Abstract

We experimentally studied the coupling between a double split ring resonator and a complementary split ring resonator. The greatest coupling occurs when the two resonators are separated by the average ring radius, and the dimensionless coupling is as large as 0.1, allowing a novel planar metamaterial based on this hybrid structure. The coupling strength can be varied up to a factor of 2 by changing the relative orientation of the split ring resonators. A 2×2 waveguide structure with −10 dB coupling factor can be achieved, and showing multi-mode plasmon-induced transparency. It can be considered one-dimensional metamaterials exhibiting negative permeability and permittivity simultaneously.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (1)

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

2015 (1)

M. Parvinnezhad Hokmabadi, E. Philip, E. Rivera, P. Kung, and S. M. Kim, “Plasmon-induced transparency by hybridizing concentric-twisted double split ring resonators,” Sci. Rep. 5, 15735 (2015).
[Crossref] [PubMed]

2014 (1)

F. Miyamaru, H. Morita, Y. Nishiyama, T. Nishida, T. Nakanishi, M. Kitano, and M. W. Takeda, “Ultrafast optical control of group delay of narrow-band terahertz waves,” Sci. Rep. 4, 4346 (2014).
[Crossref] [PubMed]

2013 (4)

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

2012 (3)

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012).
[Crossref]

2011 (5)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, and E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84, 125121 (2011).
[Crossref]

C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

2010 (1)

S. Alexander and N. Stefan, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B At. Mol. Opt. Phys. 43, 163001 (2010).
[Crossref]

2009 (1)

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

2008 (2)

2007 (3)

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. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photon. 1, 41–48 (2007).
[Crossref]

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

2006 (1)

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

2005 (3)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
[Crossref]

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[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]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

2003 (1)

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

2001 (1)

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

1991 (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE. 79, 1505–1518 (1991).
[Crossref]

Alexander, S.

S. Alexander and N. Stefan, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B At. Mol. Opt. Phys. 43, 163001 (2010).
[Crossref]

Aydin, K.

Baena, J. D.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Beruete, M.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Bonache, J.

M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials 2, 186–197 (2008).
[Crossref]

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic metamaterials: transmission line theory and microwave applications (John Wiley & Sons, 2005).

Cao, C.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Chen, J.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

Cong, L.

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Economou, E. N.

Edwards, D. J.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, and E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84, 125121 (2011).
[Crossref]

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. Status Solidi B 244, 1170–1175 (2007).
[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]

Falcone, F.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

Fan, S.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Fang, K.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

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. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

Fedotov, V. A.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Fu, Y. H.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

Garcia-Garcia, J.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

Gil, I.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

Gil, M.

M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials 2, 186–197 (2008).
[Crossref]

Gneiding, N.

E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012).
[Crossref]

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, and E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84, 125121 (2011).
[Crossref]

Hafezi, M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

Han, S.

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[Crossref] [PubMed]

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[Crossref]

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[Crossref] [PubMed]

Totsuka, K.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

Tsai, D. P.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

Tunnermann, A.

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[Crossref]

Umucalilar, R. O.

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Wan, J.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Wegener, M.

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]

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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

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C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
[Crossref] [PubMed]

Wu, J.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Xiao, B.

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Yang, H.

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

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M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
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J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Zheludev, N. I.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

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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]

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[Crossref]

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
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M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials 2, 186–197 (2008).
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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
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A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, and E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84, 125121 (2011).
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C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
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[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 (a) A photograph of a “face-to-back” coupled DSRR sample. The scale bar is 3 mm. (b) The schematic of the moving platform for the coupled DSRR and CSRR. The metal and bare PCB regions are respectively drawn in yellow and blue colors. Our setup allows us to move the lower layer in respected to the upper layer. The two layers are firmly contacted to each other during the measurement. (c) Circuit model for capacitively and inductively coupled SRRs. (d) The transmission data of the “face-to-face” coupled DSRRs. d varies from 7.7 mm to 8.2 mm, with a step of 0.1 mm, corresponding to gap between DSRRs from 0.1 mm to 0.6 mm. Two distinct resonances appear owing to the large inter-resonator coupling. When d increases to 8.2 mm, the two resonance dips gradually merge into a single dip. (e) The frequency difference fHfL of two resonance dips in (d) as a function of d for the “face-to-face”(red symbol) and the “face-to-back”(blue symbol) samples. At d = 7.7 mm, the “face-to-face” configuration gives a larger fHfL, while at d = 8.2 mm, “face-to-back” configuration results in a larger fHfL.
Fig. 2
Fig. 2 (a) The transmission data of coupled DSRR/CSRR measurement for 3 different positions, (x /mm, y/mm)=(0,0), (0, −2), and (0, −6). At the last position, two resonance absorption are clear. When the coordinates move toward +y direction, the CSRR is moved away from the microstrip and the resonance of lower frequency, to which the CSRR corresponds is getting smeared. (b) A mapping result of fD. (c) fD and fC as a function of x at y = 0 mm. (d) The coupling strengths 2gC and 2gM as a function of x at y = 0 mm. γ = 0.2 is assumed for calculation of 2gM. The maximum coupling strength is at x = 3 mm.
Fig. 3
Fig. 3 (a) The definition of orientation parameters θD and θC. (b)(c) The dimensionless coupling constants 2gC and 2gM for coupled DSRR and CSRR in various orientation configurations. Again, γ = 0.2 is assumed. The configurations is presented in (θD, θC). In (a) the 1-layer and 2-layer simulation data are multiplied by factors of 0.38 and 0.5, respectively. In (c) the factors are 0.6 and 0.44. (d) The polar plot for 2gC (red symbol) and 2gM (blue symbol) at various of θC when θD = π/2 done by 1-layer simulation. The largest coupling 2gC occurs at θC ∼ 0 and the smallest at θCπ. The angle of the minimum 2gC and maximum 2gM are the same. The red curve is a function plot 2gC = 0.088 + 0.014 cos θC.
Fig. 4
Fig. 4 The S-parameters in inter-TL coupler experiment. Transmission lines coupled by 2 DSRRs(C1) (a), 3 CSRRs(C2) (b), and interleaved DSRRs and CSRRs(C3) (c). Because of the large d between DSRRs in C1 and CSRRs in C2, resonators are not coupled to each other. C3 structure gives a inter-TL coupling than the first two. In (c) the solid curves are experimental data and the dashed curves are simulation results. (d) The schematic of C3 and the measurement ports.
Fig. 5
Fig. 5 (a)(b) Amplitude(a) and phase(b) of S21 for C3 near the resonance regime. It demonstrates a PIT spectrum with 5 modes; clear dips were found at 2740, 2865, 3027 and 3324 MHz, with an unclear one at 3220 MHz. (c) Circuit model for coupled SRR array.

Equations (16)

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Ω = ω 1 + ω 2 2 ± ( ω 1 ω 2 ) 2 2 + | κ | 2 ,
Ω 2 π = f 0 ( 1 ± g ) .
α 1 + α 2 ~ 1 ( Ω 1 Ω 2 ω 1 ω 2 ) 2 ~ 2 ( δ ω 1 ω 1 + δ ω 2 ω 2 ) .
Ω 1 2 Ω 2 2 = [ ( 1 α 2 ) ω 2 2 ( 1 α 1 ) ω 1 2 ] 2 + 4 γ ω 1 2 ( α 1 β 2 ) ( ω 2 2 α 1 ω 1 2 β 2 )
2 ω 1 2 β 2 ~ α 1 ( ω 1 2 + ω 2 2 ) ± 4 α 1 2 Δ 4 + γ 1 [ 4 Δ Ω 2 ( 2 Δ 2 α 1 ω 1 2 α 2 ω 2 2 ) 2 ]
U ( d ) = d τ E 1 ( r 1 ) E 2 ( r 2 ) ~ c 1 d τ E 1 ( r 1 ) B 1 ( r 2 ) .
= j = 1 N L j q ˙ j 2 2 ( q i q x , j + q x , j 1 ) 2 2 C j j = 1 N 1 q x , j 2 C x , j .
( ( 1 α 1 ) ω 1 2 ω 2 α 2 0 α 1 ( 1 2 α 2 ) ω 2 2 ω 2 α 1 0 α 2 ( 1 2 α 1 ) ω 1 2 ω 2 ) ( q ˜ 1 q ˜ 2 q ˜ 3 ) = 0
ω = ω 1 ( 1 α 1 α 2 ) ± α 1 2 + α 2 2 + 2 α 1 α 2 cos k d .
= j = 1 , 2 L j q ˙ j 2 2 + M q ˙ 1 q ˙ 2 ( q 1 q x ) 2 2 C 1 ( q 2 + q x ) 2 2 C 2 q x 2 C x .
d 2 d t 2 ( q 1 q 2 ) = ( L 1 M M L 2 ) 1 ( C 1 1 ( 1 C p C 1 1 ) C p C 1 1 C 2 1 C p C 1 1 C 2 1 C 2 1 ( 1 C p C 2 1 ) ) ( q 1 q 2 ) = 1 1 β 1 β 2 ( ( 1 α 1 ) ω 1 2 α 1 β 1 ω 2 2 ( 1 α 2 ) β 1 ω 2 2 + α 2 ω 1 2 ( 1 α 1 ) β 2 ω 1 2 + α 1 ω 2 2 ( 1 α 2 ) ω 2 2 α 2 β 2 ω 1 2 ) ( q 1 q 2 )
| ( 1 α 1 ) ω 1 2 α 1 β 1 ω 2 2 ( 1 β 1 β 2 ) Ω 2 ( 1 α 2 ) β 1 ω 2 2 + α 2 ω 1 2 ( 1 α 1 ) β 2 ω 1 2 + α 1 ω 2 2 ( 1 α 2 ) ω 2 2 α 2 β 2 ω 1 2 ( 1 β 1 β 2 ) Ω 2 | = 0
Ω 2 ω 1 2 = ( 1 α 1 + α 2 2 ) ± ( α 1 α 2 2 ) 2 + ( α 1 β 2 ) ( α 2 β 1 ) .
Ω = ω 1 1 α 1 ± | α 1 β 1 |
Ω 1 Ω 2 ω 1 ω 2 = 1 α 1 α 2 1 β 1 β 2 ~ 1 1 2 ( α 1 + α 2 ) = 1 2 g C
2 g C ~ 1 2 ( δ ω 1 ω 1 + δ ω 2 ω 2 ) .

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