Abstract

To better understand the resonance modes caused by the interelement couplings in the building block of metamaterials, we propose a circuit model for the hybrid resonance modes of paired split ring resonators. The model identifies the electromagnetic coupling between the paired rings by electric and magnetic coupling networks and well explains the variation of hybrid resonance modes with respect to the distance and the twist angle between the rings. The predictions of our model are further proved by experiments.

© 2014 Optical Society of America

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    [CrossRef]

2011

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

S. V. Zhukovsky, C. Kremers, D. N. Chigrin, “Plasmonic rod dimers as elementary planar chiral meta-atoms,” Opt. Lett. 36(12), 2278–2280 (2011).
[CrossRef] [PubMed]

2010

S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010).
[CrossRef]

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

2009

N. Liu, H. Liu, S. Zhu, H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

2008

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008).
[CrossRef] [PubMed]

2007

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

2006

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

L. Zhou, S. T. Chui, “Eigenmodes of metallic ring systems: a rigorous approach,” Phys. Rev. B 74(3), 035419 (2006).
[CrossRef]

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

V. G. Veselago, E. E. Narimanov, “The left hand of brightness: past, present and future of negative index materials,” Nat. Mater. 5(10), 759–762 (2006).
[CrossRef] [PubMed]

2005

M. J. Freire, R. Marques, “Planar magnetoinductive lens for threedimensional subwavelength imaging,” Appl. Phys. Lett. 86(18), 182505 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

2003

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

2002

R. Marqués, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002).
[CrossRef]

2001

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

1990

D. K. Ghodgaonkar, V. V. Varadan, V. K. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39(2), 387–394 (1990).
[CrossRef]

Aydin, K.

Bao, Y.-J.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Brandl, D. W.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

Burger, S.

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

Chen, P.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

Chigrin, D. N.

Chui, S. T.

L. Zhou, S. T. Chui, “Eigenmodes of metallic ring systems: a rigorous approach,” Phys. Rev. B 74(3), 035419 (2006).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Dolling, G.

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

Economou, E. N.

Edwards, D. J.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Engelbrecht, S.

S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Freire, M. J.

M. J. Freire, R. Marques, “Planar magnetoinductive lens for threedimensional subwavelength imaging,” Appl. Phys. Lett. 86(18), 182505 (2005).
[CrossRef]

Genov, D. A.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Ghodgaonkar, D. K.

D. K. Ghodgaonkar, V. V. Varadan, V. K. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39(2), 387–394 (1990).
[CrossRef]

Giessen, H.

N. Liu, H. Liu, S. Zhu, H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Gneiding, N.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Halas, N. J.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kafesaki, M.

Koschny, T.

Kremers, C.

Le, F.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

Lee, H.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Li, T.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

Li, T. Q.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

Li, Z.-F.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Linden, S.

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

Liu, H.

N. Liu, H. Liu, S. Zhu, H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Liu, N.

N. Liu, H. Liu, S. Zhu, H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Liu, Y. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Liu, Z. W.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

Lu, X.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Marques, R.

M. J. Freire, R. Marques, “Planar magnetoinductive lens for threedimensional subwavelength imaging,” Appl. Phys. Lett. 86(18), 182505 (2005).
[CrossRef]

Marqués, R.

R. Marqués, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002).
[CrossRef]

Medina, F.

R. Marqués, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002).
[CrossRef]

Ming, N.-B.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Narimanov, E. E.

V. G. Veselago, E. E. Narimanov, “The left hand of brightness: past, present and future of negative index materials,” Nat. Mater. 5(10), 759–762 (2006).
[CrossRef] [PubMed]

Nordlander, P.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Ozbay, E.

Penciu, R. S.

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Peng, R.-W.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Pimenov, A.

S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010).
[CrossRef]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Radkovskaya, A.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Rafii-El-Idrissi, R.

R. Marqués, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002).
[CrossRef]

Schadle, A.

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Shamonina, E.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Shao, J.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shuvaev, A. M.

S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Steele, J. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Stevens, C. J.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Sun, C.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Sun, W.-H.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Sydoruk, O.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Tatartschuk, E.

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

Varadan, V. K.

D. K. Ghodgaonkar, V. V. Varadan, V. K. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39(2), 387–394 (1990).
[CrossRef]

Varadan, V. V.

D. K. Ghodgaonkar, V. V. Varadan, V. K. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39(2), 387–394 (1990).
[CrossRef]

Veselago, V. G.

V. G. Veselago, E. E. Narimanov, “The left hand of brightness: past, present and future of negative index materials,” Nat. Mater. 5(10), 759–762 (2006).
[CrossRef] [PubMed]

Wang, F. M.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

Wang, H.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

Wang, M.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Wang, S. M.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

Wegener, M.

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

Wu, D. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Wu, R. X.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

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S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010).
[CrossRef]

Xiong, X.

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

Zhang, X.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Zhou, L.

L. Zhou, S. T. Chui, “Eigenmodes of metallic ring systems: a rigorous approach,” Phys. Rev. B 74(3), 035419 (2006).
[CrossRef]

Zhu, S.

N. Liu, H. Liu, S. Zhu, H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Zhu, S. N.

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Zhukovsky, S. V.

Appl. Phys. Lett.

M. J. Freire, R. Marques, “Planar magnetoinductive lens for threedimensional subwavelength imaging,” Appl. Phys. Lett. 86(18), 182505 (2005).
[CrossRef]

S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010).
[CrossRef]

G. Dolling, M. Wegener, A. Schadle, S. Burger, S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006).
[CrossRef]

T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008).
[CrossRef]

IEEE Trans. Instrum. Meas.

D. K. Ghodgaonkar, V. V. Varadan, V. K. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39(2), 387–394 (1990).
[CrossRef]

Nano Lett.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[CrossRef] [PubMed]

Nat. Mater.

V. G. Veselago, E. E. Narimanov, “The left hand of brightness: past, present and future of negative index materials,” Nat. Mater. 5(10), 759–762 (2006).
[CrossRef] [PubMed]

Nat. Photonics

N. Liu, H. Liu, S. Zhu, H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

R. Marqués, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002).
[CrossRef]

L. Zhou, S. T. Chui, “Eigenmodes of metallic ring systems: a rigorous approach,” Phys. Rev. B 74(3), 035419 (2006).
[CrossRef]

A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011).
[CrossRef]

X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007).
[CrossRef]

Phys. Rev. Lett.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006).
[CrossRef] [PubMed]

Science

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of unit cell of paired SRRs metamaterials; D is the vertical space between the two rings; ϕ represents the twist angle of the two rings. (b) Circuit model for the paired SRRs. The circuit is formed by two LRC-circuits of each single-ring connected by the electric coupling network which consists of mutual capacitance C1a, C1b, C2a, C2b and the magnetic coupling network with mutual inductance M. The U-shaped shadows represent the two rings in panel (a) projected on the same plane. The variation of the angle ϕ and distance D will be reflected in the changes of the coupling mutual capacitances and inductance.

Fig. 2
Fig. 2

(a) Resonant frequency changes with the space D between two rings. The inset is the transmission curve at D = 0.05, 2 and 12 mm, respectively. (b) The coupling capacitance ke = c1 and the mutual inductance km = m retrieved from resonant frequencies. (c) Circulating current distributions at D = 0.05mm. The bottom image shows the circulating currents in the lower ring where the upper ring is hidden. (d)-(e) Circulating current distributions at resonance when two rings are in parallel and anti-parallel arrangement, respectively. The circulating currents are in opposite symmetry for these two arrangements. Higher energy resonance mode is represented by ω+ and lower one by ω-.

Fig. 3
Fig. 3

Variation of hybrid resonant frequencies as a function of electric and magnetic coupling coefficients. (a) Schematic of rings rotation at fixed distance D. (b) Resonant frequencies vary with coefficients a21 and m, where load effect coefficients a11 and a22 take 0.01 and 0.1, respectively. The two resonant frequencies form two branches of surface with a gap. (c) The gap between the two branches of resonance become small as the difference of a11 and a22 is small. Solid lines are the resonant frequencies for a11 = 0.01 and a22 = 0.1 and dashed line for a11 = 0.08 and a22 = 0.1. (d) A degenerated resonant frequency is shown as the load effect coefficients a11 = a22 (here the value is 0.1) are the same. This may happen when the distance between the ring is relative large and twist angle changes.

Fig. 4
Fig. 4

(a) Resonant frequency as a function of twist angle at D = 2 mm. The exchange of circulating current symmetry happens between the angles 30° and 60°. (b) Circulating currents at twist angle ϕ = 30°, 45° and 60°, respectively.

Fig. 5
Fig. 5

Experimental results for the paired SRRs metamaterial. (a) Schematic of experimental setup. It has two focus antennas and the sample is set in the between. (b) The image of the sample in experiments of which the top uSRR array are removed. (c) Measured resonant frequencies (triangles) and simulation ones (crosses) as a function of space D between the rings. The solid square is the resonant frequency for the rings separated by Kapton film. (d) Resonant frequencies measured by changing the twist angle. The space D is 2 mm. In the panels (c)-(d), ω+ and ω- represent the higher energy and the lower energy resonance modes, respectively. (e) The typical transmission curves under different D. At D = 20 mm, the resonance dip shifts to 11.5 GHz because of the lower dielectric constant of the foam used. (f) The measured transmission curves at typical twist angles.

Fig. 6
Fig. 6

A general circuit of the paired SRR. The U-shaped shadows represent the SRR rings. L0i, C0i and R0i (i=1, 2) are the intrinsic inductance, capacitance and resistance respectively. Cja and Cjb (j=1, 2) are the mutual capacitances of the electric coupling network between the two rings, and M is the mutual inductance. Ik (k=1, 2, …, 8) in the figure stands for the current in each path of the circuit.

Equations (9)

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iω L 0 I 1 + I 1 iω C 0 a 11 I 1 iω C 0 + a 12 I 2 iω C 0 ±iωM I 2 = e 1 iω L 0 I 2 + I 2 iω C 0 a 22 I 2 iω C 0 + a 21 I 1 iω C 0 ±iωM I 1 = e 2
a 21 = a 12 = ( c 1 a c 1b c 2a c 2b ) /A a 11 = { c 1a [ c 1b (1+ c 2a + c 2b )+ c 2b (1+ c 2a ) ]+ c 2a ( c 1b + c 2b + c 1b c 2b ) } /A a 22 = { c 1a [ c 2a (1+ c 2b )+ c 1b (1+ c 2a + c 2b ) ]+ c 2b ( c 1b + c 2a + c 2a c 1b ) } /A
ω 01 2 ω 0 2 = 2 a 11 a 22 2 a 21 m+SQ 2( 1 m 2 ) ω 02 2 ω 0 2 = 2 a 11 a 22 2 a 21 mSQ 2( 1 m 2 )
a 11 = a 22 = 1 2 c 1 + c 2 +2 c 1 c 2 ( c 1 +1 )( c 2 +1 ) , a 12 = a 21 = 1 2 c 1 c 2 ( c 1 +1 )( c 2 +1 )
ω 01 ω 0 = 1 a 11 a 21 1+m , ω 02 ω 0 = 1 a 11 ± a 21 1m
ω 01 ω 0 = 1( c 1 /2 ) 1+m , ω 02 ω 0 = 1±( c 1 /2 ) 1m
I 1 = I 3 + I 4 + I 5 , I 2 + I 4 = I 6 + I 7 , I 5 + I 6 = I 2 + I 8 , I 1 = I 3 + I 7 + I 8 .
I 3 j ω C 01 I 4 j ω C 1 a I 6 j ω C 02 I 8 j ω C 1 b = 0 , I 4 j ω C 1 a + I 5 j ω C 2 a I 6 j ω C 02 = 0 , I 6 j ω C 02 + I 7 j ω C 2 b I 8 j ω C 1 b = 0 , I 3 j ω C 01 I 5 j ω C 2 a I 8 j ω C 1 b = 0 , I 3 j ω C 01 I 4 j ω C 1 a I 7 j ω C 2 b = 0 ,
I 1 R 01 + I 3 j ω C 01 + I 1 j ω L 01 I 2 j ω M = e 1 , I 2 R 02 + I 6 j ω C 02 + I 2 j ω L 02 I 1 j ω M = e 2 ,

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