Abstract

Plasmonic structures with high symmetry, such as double-identical gap split ring resonators, possess dark eigenmodes. These dark eigenmodes are dominated by magnetic dipole and/or higher-order multi-poles such as electric quadrapoles. Consequently these dark modes interact very weakly with the surrounding environment, and can have very high quality factors (Q). In this work, we have studied, experimentally as well as theoretically, these dark eigenmodes in terahertz metamaterials. Theoretical investigations with the help of classical perturbation theory clearly indicate the existence of these dark modes in symmetric plasmonic metamaterials. However, these dark modes can be excited experimentally by breaking the symmetry within the constituting metamaterial resonators cell, resulting in high quality factor resonance mode. The symmetry broken metamaterials with such high quality factor can pave the way in realizing high sensitivity sensors, in addition to other applications.

© 2014 Optical Society of America

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2013 (1)

D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

2012 (4)

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
[CrossRef]

Y. Zeng, D. A. R. Dalvit, J. O’Hara, and S. A. Trugman, “A modal analysis method to describe weak nonlinear effects in metamaterials,” Phys. Rev. B 85, 125107 (2012).
[CrossRef]

W. Cao, R. Singh, I. A. I. Al-Naib, M. He, A. J. Taylor, and W. Zhang, “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37, 3366–3368 (2012).
[CrossRef]

2011 (4)

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

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

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

M. T. Reiten, D. Roy Chowdhury, J. Zhou, A. J. Taylor, J. O’Hara, and A. K. Azad, “Resonance tuning behavior in closely spaced inhomogeneous bilayer metamaterials,” Appl. Phys. Lett. 98, 131105 (2011).
[CrossRef]

2010 (4)

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

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257 (2010).
[CrossRef]

A. Raman and S. Fan, “Photonic band structure of dispersive meta-materials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett. 104, 087401 (2010).
[CrossRef]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

2009 (2)

O. Sydoruk, E. Tarartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105, 014903 (2009).
[CrossRef]

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. 94, 131901 (2009).
[CrossRef]

2008 (5)

2007 (2)

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

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

2006 (3)

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[CrossRef]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

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

2004 (3)

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

N. 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 (2004).
[CrossRef]

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

2002 (1)

R. D. Averitt and A. J. Taylor, “Ultrafast optical and far infra-red quasiparticle dynamics in correlated electron materials,” J. Phys.: Cond. Matter. 14, R1357 (2002).

2001 (1)

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

2000 (2)

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 (2000).
[CrossRef] [PubMed]

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

1999 (1)

J. B. 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]

1990 (1)

1968 (1)

V. G. Vaselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Adato, R.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
[CrossRef]

Ågren, H.

Al-Naib, I.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

Al-Naib, I. A. I.

W. Cao, R. Singh, I. A. I. Al-Naib, M. He, A. J. Taylor, and W. Zhang, “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37, 3366–3368 (2012).
[CrossRef]

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

Altissimo, M.

D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

Altug, H.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
[CrossRef]

Arju, N.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
[CrossRef]

Averitt, R. D.

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for terahertz regime: Design fabrication and characterization,” Opt. Express 16, 7181–7188 (2008).
[CrossRef] [PubMed]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

R. D. Averitt and A. J. Taylor, “Ultrafast optical and far infra-red quasiparticle dynamics in correlated electron materials,” J. Phys.: Cond. Matter. 14, R1357 (2002).

Azad, A. K.

M. T. Reiten, D. Roy Chowdhury, J. Zhou, A. J. Taylor, J. O’Hara, and A. K. Azad, “Resonance tuning behavior in closely spaced inhomogeneous bilayer metamaterials,” Appl. Phys. Lett. 98, 131105 (2011).
[CrossRef]

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

Bengtsson, M.

Bingham, C. M.

Boltasseva, A.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

Boreman, G. D.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

Brener, I.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Opt. Express 16, 1786–1795 (2008).
[CrossRef] [PubMed]

Cao, W.

W. Cao, R. Singh, I. A. I. Al-Naib, M. He, A. J. Taylor, and W. Zhang, “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37, 3366–3368 (2012).
[CrossRef]

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

Chaker, M.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

Chen, H. T.

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

Chen, X. S.

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[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,” Nat. Mat. 9, 707–715 (2010).
[CrossRef]

Chowdhury, D. R.

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

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

Coffey, K. R.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

Cole, R.

R. Cole, Theory of Ordinary Differential Equations (Appleton-Century-Crofts, New York, 1968).

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[CrossRef]

Dalvit, D. A. R.

Y. Zeng, D. A. R. Dalvit, J. O’Hara, and S. A. Trugman, “A modal analysis method to describe weak nonlinear effects in metamaterials,” Phys. Rev. B 85, 125107 (2012).
[CrossRef]

Davis, T. J.

D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

Delprat, S.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

Earl, S.

D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

Economou, E. N.

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 (2004).
[CrossRef]

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Emani, N. K.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

Enkrich, C.

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]

Etrich, C.

Fan, S.

A. Raman and S. Fan, “Photonic band structure of dispersive meta-materials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett. 104, 087401 (2010).
[CrossRef]

Fattinger, C.

Fedotov, V. A.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. 94, 131901 (2009).
[CrossRef]

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

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A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257 (2010).
[CrossRef]

Fu, Y.

Giessen, H.

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

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

Ginn, J. C.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

Gomez, D. E.

D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

Gossard, A. C.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

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A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

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Halas, N. J.

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

Han, J.

He, M.

Holden, A. J.

J. B. 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]

Ishii, S.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

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I. A. I. Naib, C. Jansen, and M. Koch, “Thin film sensing with planar asymetric metamaterial resonators,” Appl. Phys. Lett. 93, 083507 (2008).
[CrossRef]

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R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[CrossRef]

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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 (2004).
[CrossRef]

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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 (2004).
[CrossRef]

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Khanikaev, A. B.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
[CrossRef]

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A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257 (2010).
[CrossRef]

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I. A. I. Naib, C. Jansen, and M. Koch, “Thin film sensing with planar asymetric metamaterial resonators,” Appl. Phys. Lett. 93, 083507 (2008).
[CrossRef]

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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 (2004).
[CrossRef]

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]

Kuehl, J.

Landy, N. I.

Lederer, F.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

R. Singh, C. Rocksuhl, F. Lederer, and W. Zhang, “Coupling between a dark and a bright eigenmode in a terahertz metamaterial,” Phys. Rev. B. 79, 085111 (2008).
[CrossRef]

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

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[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]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[CrossRef]

Lu, W.

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

Maier, S. A.

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

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257 (2010).
[CrossRef]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[CrossRef]

Morandotti, R.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

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

Naib, I. A. I.

I. A. I. Naib, C. Jansen, and M. Koch, “Thin film sensing with planar asymetric metamaterial resonators,” Appl. Phys. Lett. 93, 083507 (2008).
[CrossRef]

Naik, G. V.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

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M. Naimark, Linear Differential Operators, Part 1, W. Everitt, ed. (Ungar, New York, 1968).

Nemat-Nasser, S. C.

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 (2000).
[CrossRef] [PubMed]

Nordlander, P.

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

O’Hara, J.

Y. Zeng, D. A. R. Dalvit, J. O’Hara, and S. A. Trugman, “A modal analysis method to describe weak nonlinear effects in metamaterials,” Phys. Rev. B 85, 125107 (2012).
[CrossRef]

M. T. Reiten, D. Roy Chowdhury, J. Zhou, A. J. Taylor, J. O’Hara, and A. K. Azad, “Resonance tuning behavior in closely spaced inhomogeneous bilayer metamaterials,” Appl. Phys. Lett. 98, 131105 (2011).
[CrossRef]

O’Hara, J. F.

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Opt. Express 16, 1786–1795 (2008).
[CrossRef] [PubMed]

Ozaki, T.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

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

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H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for terahertz regime: Design fabrication and characterization,” Opt. Express 16, 7181–7188 (2008).
[CrossRef] [PubMed]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

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 (2000).
[CrossRef] [PubMed]

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 brocken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

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

J. B. 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]

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D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

Plum, E.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Planar metamaterial with transmission and reflection that depend on the direction of incidence,” Appl. Phys. Lett. 94, 131901 (2009).
[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 brocken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

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A. Raman and S. Fan, “Photonic band structure of dispersive meta-materials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett. 104, 087401 (2010).
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M. T. Reiten, D. Roy Chowdhury, J. Zhou, A. J. Taylor, J. O’Hara, and A. K. Azad, “Resonance tuning behavior in closely spaced inhomogeneous bilayer metamaterials,” Appl. Phys. Lett. 98, 131105 (2011).
[CrossRef]

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J. B. 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]

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D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

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I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

Rockstuhl, C.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

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

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

Rocksuhl, C.

R. Singh, C. Rocksuhl, F. Lederer, and W. Zhang, “Coupling between a dark and a bright eigenmode in a terahertz metamaterial,” Phys. Rev. B. 79, 085111 (2008).
[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 brocken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

Roy Chowdhury, D.

M. T. Reiten, D. Roy Chowdhury, J. Zhou, A. J. Taylor, J. O’Hara, and A. K. Azad, “Resonance tuning behavior in closely spaced inhomogeneous bilayer metamaterials,” Appl. Phys. Lett. 98, 131105 (2011).
[CrossRef]

Schultz, S.

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

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 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

Shamonina, E.

O. Sydoruk, E. Tarartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105, 014903 (2009).
[CrossRef]

Shelby, R. A.

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

Shelton, D. J.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

Shvets, G.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
[CrossRef]

Sinclair, M. B.

D. J. Shelton, I. Brener, J. C. Ginn, M. B. Sinclair, D. W. Peters, K. R. Coffey, and G. D. Boreman, “Strong coupling between nanoscale metamaterials and phonons,” Nano Lett. 11, 2104–2108 (2011).
[CrossRef] [PubMed]

Singh, R.

I. Al-Naib, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phy. Lett. 101, 071108 (2012).
[CrossRef]

W. Cao, R. Singh, I. A. I. Al-Naib, M. He, A. J. Taylor, and W. Zhang, “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37, 3366–3368 (2012).
[CrossRef]

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

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

R. Singh, C. Rocksuhl, F. Lederer, and W. Zhang, “Coupling between a dark and a bright eigenmode in a terahertz metamaterial,” Phys. Rev. B. 79, 085111 (2008).
[CrossRef]

J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Opt. Express 16, 1786–1795 (2008).
[CrossRef] [PubMed]

Smirnova, E.

smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2006).
[CrossRef]

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

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 (2000).
[CrossRef] [PubMed]

Solymar, L.

O. Sydoruk, E. Tarartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105, 014903 (2009).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

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

N. 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 (2004).
[CrossRef]

Stewart, W. J.

J. B. 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]

Sydoruk, O.

O. Sydoruk, E. Tarartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105, 014903 (2009).
[CrossRef]

Tao, H.

Tarartschuk, E.

O. Sydoruk, E. Tarartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105, 014903 (2009).
[CrossRef]

Taylor, A. J.

W. Cao, R. Singh, I. A. I. Al-Naib, M. He, A. J. Taylor, and W. Zhang, “Low-loss ultra-high-Q dark mode plasmonic Fano metamaterials,” Opt. Lett. 37, 3366–3368 (2012).
[CrossRef]

M. T. Reiten, D. Roy Chowdhury, J. Zhou, A. J. Taylor, J. O’Hara, and A. K. Azad, “Resonance tuning behavior in closely spaced inhomogeneous bilayer metamaterials,” Appl. Phys. Lett. 98, 131105 (2011).
[CrossRef]

D. R. Chowdhury, R. Singh, J. F. O’Hara, H. T. Chen, A. J. Taylor, and A. K. Azad, “Dynamically reconfigurable terahertz metamaterials through photo doped semiconductors,” Appl. Phys. Lett. 99, 231101 (2011).
[CrossRef]

J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Opt. Express 16, 1786–1795 (2008).
[CrossRef] [PubMed]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
[CrossRef] [PubMed]

R. D. Averitt and A. J. Taylor, “Ultrafast optical and far infra-red quasiparticle dynamics in correlated electron materials,” J. Phys.: Cond. Matter. 14, R1357 (2002).

Teo, Z. Q.

D. E. Gomez, Z. Q. Teo, M. Altissimo, T. J. Davis, S. Earl, and A. Roberts, “The drak side of plasmonics,” Nano Lett. 13, 3722–3728 (2013).
[CrossRef]

Trugman, S. A.

Y. Zeng, D. A. R. Dalvit, J. O’Hara, and S. A. Trugman, “A modal analysis method to describe weak nonlinear effects in metamaterials,” Phys. Rev. B 85, 125107 (2012).
[CrossRef]

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Vaselago, V. G.

V. G. Vaselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

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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 (2000).
[CrossRef] [PubMed]

Wegener, M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[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]

West, P. R.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[CrossRef]

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C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymetric metamaterials for ultrasensitive spectroscopy and identification of moleculer monolayers,” Nat. Mat. 11, 69–75 (2012).
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R. Singh, I. A. I. Al-Naib, Y. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phy. Lett. 99, 201107 (2011).
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H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz meta-material devices,” Nature 444, 597–600 (2006).
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Nature (1)

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Figures (5)

Fig. 1
Fig. 1

The optical images of (a) MM1, (b) MM2, (c) MM3, and (d) an array of metamaterial sample MM3. The lengths of metal arms are 60 μm in both the x- and y- directions. Width of the split gap (g) and metal arms (w) are 16 μm and 8 μm, respectively. Px and Py are periodicity in the x and y directions, respectively with value as 88 μm. d is the plate length of the split gap in one arm which is 16 μm for MM2 and 24 μm for MM3.

Fig. 2
Fig. 2

Measured and simulated transmission spectra for (a) MM1, (b) MM2 and (c) MM3 with electric field parallel to the split gaps, and for (d) MM1, (e) MM2 and (f) MM3 with electric field perpendicular to the split gap.

Fig. 3
Fig. 3

(a) The electric field distributions (|E x |, |E y |) for the ee eigenmode of MM1 structure, excited by a plane wave whose electric field polarization is parallel to the split gap (along the x direction). When week asymmetry is introduced, this mode is modified to different frequency and slightly different field distributions. (b) and (c) shows the corresponding field for the asymmetrical MM2 and MM3 structures.

Fig. 4
Fig. 4

(a) The electric field distributions (|E x |, |E y |) for the oo eigenmode of MM1 structure, excited by a plane wave whose electric field polarization is along the y direction. When week asymmetry is introduced, this mode is modified to different frequency and slightly different field distributions. (b) and (c) shows the corresponding field for the asymmetrical MM2 and MM3 structures.

Fig. 5
Fig. 5

The eo eigenmode: (a) The electric field distributions (|E x |, |E y |) for the eo eigen-mode of MM1 structure, which cannot be excited by a plane wave whose electric field polarization is along the y direction. However, it can be excited when weak asymmetry is introduced. (b) and (c) depict the corresponding field distribution for the asymmetrical MM2 and MM3 structures. Due to different excitation source, one can not compare directly the electric field strength shown in (a) with that shown in (b) and (c).

Equations (17)

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ω u = u ,
λ u = u .
E x ( x , y , z ) = E x ( x , y , z ) = E x ( x , y , z ) , E y ( x , y , z ) = E y ( x , y , z ) = E y ( x , y , z ) .
E x ( x , y , z ) = E x ( x , y , z ) = E x ( x , y , z ) , E y ( x , y , z ) = E y ( x , y , z ) = E y ( x , y , z ) .
E x ( x , y , z ) = E x ( x , y , z ) = E x ( x , y , z ) , E y ( x , y , z ) = E y ( x , y , z ) = E y ( x , y , z ) .
E x ( x , y , z ) = E x ( x , y , z ) = E x ( x , y , z ) , E y ( x , y , z ) = E y ( x , y , z ) = E y ( x , y , z ) .
u 1 , e e = , u 1 , e o = , u 1 , o e = , u 1 , o o = .
ω v = ( + 𝒳 ) v ,
E x ( x , y , z ) = E x ( x , y , z ) , E y ( x , y , z ) = E y ( x , y , z ) ,
E x ( x , y , z ) = E x ( x , y , z ) , E y ( x , y , z ) = E y ( x , y , z ) .
v = n [ α n , e e u n , e e + α n , o e u n , o e + α u , e o u n , e o + α u , o o u n , o o ] .
ω ( α 1 , e e α 1 , o e α 1 , e o α 1 , o o ) = ( ω 1 , e e ω 1 , o e ω 1 , e o ω 1 , o o ) ( α 1 , e e α 1 , o e α 1 , e o α 1 , o o ) + X ( α 1 , e e α 1 , o e α 1 , e o α 1 , o o ) .
v m , e = n [ α n , e e m u n , e e + α n , o e m u n , o e ] , v m , o = n [ α n , e o m u n , e o + α n , o o m u n , o o ] .
ω V = ( + 𝒳 ) V + S ,
V ( r , ω ) = m v m | S ω Ω m v m ( r ) ,
m v m , e | S ω Ω m , e v m , e ( r ) = m v m , e ω Ω m , e [ n α n , e e m , * u n , e e | S ] .
m v m , o | S ω Ω m , o v m , o ( r ) = m v m , o ω Ω m , o [ n α n , o o m , * u n , o o | S ] .

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