 195128 (2010).

Cui, Y.

de Lustrac, A.

B. Kanté, S. N. Burokur, A. Sellier, A. de Lustrac, and J. M. Lourtioz, “Controlling plasmon hybridization for negative refraction metamaterials,” Phys. Rev. B 79, 075121 (2009).
[CrossRef]

Ding, P.

Economou, E. N.

Eigenthaler, U.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[CrossRef]

Ekinci, Y.

A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171–2175 (2008).
[CrossRef]

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

Fan, C.

Fedotov, V. A.

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

Fleischhauer, M.

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]

Fu, J. H.

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “An approach to configure low-loss and full transmission metamaterial based on electromagnetically induced transparency,” IEEE Trans. Magn. 48, 4285–4288 (2012).

L. Zhu, F. Y. Meng, J. H. Fu, and Q. Wu, “Electromagnetically induced transparency metamaterial with polarization insensitivity based on multi-quasi-dark modes,” J. Phys. D 45, 445105 (2012).
[CrossRef]

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “Multi-band slow light metamaterial,” Opt. Express 20, 4494–4502 (2012).
[CrossRef]

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Fu, L.

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]

Giessen, H.

B. Lukiyanchuk, 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, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (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,” Nature 9, 707–715 (2010).
[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]

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]

Gippius, N. A.

A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171–2175 (2008).
[CrossRef]

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

Guo, H.

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]

Halas, N. J.

B. Lukiyanchuk, 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]

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,” Nature 9, 707–715 (2010).
[CrossRef]

Hamm, J. M.

K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A 81, 195128 (2010).

Hassani, A.

Hess, O.

K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A 81, 195128 (2010).

Hirscher, M.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[CrossRef]

Hong, M. H.

Hua, J.

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “Multi-band slow light metamaterial,” Opt. Express 20, 4494–4502 (2012).
[CrossRef]

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “An approach to configure low-loss and full transmission metamaterial based on electromagnetically induced transparency,” IEEE Trans. Magn. 48, 4285–4288 (2012).

Jang, W. H.

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

X. R. Jin, J. Park, H. Y. Zheng, S. Lee, Y. Lee, J. Y. Rhee, K. W. Kim, H. S. Cheong, and W. H. Jang, “Highly-dispersive transparency at optical frequencies in planar metamaterials based on two-bright-mode coupling,” Opt. Express 19, 21652–21657 (2011).
[CrossRef]

Jin, X. R.

X. R. Jin, J. Park, H. Y. Zheng, S. Lee, Y. Lee, J. Y. Rhee, K. W. Kim, H. S. Cheong, and W. H. Jang, “Highly-dispersive transparency at optical frequencies in planar metamaterials based on two-bright-mode coupling,” Opt. Express 19, 21652–21657 (2011).
[CrossRef]

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

Kaiser, S.

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]

Kang, M.

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Kanté, B.

B. Kanté, S. N. Burokur, A. Sellier, A. de Lustrac, and J. M. Lourtioz, “Controlling plasmon hybridization for negative refraction metamaterials,” Phys. Rev. B 79, 075121 (2009).
[CrossRef]

Kästel, J.

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]

Kim, J. Y.

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Kim, K. W.

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

X. R. Jin, J. Park, H. Y. Zheng, S. Lee, Y. Lee, J. Y. Rhee, K. W. Kim, H. S. Cheong, and W. H. Jang, “Highly-dispersive transparency at optical frequencies in planar metamaterials based on two-bright-mode coupling,” Opt. Express 19, 21652–21657 (2011).
[CrossRef]

Koschny, T.

Langguth, L.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[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]

Lederer, F.

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

Lee, B.

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Lee, J. C.

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Lee, S.

Lee, Y.

Lee, Y. P.

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

Li, K.

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

Li, Y. N.

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Liang, E.

Liew, Y. F.

Liu, N.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[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]

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]

Lourtioz, J. M.

B. Kanté, S. N. Burokur, A. Sellier, A. de Lustrac, and J. M. Lourtioz, “Controlling plasmon hybridization for negative refraction metamaterials,” Phys. Rev. B 79, 075121 (2009).
[CrossRef]

Lu, Y. H.

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

Luk’yanchuk, B.

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,” Nature 9, 707–715 (2010).
[CrossRef]

Lukiyanchuk, B.

Maier, S. A.

B. Lukiyanchuk, 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]

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,” Nature 9, 707–715 (2010).
[CrossRef]

Martin, O. J. F.

A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171–2175 (2008).
[CrossRef]

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

Martinez, M. A. G.

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37–41 (2002).
[CrossRef]

Meng, F. Y.

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “An approach to configure low-loss and full transmission metamaterial based on electromagnetically induced transparency,” IEEE Trans. Magn. 48, 4285–4288 (2012).

L. Zhu, F. Y. Meng, J. H. Fu, and Q. Wu, “Electromagnetically induced transparency metamaterial with polarization insensitivity based on multi-quasi-dark modes,” J. Phys. D 45, 445105 (2012).
[CrossRef]

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “Multi-band slow light metamaterial,” Opt. Express 20, 4494–4502 (2012).
[CrossRef]

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

F. Y. Meng, F. Zhang, K. Zhang, and Q. Wu, “Low-loss magnetic metamaterial based on analog of electromagnetically induced transparency,” IEEE Trans. Magn. 47, 3347–3350 (2011).
[CrossRef]

Mesch, M.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[CrossRef]

Morandotti, R.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Ng, B.

Nordlander, P.

B. Lukiyanchuk, 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]

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,” Nature 9, 707–715 (2010).
[CrossRef]

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

Nussenzveig, P.

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37–41 (2002).
[CrossRef]

Oubre, C.

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

Ourir, A.

Ozaki, T.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Papasimakis, N.

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

Park, J.

Pendry, J. B.

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

Pfau, T.

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]

Prodan, E.

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

Prosvirnin, S. L.

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

Pryce, I. M.

Rahmani, M.

Rhee, J. Y.

X. R. Jin, J. Park, H. Y. Zheng, S. Lee, Y. Lee, J. Y. Rhee, K. W. Kim, H. S. Cheong, and W. H. Jang, “Highly-dispersive transparency at optical frequencies in planar metamaterials based on two-bright-mode coupling,” Opt. Express 19, 21652–21657 (2011).
[CrossRef]

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

Rockstuhl, C.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

Rose, M.

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

Rosny, J. d.

Schultz, S.

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

Schweizer, H.

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]

Sellier, A.

B. Kanté, S. N. Burokur, A. Sellier, A. de Lustrac, and J. M. Lourtioz, “Controlling plasmon hybridization for negative refraction metamaterials,” Phys. Rev. B 79, 075121 (2009).
[CrossRef]

Shelby, R. A.

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

Singh, R.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

Skorobogatiy, M.

Smith, D. R.

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

Solak, H. H.

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

Sönnichsen, C.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[CrossRef]

Soukoulis, C. M.

Stockman, M. I.

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

Sun, L. K.

Tassin, P.

Tikhodeev, S. G.

A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171–2175 (2008).
[CrossRef]

A. Christ, Y. Ekinci, H. H. Solak, N. A. Gippius, S. G. Tikhodeev, and O. J. F. Martin, “Controlling the Fano interference in a plasmonic lattice,” Phys. Rev. B 76, 201405(2007).
[CrossRef]

Tsakmakidis, K. L.

K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A 81, 195128 (2010).

Wang, H. T.

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Wang, J.

Wartak, M. S.

K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A 81, 195128 (2010).

Weiss, T.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10, 1103–1107 (2010).
[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]

Wu, P. H.

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Wu, Q.

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “An approach to configure low-loss and full transmission metamaterial based on electromagnetically induced transparency,” IEEE Trans. Magn. 48, 4285–4288 (2012).

L. Zhu, F. Y. Meng, J. H. Fu, and Q. Wu, “Electromagnetically induced transparency metamaterial with polarization insensitivity based on multi-quasi-dark modes,” J. Phys. D 45, 445105 (2012).
[CrossRef]

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “Multi-band slow light metamaterial,” Opt. Express 20, 4494–4502 (2012).
[CrossRef]

F. Y. Meng, F. Zhang, K. Zhang, and Q. Wu, “Low-loss magnetic metamaterial based on analog of electromagnetically induced transparency,” IEEE Trans. Magn. 47, 3347–3350 (2011).
[CrossRef]

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Xue, Q.

Yang, Y. P.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Yun, B.

Zhang, F.

F. Y. Meng, F. Zhang, K. Zhang, and Q. Wu, “Low-loss magnetic metamaterial based on analog of electromagnetically induced transparency,” IEEE Trans. Magn. 47, 3347–3350 (2011).
[CrossRef]

Zhang, K.

F. Y. Meng, F. Zhang, K. Zhang, and Q. Wu, “Low-loss magnetic metamaterial based on analog of electromagnetically induced transparency,” IEEE Trans. Magn. 47, 3347–3350 (2011).
[CrossRef]

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

Zhang, L.

Zhang, W.

S. Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[CrossRef]

Zhang, W. L.

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

Zheludev, N. I.

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,” Nature 9, 707–715 (2010).
[CrossRef]

B. Lukiyanchuk, 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]

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

Zheng, H. Y.

X. R. Jin, Y. H. Lu, H. Y. Zheng, Y. P. Lee, J. Y. Rhee, K. W. Kim, and W. H. Jang, “Plasmonic electromagnetically-induced transparency in metamaterial based on second-order plasmonic resonance,” Opt. Commun. 284, 4766–4768 (2011).
[CrossRef]

X. R. Jin, J. Park, H. Y. Zheng, S. Lee, Y. Lee, J. Y. Rhee, K. W. Kim, H. S. Cheong, and W. H. Jang, “Highly-dispersive transparency at optical frequencies in planar metamaterials based on two-bright-mode coupling,” Opt. Express 19, 21652–21657 (2011).
[CrossRef]

Zhou, Y. J.

Zhu, L.

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “Multi-band slow light metamaterial,” Opt. Express 20, 4494–4502 (2012).
[CrossRef]

L. Zhu, F. Y. Meng, J. H. Fu, and Q. Wu, “Electromagnetically induced transparency metamaterial with polarization insensitivity based on multi-quasi-dark modes,” J. Phys. D 45, 445105 (2012).
[CrossRef]

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “An approach to configure low-loss and full transmission metamaterial based on electromagnetically induced transparency,” IEEE Trans. Magn. 48, 4285–4288 (2012).

Adv. Mater. (1)

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]

Am. J. Phys. (1)

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37–41 (2002).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

M. Kang, Y. N. Li, J. Chen, J. Chen, Q. Bai, H. T. Wang, and P. H. Wu, “Slow light in a simple metamaterial structure constructed by cut and continuous metal strips,” Appl. Phys. B 100, 699–703 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

R. Singh, I. A. I. Al-Naib, Y. P. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. L. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99, 201107 (2011).
[CrossRef]

IEEE Trans. Magn. (2)

L. Zhu, F. Y. Meng, J. H. Fu, Q. Wu, and J. Hua, “An approach to configure low-loss and full transmission metamaterial based on electromagnetically induced transparency,” IEEE Trans. Magn. 48, 4285–4288 (2012).

F. Y. Meng, F. Zhang, K. Zhang, and Q. Wu, “Low-loss magnetic metamaterial based on analog of electromagnetically induced transparency,” IEEE Trans. Magn. 47, 3347–3350 (2011).
[CrossRef]

J. Phys. D (2)

F. Y. Meng, J. H. Fu, K. Zhang, Q. Wu, J. Y. Kim, J. J. Choi, B. Lee, and J. C. Lee, “Metamaterial analogue of electromagnetically induced transparency in two orthogonal directions,” J. Phys. D 44, 265402 (2011).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Unit cell structures for three different metamaterials are displayed in panels (a)–(c). (d) The overall metamaterial structure. The geometric parameters are l1=13mm, l2=12mm, l3=11mm, s1=3.5mm, s2=4mm, s3=4.5mm, d1=1mm and d2=2mm.

Fig. 2.
Fig. 2.

Transmission spectra of three metamaterials displayed in Figs. 1(a)1(c) are shown in left panels (a)–(c). Current density distribution at 11.88 GHz in (a) is shown in right panel (d). Current density distributions at 9.63, 9.89, and 12.01 GHz in (b) are shown in right panels (e)–(g), respectively. Current density distributions at 9.02, 9.10, 9.85, and 9.97 GHz in (c) are shown in right panels (h)–(k), respectively.

Fig. 3.
Fig. 3.

Quality factor and transmission amplitude as a function of (a) the separation distance g, and (b) the length difference between the wires I, II and the wire III. The g refers to the separation distance between the wires I, II, and the wire III as shown in the inset of (a).

Fig. 4.
Fig. 4.

Influence of the variation of the relative permittivity of the surrounding material on the transmission spectrum.

Fig. 5.
Fig. 5.

Simulated absorption power by CST MICROWAVE STUDIO (blue solid curve) and the calculated absorption power by the three-particle analytical model (red dotted–dashed curve).

Equations (5)

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FOM=m(mm/RIU)FWHM(mm),
x¨1(t)+γ1x¨1(t)+ω12x1(t)Ω2x2(t)=Fmeiωt,
x¨2(t)+γ2x¨2(t)+ω22x2(t)Ω2x1(t)Ω2x3(t)=Fmeiωt,
x¨3(t)+γ3x¨3(t)+ω32x3(t)Ω2x2(t)=Fmeiωt,
P(ω)=(iF2ω[(ω12ω2iγ1ω)(ω32ω2iγ3ω)+Ω2((ω12ω2iγ1ω)+(ω32ω2iγ3ω))])/(2m[(ω12ω2iγ1ω)(ω22ω2iγ2ω)(ω32ω2iγ3ω)Ω4((ω12ω2iγ1ω)+(ω32ω2iγ3ω))]).

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