Abstract

Magnetic resonance coupling between connected split ring resonators (SRRs) and magnetic plasmon (MP) excitations in the connected SRR chains were theoretically studied. By changing the connection configuration, two different coupling behaviors were observed, and therefore two kinds of MP bands were formed in the connected ring chains accordingly. From the extracted dispersion properties of MPs, forward and backward characteristics of the guided waves are well exhibited corresponding to the homo- and hetero-connected chains. Notably, thanks to the conductive coupling the revealed MP waves both have wide bandwidth even starting from the zero frequency. These results are suggested to provide instructions to build new kinds of subwavelength waveguides.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  5. W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70,125429 (2004).
    [CrossRef]
  6. W. N. Hardy and L. A. Whitehead, "Split-ring resonator for use in magnetic resonance from 200-2000 MHz," Rev. SCI. Instr. 52,213-216 (1981).
    [CrossRef]
  7. 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]
  8. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  23. T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
    [CrossRef]
  24. T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
    [CrossRef]
  25. F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
    [CrossRef]
  26. N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008).
    [CrossRef]
  27. N. Liu and H. Giessen, "Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling," Opt. Express 16, 21233-21238 (2008).
    [CrossRef] [PubMed]
  28. N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
    [CrossRef]

2009

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
[CrossRef]

2008

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

N. Liu and H. Giessen, "Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling," Opt. Express 16, 21233-21238 (2008).
[CrossRef] [PubMed]

2007

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

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

V. M. Shalaev, "Optical negative-index metamaterial," Nat. Photon. 1, 41-48 (2007).
[CrossRef]

2006

O. Sydoruk, A. Radkovskaya O. Zhuromskyy, E. Shamonina, M. Shamonin,C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, "Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell," Phys. Rev. B 73,224406 (2006).
[CrossRef]

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic plasmon resonance," Phys. Rev. B 73, 036609 (2006).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (2006).
[CrossRef] [PubMed]

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

2005

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005).
[CrossRef]

2004

W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70,125429 (2004).
[CrossRef]

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

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

2003

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

2002

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
[CrossRef]

2001

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

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

2000

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, R16356-R16359 (2000).
[CrossRef]

1999

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]

A. Yariv, Y. Xu. R. K. Lee, and A. Scherer, "Coupled resonator optical waveguides: a proposal and analysis," Opt. Lett. 24,711-713 (1999).
[CrossRef]

1981

W. N. Hardy and L. A. Whitehead, "Split-ring resonator for use in magnetic resonance from 200-2000 MHz," Rev. SCI. Instr. 52,213-216 (1981).
[CrossRef]

Ari, S.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, R16356-R16359 (2000).
[CrossRef]

Baena, J. D.

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

Basov, D. N.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Beruete, M.

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, R16356-R16359 (2000).
[CrossRef]

Dong, Z. G.

Edwards, D. J.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

Enkrich, C.

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

Falcone, F.

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

Fang, N.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Faulkner, G.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

Ford, G. W.

W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70,125429 (2004).
[CrossRef]

Freire, M. J.

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

Genov, D. A.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Giessen, H.

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
[CrossRef]

N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008).
[CrossRef]

N. Liu and H. Giessen, "Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling," Opt. Express 16, 21233-21238 (2008).
[CrossRef] [PubMed]

Hao, T.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

Hardy, W. N.

W. N. Hardy and L. A. Whitehead, "Split-ring resonator for use in magnetic resonance from 200-2000 MHz," Rev. SCI. Instr. 52,213-216 (1981).
[CrossRef]

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, R16356-R16359 (2000).
[CrossRef]

Hesmer, F.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

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]

Kaiser, S.

N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008).
[CrossRef]

Kalinin, V. A.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Koschny, T.

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

Li, J. Q.

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

Li, T.

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (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. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Liu, H.

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
[CrossRef]

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (2006).
[CrossRef] [PubMed]

Liu, N.

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
[CrossRef]

N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008).
[CrossRef]

N. Liu and H. Giessen, "Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling," Opt. Express 16, 21233-21238 (2008).
[CrossRef] [PubMed]

Liu, Y. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

Marques, R.

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

Padilla, W. J.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[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]

Radkovskaya, A. A.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

Requicha, A. A.G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Ringhofer, K. H.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002).
[CrossRef]

Robbins, D. 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]

Sarychev, A. K.

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic plasmon resonance," Phys. Rev. B 73, 036609 (2006).
[CrossRef]

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] [PubMed]

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterial," Nat. Photon. 1, 41-48 (2007).
[CrossRef]

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic plasmon resonance," Phys. Rev. B 73, 036609 (2006).
[CrossRef]

Shamonin, M.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

Shamonina, E.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002).
[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] [PubMed]

Shvets, G.

A. K. Sarychev, G. Shvets, and V. M. Shalaev, "Magnetic plasmon resonance," Phys. Rev. B 73, 036609 (2006).
[CrossRef]

Smith, D. R.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

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

Solymar, L.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002).
[CrossRef]

Solymara, L.

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005).
[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. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Steele, J. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Stevens, C. J.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[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]

Sun, C.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Sydoruk, O.

O. Sydoruk, A. Radkovskaya O. Zhuromskyy, E. Shamonina, M. Shamonin,C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, "Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell," Phys. Rev. B 73,224406 (2006).
[CrossRef]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005).
[CrossRef]

Tatartschuk, E.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

Vier, D. C.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Wang, F. M.

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (2006).
[CrossRef] [PubMed]

Wang, F.M.

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

Wang, Q. J.

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

Wang, S. M.

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

Weber, W. H.

W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70,125429 (2004).
[CrossRef]

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. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Whitehead, L. A.

W. N. Hardy and L. A. Whitehead, "Split-ring resonator for use in magnetic resonance from 200-2000 MHz," Rev. SCI. Instr. 52,213-216 (1981).
[CrossRef]

Wu, D. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Xu, Y.

Yariv, A.

Yen, T. Y.

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Zhang, X.

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (2006).
[CrossRef] [PubMed]

T. Y. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Zhou, J. F.

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

Zhu, S. N.

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
[CrossRef]

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Magnetic plasmon modes in periodic chains of nanosandwiches," Opt. Express 16,3560-3565 (2008).
[CrossRef] [PubMed]

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, "Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies," Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, "Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission," Opt. Express 14, 11155-11163 (2006).
[CrossRef] [PubMed]

Zhu, S.N.

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

Zhu, Y. Y.

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

Zhuromskyy, O.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, "Coupling mechanisms for split ring resonators: Theory and experiment," Phys. Stat. Sol. (B) 244 (4), 1170-1175 (2007).
[CrossRef]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005).
[CrossRef]

Adv. Mater.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, AriA. G. Requicha, and H. A. Atwater, "Plasmonics - a route to nanoscale optical devices," Adv. Mater. 13,1501 (2001).
[CrossRef]

N. Liu, S. Kaiser, and H. Giessen, "Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules," Adv. Mater. 20, 1-5 (2008).
[CrossRef]

Appl. Phys. Lett.

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, "Selective switch made from a graded anosandwich chain," Appl. Phys. Lett. 93,233102 (2008).
[CrossRef]

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, "Electroinductive waves in chains of complementary metamaterial elements," Appl. Phys. Lett. 88,083503 (2006).
[CrossRef]

T. Li, J. Q. Li, F.M. Wang, Q. J. Wang, H. Liu, S.N. Zhu, and Y. Y. Zhu, "Exploring magnetic plasmon polaritons in optical transmission through hole arrays perforated in trilayer structures", Appl. Phys. Lett. 90, 251112 (2007).
[CrossRef]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymara, "Phonon-like dispersion curves of magnetoinductive waves," Appl. Phys. Lett. 87,072501 (2005).
[CrossRef]

Electron. lett.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magneto-inductive waveguide," Electron. lett. 38,371-373 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

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]

J. Appl. Phys.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, "Magnetoinductive waves in one, two, and three dimensions," J. Appl. Phys. 92, 6252-6261 (2002).
[CrossRef]

T. Li, S. M. Wang, H. Liu, J. Q. Li, F. M. Wang, S. N. Zhu, and X. Zhang, "Dispersion of magnetic plasmon polaritons in perforated trilayer metamaterials," J. Appl. Phys. 103,023104 (2008).
[CrossRef]

Nat. Mater.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A.G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Nat. Photon.

V. M. Shalaev, "Optical negative-index metamaterial," Nat. Photon. 1, 41-48 (2007).
[CrossRef]

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, "Stereometamaterials," Nat. Photon. 3, 157-162 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

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Supplementary Material (1)

» Media 1: MPG (445 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) Scheme of the geometry of single SRR with the parameters marked on. (b) and (c) are the connect SRR pairs with homo- and hetero-connection configurations, respectively.

Fig. 2.
Fig. 2.

The magnetic field amplitude intensity detected at the center of the resonators of a single SRR (black curve) and homo-connected SRR pair (red curve) for (a), and hetero-connected pair for (b); The insets correspond to the magnetic field maps for the split higher and lower modes, respectively.

Fig. 3.
Fig. 3.

Equivalent circuits of the two SRR pairs with different connection configurations: (a) homo-connected and (b) hetero-connected; Effective configurations of the magnetic and electric dipoles corresponding to the split eigen modes for these two connections, (c) homo-connection and (d) hetero-connection.

Fig. 4.
Fig. 4.

Subwavelength waveguides constituted by the SRR chains with (a) homo-connection and (b) amti-connection. (c) and (d) are the calculated corresponding magnetic field map in the ω-k space with respect to the waveguides (a) and (b) respectively, in which dispersions of MP waves propagating with the waveguides are manifested clearly. (Media 1)

Fig. 5.
Fig. 5.

Magnetic field distributions along homo-connected SRR chain at different frequencies, where the dipole source is placed at the left side.

Equations (7)

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=12L(Q˙12+Q˙22)12Lω02(Q12+Q22)+12L(Q˙1Q˙2)2MmQ˙1Q˙2Meω02Q1Q2
{(L+L)Q̈1+Lω02Q1(Mm+L)Q̈2+M2ω02Q2=0(Mm+L)Q̈1+Meω02Q1+(L+L)Q̈2+Lω02Q2=0·
{ω1=ω01Ke1+2η+Km,withQ1=Q2.ω2=ω01+Ke1Km,withQ1=Q2
ω+=ω01Ke11+2η+Km, ω=ω01+Ke11Km,
ω+=ω01+Ke21Km, ω=ω01Ke21+2η+Km,
Edipole=d1·d2r33(d1·r)(d2·r)r5,
H(ω,k)=H(ω,x)eikxdx·

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