Abstract

The physical mechanism of the interface states in layered structures consisting of single-negative metamaterials is investigated using a simple resonant cavity model. We found that the interface states and their corresponding tunneling transmission modes appeared when the resonant condition is satisfied. Such resonant condition depends on the phase changes inside the resonant cavity. Based on these results, we proposed an efficient method to precisely predict the frequencies of the tunneling interface states inside the single-negative metamaterial layers. Our method is effective for interface states corresponding to perfect or imperfect tunneling transmission. Composite right/left-handed transmission lines were used to realize the pair and sandwich metamaterial layered structures in the microwave region. Electromagnetic tunneling interface states were observed in the measurements, which agreed well with the theory. Our study offers a way for effectively designing metamaterial devices with novel electromagnetic tunneling properties.

© 2013 OSA

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2012 (2)

Z. F. Chen, P. Han, C. W. Leung, Y. Wang, M. Hu, and Y. Chen, “Study of optical Tamm states based on the phase properties of one-dimensional photonic crystals,” Opt. Express20(19), 21618–21626 (2012).
[CrossRef] [PubMed]

A. V. Baryshev, K. Kawasaki, P. B. Lim, and M. Inoue, “Interplay of surface resonances in one-dimensional plasmonic magnetophotonic crystal slabs,” Phys. Rev. B85(20), 205130 (2012).
[CrossRef]

2011 (2)

D. Ö. Güney, T. Koschny, and C. M. Soukoulis, “Surface plasmon driven electric and magnetic resonators for metamaterials,” Phys. Rev. B83(4), 045107 (2011).
[CrossRef]

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

2009 (3)

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics3(7), 388–394 (2009).
[CrossRef]

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

2008 (1)

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

2007 (1)

R. P. Liu, B. Zhao, X. Q. Lin, Q. Cheng, and T. J. Cui, “Evanescent-wave amplification studied using a bilayer periodic circuit structure and its effective medium model,” Phys. Rev. B75(12), 125118 (2007).
[CrossRef]

2006 (4)

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including Metals) Transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett.89(14), 141101 (2006).
[CrossRef]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett.96(7), 073907 (2006).
[CrossRef] [PubMed]

A. Alù, N. Engheta, and R. W. Ziolkowski, “Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of ε-negative and μ-negative slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(1), 016604 (2006).
[CrossRef] [PubMed]

2005 (5)

A. V. Krasavin, A. V. Zayats, and N. I. Zheludev, “Active control of surface plasmon-polariton waves,” J. Opt. A, Pure Appl. Opt.7(2), S85–S89 (2005).
[CrossRef]

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B72(23), 233102 (2005).
[CrossRef]

A. Kavokin, I. Shelykh, and G. Malpuech, “Optical Tamm states for the fabrication of polariton lasers,” Appl. Phys. Lett.87(26), 261105 (2005).
[CrossRef]

T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the ENG-MNG pair in a CRLH transmission line implementation,” Microw. Opt. Technol. Lett.46(5), 476–481 (2005).
[CrossRef]

C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys.7, 167 (2005).
[CrossRef]

2004 (4)

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
[CrossRef] [PubMed]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

D. K. Qing and G. Chen, “Enhancement of evanescent waves in waveguides using metamaterials of negative permittivity and permeability,” Appl. Phys. Lett.84(5), 669–671 (2004).
[CrossRef]

2003 (4)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag.51(10), 2558–2571 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Microw. Theory Tech.51, 2558–2571 (2003).

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science302(5644), 419–422 (2003).
[CrossRef] [PubMed]

2002 (3)

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech.50(12), 2702–2712 (2002).
[CrossRef]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(44 Pt 2B), 046607 (2002).
[CrossRef] [PubMed]

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

2001 (2)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter13(9), 1811–1818 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

1999 (1)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B Chem.54(1-2), 3–15 (1999).
[CrossRef]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett.76(25), 4773–4776 (1996).
[CrossRef] [PubMed]

Alù, A.

A. Alù, N. Engheta, and R. W. Ziolkowski, “Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of ε-negative and μ-negative slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(1), 016604 (2006).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag.51(10), 2558–2571 (2003).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Microw. Theory Tech.51, 2558–2571 (2003).

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Baryshev, A. V.

A. V. Baryshev, K. Kawasaki, P. B. Lim, and M. Inoue, “Interplay of surface resonances in one-dimensional plasmonic magnetophotonic crystal slabs,” Phys. Rev. B85(20), 205130 (2012).
[CrossRef]

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Basov, D. N.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Bellessa, J.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Bloch, J.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Caloz, C.

C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys.7, 167 (2005).
[CrossRef]

T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the ENG-MNG pair in a CRLH transmission line implementation,” Microw. Opt. Technol. Lett.46(5), 476–481 (2005).
[CrossRef]

Chen, G.

D. K. Qing and G. Chen, “Enhancement of evanescent waves in waveguides using metamaterials of negative permittivity and permeability,” Appl. Phys. Lett.84(5), 669–671 (2004).
[CrossRef]

Chen, H.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(44 Pt 2B), 046607 (2002).
[CrossRef] [PubMed]

Chen, Y.

Chen, Y. H.

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett.89(14), 141101 (2006).
[CrossRef]

Chen, Z. F.

Cheng, Q.

R. P. Liu, B. Zhao, X. Q. Lin, Q. Cheng, and T. J. Cui, “Evanescent-wave amplification studied using a bilayer periodic circuit structure and its effective medium model,” Phys. Rev. B75(12), 125118 (2007).
[CrossRef]

Cui, T. J.

R. P. Liu, B. Zhao, X. Q. Lin, Q. Cheng, and T. J. Cui, “Evanescent-wave amplification studied using a bilayer periodic circuit structure and its effective medium model,” Phys. Rev. B75(12), 125118 (2007).
[CrossRef]

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

de Vasconcellos, S. M.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Dong, J. W.

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett.89(14), 141101 (2006).
[CrossRef]

Dorofeenko, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Eleftheriades, G. V.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech.50(12), 2702–2712 (2002).
[CrossRef]

Engheta, N.

A. Alù, N. Engheta, and R. W. Ziolkowski, “Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of ε-negative and μ-negative slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(1), 016604 (2006).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag.51(10), 2558–2571 (2003).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Microw. Theory Tech.51, 2558–2571 (2003).

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett.96(7), 073907 (2006).
[CrossRef] [PubMed]

Fang, N.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

Feng, T. H.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

Fujishige, T.

T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the ENG-MNG pair in a CRLH transmission line implementation,” Microw. Opt. Technol. Lett.46(5), 476–481 (2005).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B Chem.54(1-2), 3–15 (1999).
[CrossRef]

Gauthron, K.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Gazzano, O.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Giessen, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Goto, T.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Granovsky, A. B.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Güney, D. Ö.

D. Ö. Güney, T. Koschny, and C. M. Soukoulis, “Surface plasmon driven electric and magnetic resonators for metamaterials,” Phys. Rev. B83(4), 045107 (2011).
[CrossRef]

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Han, P.

He, L.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett.76(25), 4773–4776 (1996).
[CrossRef] [PubMed]

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B Chem.54(1-2), 3–15 (1999).
[CrossRef]

Hooper, I. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including Metals) Transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Hu, M.

Inoue, M.

A. V. Baryshev, K. Kawasaki, P. B. Lim, and M. Inoue, “Interplay of surface resonances in one-dimensional plasmonic magnetophotonic crystal slabs,” Phys. Rev. B85(20), 205130 (2012).
[CrossRef]

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Inouye, Y.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics3(7), 388–394 (2009).
[CrossRef]

Itoh, T.

C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys.7, 167 (2005).
[CrossRef]

T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the ENG-MNG pair in a CRLH transmission line implementation,” Microw. Opt. Technol. Lett.46(5), 476–481 (2005).
[CrossRef]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

Iyer, A. K.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech.50(12), 2702–2712 (2002).
[CrossRef]

Jiang, H. T.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

Kavokin, A.

A. Kavokin, I. Shelykh, and G. Malpuech, “Optical Tamm states for the fabrication of polariton lasers,” Appl. Phys. Lett.87(26), 261105 (2005).
[CrossRef]

Kavokin, A. V.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B72(23), 233102 (2005).
[CrossRef]

Kawasaki, K.

A. V. Baryshev, K. Kawasaki, P. B. Lim, and M. Inoue, “Interplay of surface resonances in one-dimensional plasmonic magnetophotonic crystal slabs,” Phys. Rev. B85(20), 205130 (2012).
[CrossRef]

Kawata, S.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics3(7), 388–394 (2009).
[CrossRef]

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

Koschny, T.

D. Ö. Güney, T. Koschny, and C. M. Soukoulis, “Surface plasmon driven electric and magnetic resonators for metamaterials,” Phys. Rev. B83(4), 045107 (2011).
[CrossRef]

Krasavin, A. V.

A. V. Krasavin, A. V. Zayats, and N. I. Zheludev, “Active control of surface plasmon-polariton waves,” J. Opt. A, Pure Appl. Opt.7(2), S85–S89 (2005).
[CrossRef]

Kremer, P. C.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech.50(12), 2702–2712 (2002).
[CrossRef]

Lai, A.

C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys.7, 167 (2005).
[CrossRef]

Lemaître, A.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Leung, C. W.

Li, H. Q.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

Li, Y. H.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

Lim, P. B.

A. V. Baryshev, K. Kawasaki, P. B. Lim, and M. Inoue, “Interplay of surface resonances in one-dimensional plasmonic magnetophotonic crystal slabs,” Phys. Rev. B85(20), 205130 (2012).
[CrossRef]

Lin, X. Q.

R. P. Liu, B. Zhao, X. Q. Lin, Q. Cheng, and T. J. Cui, “Evanescent-wave amplification studied using a bilayer periodic circuit structure and its effective medium model,” Phys. Rev. B75(12), 125118 (2007).
[CrossRef]

Lisyansky, A. A.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Liu, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Liu, N.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Liu, N. H.

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(44 Pt 2B), 046607 (2002).
[CrossRef] [PubMed]

Liu, R. P.

R. P. Liu, B. Zhao, X. Q. Lin, Q. Cheng, and T. J. Cui, “Evanescent-wave amplification studied using a bilayer periodic circuit structure and its effective medium model,” Phys. Rev. B75(12), 125118 (2007).
[CrossRef]

Malpuech, G.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B72(23), 233102 (2005).
[CrossRef]

A. Kavokin, I. Shelykh, and G. Malpuech, “Optical Tamm states for the fabrication of polariton lasers,” Appl. Phys. Lett.87(26), 261105 (2005).
[CrossRef]

Merzlikin, A. M.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Padilla, W. J.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
[CrossRef] [PubMed]

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett.76(25), 4773–4776 (1996).
[CrossRef] [PubMed]

Preist, T. W.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including Metals) Transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Qing, D. K.

D. K. Qing and G. Chen, “Enhancement of evanescent waves in waveguides using metamaterials of negative permittivity and permeability,” Appl. Phys. Lett.84(5), 669–671 (2004).
[CrossRef]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A Hybridization Model for the Plasmon Response of Complex Nanostructures,” Science302(5644), 419–422 (2003).
[CrossRef] [PubMed]

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter13(9), 1811–1818 (2001).
[CrossRef]

Sambles, J. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including Metals) Transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Senellart, P.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shelykh, I.

A. Kavokin, I. Shelykh, and G. Malpuech, “Optical Tamm states for the fabrication of polariton lasers,” Appl. Phys. Lett.87(26), 261105 (2005).
[CrossRef]

Shelykh, I. A.

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B72(23), 233102 (2005).
[CrossRef]

Shi, Y. L.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett.96(7), 073907 (2006).
[CrossRef] [PubMed]

Smith, D. R.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

D. Ö. Güney, T. Koschny, and C. M. Soukoulis, “Surface plasmon driven electric and magnetic resonators for metamaterials,” Phys. Rev. B83(4), 045107 (2011).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett.76(25), 4773–4776 (1996).
[CrossRef] [PubMed]

Sun, Y.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

Symonds, C.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Verma, P.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics3(7), 388–394 (2009).
[CrossRef]

Vier, D. C.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Vinogradov, A. P.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett.101(11), 113902 (2008).
[CrossRef] [PubMed]

Voisin, P.

O. Gazzano, S. M. de Vasconcellos, K. Gauthron, C. Symonds, J. Bloch, P. Voisin, J. Bellessa, A. Lemaître, and P. Senellart, “Evidence for confined Tamm plasmon modes under metallic microdisks and application to the control of spontaneous optical emission,” Phys. Rev. Lett.107(24), 247402 (2011).
[CrossRef] [PubMed]

Wang, H. Z.

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett.89(14), 141101 (2006).
[CrossRef]

Wang, Y.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Wu, X.

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(44 Pt 2B), 046607 (2002).
[CrossRef] [PubMed]

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B Chem.54(1-2), 3–15 (1999).
[CrossRef]

Yen, T. J.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett.76(25), 4773–4776 (1996).
[CrossRef] [PubMed]

Zayats, A. V.

A. V. Krasavin, A. V. Zayats, and N. I. Zheludev, “Active control of surface plasmon-polariton waves,” J. Opt. A, Pure Appl. Opt.7(2), S85–S89 (2005).
[CrossRef]

Zhang, X.

T. J. 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,” Science303(5663), 1494–1496 (2004).
[CrossRef] [PubMed]

Zhang, Y. W.

T. H. Feng, Y. H. Li, H. T. Jiang, Y. Sun, L. He, H. Q. Li, Y. W. Zhang, Y. L. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026601 (2009).
[CrossRef] [PubMed]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

Zhao, B.

R. P. Liu, B. Zhao, X. Q. Lin, Q. Cheng, and T. J. Cui, “Evanescent-wave amplification studied using a bilayer periodic circuit structure and its effective medium model,” Phys. Rev. B75(12), 125118 (2007).
[CrossRef]

Zheludev, N. I.

A. V. Krasavin, A. V. Zayats, and N. I. Zheludev, “Active control of surface plasmon-polariton waves,” J. Opt. A, Pure Appl. Opt.7(2), S85–S89 (2005).
[CrossRef]

Zhu, S.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Zhu, S. Y.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(44 Pt 2B), 046607 (2002).
[CrossRef] [PubMed]

Zi, J.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.69(6), 066607 (2004).
[CrossRef] [PubMed]

Ziolkowski, R. W.

A. Alù, N. Engheta, and R. W. Ziolkowski, “Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of ε-negative and μ-negative slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(1), 016604 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

A. Kavokin, I. Shelykh, and G. Malpuech, “Optical Tamm states for the fabrication of polariton lasers,” Appl. Phys. Lett.87(26), 261105 (2005).
[CrossRef]

D. K. Qing and G. Chen, “Enhancement of evanescent waves in waveguides using metamaterials of negative permittivity and permeability,” Appl. Phys. Lett.84(5), 669–671 (2004).
[CrossRef]

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett.89(14), 141101 (2006).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag.51(10), 2558–2571 (2003).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (2)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Microw. Theory Tech.51, 2558–2571 (2003).

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech.50(12), 2702–2712 (2002).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

A. V. Krasavin, A. V. Zayats, and N. I. Zheludev, “Active control of surface plasmon-polariton waves,” J. Opt. A, Pure Appl. Opt.7(2), S85–S89 (2005).
[CrossRef]

J. Phys. Condens. Matter (2)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter14(18), R597–R624 (2002).
[CrossRef]

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter13(9), 1811–1818 (2001).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

T. Fujishige, C. Caloz, and T. Itoh, “Experimental demonstration of transparency in the ENG-MNG pair in a CRLH transmission line implementation,” Microw. Opt. Technol. Lett.46(5), 476–481 (2005).
[CrossRef]

Nat. Photonics (2)

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics3(7), 388–394 (2009).
[CrossRef]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003).
[CrossRef] [PubMed]

New J. Phys. (1)

C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys.7, 167 (2005).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (4)

A. V. Baryshev, K. Kawasaki, P. B. Lim, and M. Inoue, “Interplay of surface resonances in one-dimensional plasmonic magnetophotonic crystal slabs,” Phys. Rev. B85(20), 205130 (2012).
[CrossRef]

D. Ö. Güney, T. Koschny, and C. M. Soukoulis, “Surface plasmon driven electric and magnetic resonators for metamaterials,” Phys. Rev. B83(4), 045107 (2011).
[CrossRef]

A. V. Kavokin, I. A. Shelykh, and G. Malpuech, “Lossless interface modes at the boundary between two periodic dielectric structures,” Phys. Rev. B72(23), 233102 (2005).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

A photonic well formed by two SNG metamaterials barriers.

Fig. 2
Fig. 2

Transmittance and phase changes as a function of frequency for pair SNG structures. The thicknesses of the ENG and MNG layers are 30 mm and 40 mm, respectively. The inset shows the electric field distribution at 0.796 GHz.

Fig. 3
Fig. 3

Transmittance and phase change as a function of frequency for ENG-MNG-ENG sandwich structures, for different thicknesses of the MNG layer: (a) 15 mm, (b) 30 mm, (c) 45 mm, and (d) 60 mm.

Fig. 4
Fig. 4

Calculated effective permittivity and permeability, and the simulated and measured S parameters of the CRLH TLs A12 and B12, respectively.

Fig. 5
Fig. 5

Photograph of the pair structures A6B6 fabricated based on CRLH TLs.

Fig. 6
Fig. 6

(a) The simulated and measured S parameters and (b) the simulated reflection phase changes φ1(ω) + φ2(ω) as a function of frequency for structure A6B6. The grey areas represent the common SNG frequency region of TLs A and B.

Fig. 7
Fig. 7

Photograph of the sandwich structures (a) A6B2A6, (b) A6B4A6, (c) A6B6A6 and (d) A6B12A6 fabricated using the CRLH TLs, respectively.

Fig. 8
Fig. 8

The simulated and measured S21 parameters and the simulated values of φ1(ω) + φ2(ω) for SNG sandwich structures A6B2A6 (a & b), A6B4A6 (c & d), A6B6A6 (e & f) and A6B12A6 (g & h), respectively. The grey regions correspond to the common SNG frequency range of TLs A and B.

Equations (7)

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φ 1 ( ω )+2φ+ φ 2 ( ω )=2mπ,
φ 1 ( ω )+ φ 2 ( ω )=2mπ.
M k =[ X 11 k X 12 k X 21 k X 22 k ],
r k =| r k | e i φ k = X 22 k X 11 k +i( X 12 k + X 21 k ) X 22 k + X 11 k +i( X 12 k X 21 k ) .
ε ENG =1 ω ep 2 / ω 2 , μ ENG =3
ε MNG =3, μ MNG =1 ω mp 2 / ω 2
ε j ( C 0 1/ ω 2 L i d i )/( ε 0 p ), μ j p( L 0 1/ ω 2 C i d i )/ μ 0 ,

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