Abstract

The interaction of specific surface plasmon modes in metal-dielectric-metal arrangements is investigated, motivated by their relevance to device-based configurations. The absorption spectra of the relevant nanostructures considering geometrical variation, such as the width and height of the metal or dielectric, are probed considering such interactions. Frequency domain simulations are used to study related multiple surface plasmon polariton resonance modes. It is indicated that the resonant energy level interaction due to the coupling between modes in a horizontal dielectric layer and those in a vertical groove can be engineered and understood in terms of energy level hybridization.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2018 (1)

2017 (1)

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

2016 (3)

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Z. Yong, S. Zhang, C. Gong, and S. He, “Narrow band perfect absorber for maximum localized magnetic and electric field enhancement and sensing applications,” Sci. Rep. 6(1), 1–7 (2016).
[Crossref]

2015 (3)

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

L. Lin and Y. Zheng, “Optimizing plasmonic nanoantennas via coordinated multiple coupling,” Sci. Rep. 5(1), 14788 (2015).
[Crossref]

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

2014 (2)

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

2013 (3)

S. V. Boriskina, H. Ghasemi, and G. Chen, “Plasmonic materials for energy: From physics to applications,” Mater. Today 16(10), 375–386 (2013).
[Crossref]

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Hybridization of Localized and Guided Modes in 2D Metal–Insulator–Metal Nanocavity Arrays,” J. Phys. Chem. C 117(6), 2541–2546 (2013).
[Crossref]

2012 (1)

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

2011 (2)

Z. Sun and X. Zuo, “Tunable Absorption of Light via Localized Plasmon Resonances on a Metal Surface with Interspaced Ultra-thin Metal Gratings,” Plasmonics 6(1), 83–89 (2011).
[Crossref]

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2(1), 517 (2011).
[Crossref]

2010 (1)

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

2009 (1)

2008 (3)

2007 (2)

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

K. A. Willets and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy and Sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

2005 (1)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

1997 (2)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref]

1974 (1)

1969 (1)

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

1965 (1)

Akselrod, G. M.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Atwater, H. A.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2(1), 517 (2011).
[Crossref]

Aydin, K.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2(1), 517 (2011).
[Crossref]

Barbara, A.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why Metallic Surfaces with Grooves a Few Nanometers Deep and Wide May Strongly Absorb Visible Light,” Phys. Rev. Lett. 100(6), 066408 (2008).
[Crossref]

Barnard, E. S.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Bonod, N.

Boriskina, S. V.

S. V. Boriskina, H. Ghasemi, and G. Chen, “Plasmonic materials for energy: From physics to applications,” Mater. Today 16(10), 375–386 (2013).
[Crossref]

Bowen, P. T.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Briggs, R. M.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2(1), 517 (2011).
[Crossref]

Brongersma, M. L.

Cao, F.

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

Chandran, A.

Chen, G.

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

S. V. Boriskina, H. Ghasemi, and G. Chen, “Plasmonic materials for energy: From physics to applications,” Mater. Today 16(10), 375–386 (2013).
[Crossref]

Chilkoti, A.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Ciracì, C.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Cui, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

de Abajo, F. J.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Ding, F.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Ding, S. Y.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Economou, E. N.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

Edee, K.

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

Emory, S. R.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref]

Enoch, S.

Fei Guo, C.

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

Feld, M. S.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Ferry, V. E.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2(1), 517 (2011).
[Crossref]

Gan, Q.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Ghasemi, H.

S. V. Boriskina, H. Ghasemi, and G. Chen, “Plasmonic materials for energy: From physics to applications,” Mater. Today 16(10), 375–386 (2013).
[Crossref]

Gong, C.

Z. Yong, S. Zhang, C. Gong, and S. He, “Narrow band perfect absorber for maximum localized magnetic and electric field enhancement and sensing applications,” Sci. Rep. 6(1), 1–7 (2016).
[Crossref]

Granet, G.

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

Han, Z.

Hao, J.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

He, S.

Z. Yong, S. Zhang, C. Gong, and S. He, “Narrow band perfect absorber for maximum localized magnetic and electric field enhancement and sensing applications,” Sci. Rep. 6(1), 1–7 (2016).
[Crossref]

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645 (2005).
[Crossref]

He, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Hill, R. T.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Hoang, T. B.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Hu, H.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Hu, J.

Hu, J.-G.

Hua, Y.

W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Hybridization of Localized and Guided Modes in 2D Metal–Insulator–Metal Nanocavity Arrays,” J. Phys. Chem. C 117(6), 2541–2546 (2013).
[Crossref]

Huang, J.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Itzkan, I.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Ji, D.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Jin, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Kaminow, I. P.

Kneipp, H.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Kneipp, K.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Kuttge, M.

Lafarge, C.

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

Lassiter, J. B.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

Laurent, N.

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

Le Perchec, J.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why Metallic Surfaces with Grooves a Few Nanometers Deep and Wide May Strongly Absorb Visible Light,” Phys. Rev. Lett. 100(6), 066408 (2008).
[Crossref]

Li, H.

Li, H.-J.

Li, J. F.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Li, X.

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

Lin, L.

L. Lin and Y. Zheng, “Optimizing plasmonic nanoantennas via coordinated multiple coupling,” Sci. Rep. 5(1), 14788 (2015).
[Crossref]

Lin, Q.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Lin, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Liu, G. D.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Liu, J. P.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Liu, K.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Liu, L.

Liu, Q.

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

Liu, X.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

López-Ríos, T.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why Metallic Surfaces with Grooves a Few Nanometers Deep and Wide May Strongly Absorb Visible Light,” Phys. Rev. Lett. 100(6), 066408 (2008).
[Crossref]

Luo, X.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, US, 2007).

Malitson, I. H.

Mammel, W. L.

Maystre, D.

McGuire, F.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

Mikkelsen, M. H.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Mock, J. J.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Moreau, A.

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

Nie, S.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref]

Odom, T. W.

W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Hybridization of Localized and Guided Modes in 2D Metal–Insulator–Metal Nanocavity Arrays,” J. Phys. Chem. C 117(6), 2541–2546 (2013).
[Crossref]

Padilla, W. J.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

Panneerselvam, R.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Perelman, L. T.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Polman, A.

Popov, E.

Qi, Z.

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

Qin, M.

Qiu, M.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

Quémerais, P.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why Metallic Surfaces with Grooves a Few Nanometers Deep and Wide May Strongly Absorb Visible Light,” Phys. Rev. Lett. 100(6), 066408 (2008).
[Crossref]

Ren, B.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Ren, Y.

Ren, Y.-Z.

Ren, Z.

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

Smith, D. R.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Song, H.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Su, L.

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Suh, J. Y.

W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Hybridization of Localized and Guided Modes in 2D Metal–Insulator–Metal Nanocavity Arrays,” J. Phys. Chem. C 117(6), 2541–2546 (2013).
[Crossref]

Sun, T.

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

Sun, Z.

Z. Sun and X. Zuo, “Tunable Absorption of Light via Localized Plasmon Resonances on a Metal Surface with Interspaced Ultra-thin Metal Gratings,” Plasmonics 6(1), 83–89 (2011).
[Crossref]

Tayeb, G.

Tian, Z. Q.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Van Duyne, R. P.

K. A. Willets and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy and Sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

Wang, J.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

Wang, L. L.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Wang, L.-L.

Wang, Q.

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Wang, Y.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Weber, H. P.

White, J. S.

Wiley, B. J.

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

Willets, K. A.

K. A. Willets and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy and Sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

Wu, D. Y.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Xia, S. X.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Xu, J.

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

Yang, L.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Ye, Y.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Yi, J.

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Yong, Z.

Z. Yong, S. Zhang, C. Gong, and S. He, “Narrow band perfect absorber for maximum localized magnetic and electric field enhancement and sensing applications,” Sci. Rep. 6(1), 1–7 (2016).
[Crossref]

Zeng, X.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Zhai, X.

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Zhai, Y.

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

Zhang, N.

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Zhang, S.

Z. Yong, S. Zhang, C. Gong, and S. He, “Narrow band perfect absorber for maximum localized magnetic and electric field enhancement and sensing applications,” Sci. Rep. 6(1), 1–7 (2016).
[Crossref]

Zheng, Y.

L. Lin and Y. Zheng, “Optimizing plasmonic nanoantennas via coordinated multiple coupling,” Sci. Rep. 5(1), 14788 (2015).
[Crossref]

Zhong, S.

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Zhou, L.

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

Zhou, W.

W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Hybridization of Localized and Guided Modes in 2D Metal–Insulator–Metal Nanocavity Arrays,” J. Phys. Chem. C 117(6), 2541–2546 (2013).
[Crossref]

Zuo, X.

Z. Sun and X. Zuo, “Tunable Absorption of Light via Localized Plasmon Resonances on a Metal Surface with Interspaced Ultra-thin Metal Gratings,” Plasmonics 6(1), 83–89 (2011).
[Crossref]

Adv. Mater. (1)

G. M. Akselrod, J. Huang, T. B. Hoang, P. T. Bowen, L. Su, D. R. Smith, and M. H. Mikkelsen, “Large-Area Metasurface Perfect Absorbers from Visible to Near-Infrared,” Adv. Mater. 27(48), 8028–8034 (2015).
[Crossref]

Annu. Rev. Phys. Chem. (1)

K. A. Willets and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy and Sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Express (1)

Q. Lin, X. Zhai, L. L. Wang, X. Luo, G. D. Liu, J. P. Liu, and S. X. Xia, “A novel design of plasmon-induced absorption sensor,” Appl. Phys. Express 9(6), 062002 (2016).
[Crossref]

Appl. Phys. Lett. (1)

J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. 96(25), 251104 (2010).
[Crossref]

J. Lightwave Technol. (1)

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

A. Moreau, C. Lafarge, N. Laurent, K. Edee, and G. Granet, “Enhanced transmission of slit arrays in an extremely thin metallic film,” J. Opt. A: Pure Appl. Opt. 9(2), 165–169 (2007).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. C (1)

W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Hybridization of Localized and Guided Modes in 2D Metal–Insulator–Metal Nanocavity Arrays,” J. Phys. Chem. C 117(6), 2541–2546 (2013).
[Crossref]

Laser Photonics Rev. (1)

Y. Cui, Y. He, Y. Jin, F. Ding, L. Yang, Y. Ye, S. Zhong, Y. Lin, and S. He, “Plasmonic and metamaterial structures as electromagnetic absorbers,” Laser Photonics Rev. 8(4), 495–520 (2014).
[Crossref]

Light: Sci. Appl. (1)

C. Fei Guo, T. Sun, F. Cao, Q. Liu, and Z. Ren, “Metallic nanostructures for light trapping in energy-harvesting devices,” Light: Sci. Appl. 3(4), e161 (2014).
[Crossref]

Mater. Today (1)

S. V. Boriskina, H. Ghasemi, and G. Chen, “Plasmonic materials for energy: From physics to applications,” Mater. Today 16(10), 375–386 (2013).
[Crossref]

Nano Lett. (1)

J. B. Lassiter, F. McGuire, J. J. Mock, C. Ciracì, R. T. Hill, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Plasmonic waveguide modes of film-coupled metallic nanocubes,” Nano Lett. 13(12), 5866–5872 (2013).
[Crossref]

Nat. Commun. (1)

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2(1), 517 (2011).
[Crossref]

Nat. Rev. Mater. (1)

S. Y. Ding, J. Yi, J. F. Li, B. Ren, D. Y. Wu, R. Panneerselvam, and Z. Q. Tian, “Nanostructure-based plasmon-enhanced Raman spectroscopy for surface analysis of materials,” Nat. Rev. Mater. 1(6), 16021 (2016).
[Crossref]

Nature (2)

A. Moreau, C. Ciracì, J. J. Mock, D. R. Smith, R. T. Hill, A. Chilkoti, Q. Wang, and B. J. Wiley, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492(7427), 86–89 (2012).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Opt. Commun. (1)

Y. Zhai, G. Chen, J. Xu, Z. Qi, X. Li, and Q. Wang, “Multiple-band perfect absorbers based on the combination of Fabry-Perot resonance and the gap plasmon resonance,” Opt. Commun. 399, 28–33 (2017).
[Crossref]

Opt. Express (5)

Phys. Rev. (1)

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

Phys. Rev. Lett. (2)

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why Metallic Surfaces with Grooves a Few Nanometers Deep and Wide May Strongly Absorb Visible Light,” Phys. Rev. Lett. 100(6), 066408 (2008).
[Crossref]

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Plasmonics (1)

Z. Sun and X. Zuo, “Tunable Absorption of Light via Localized Plasmon Resonances on a Metal Surface with Interspaced Ultra-thin Metal Gratings,” Plasmonics 6(1), 83–89 (2011).
[Crossref]

Sci. Rep. (3)

D. Ji, H. Song, X. Zeng, H. Hu, K. Liu, N. Zhang, and Q. Gan, “Broadband absorption engineering of hyperbolic metafilm patterns,” Sci. Rep. 4(1), 4498 (2015).
[Crossref]

Z. Yong, S. Zhang, C. Gong, and S. He, “Narrow band perfect absorber for maximum localized magnetic and electric field enhancement and sensing applications,” Sci. Rep. 6(1), 1–7 (2016).
[Crossref]

L. Lin and Y. Zheng, “Optimizing plasmonic nanoantennas via coordinated multiple coupling,” Sci. Rep. 5(1), 14788 (2015).
[Crossref]

Science (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref]

Other (1)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, US, 2007).

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic of a metal grating on the top of dielectric spacer (chosen to be SiO2) and metal (Ag) substrate, with normally incident p-polarized light. (b) The configuration of the unit cell (with periodic boundaries – $\color{red}{\textrm{red}}$ vertical lines), indicating the metal/dielectric/metal (M/D/M) as well as the metal/air/metal (M/A/M) constituents. The geometrical parameters studied include the length of the metal cavity (w), groove width (g), periodicity (p) = w + g, height of the grating ($h$), and the dielectric spacer thickness (t). The two ports (port 1: input, port 2: output) were used for the simulations.
Fig. 2.
Fig. 2. (a) Absorption (A) spectra of Ag grating with (t = 5 nm, $\color{orange}{\textrm{orange dotted line}}$) and without (t = 0 nm, $\color{blue}{\textrm{blue solid line}}$) $\textrm{Si}{\textrm{O}_2}$ dielectric layer spacer. Here p = 200 nm, h = 40 nm and g = 40 nm. For the labeled absorption peaks in (a), the magnitude of the out-of-plane magnetic field are plotted for structure with dielectric spacer in (b) E ∼ 1.62 eV: peak 1, (c) E ∼ 2.15 eV: peak 2, and (d) E ∼ 2.57 eV: peak 3. The magnitude of the magnetic field is indicated at the right.
Fig. 3.
Fig. 3. (a) Absorption (A) spectra for g = 40 nm ($\color{blue}{\textrm{blue solid line}}$), with absorption peaks (1, 2, and 3) and g = 20 nm ($\color{orange}{\textrm{orange dotted line}}$), with absorption peaks (a, b, and c). The magnitude of the out-of-plane magnetic field are plotted for the lower energy peak (b) E ∼ 2.33 eV, and the higher energy peak (c) E ∼ 2.44 eV. The magnitude of the magnetic field is indicated at the right.
Fig. 4.
Fig. 4. (a) The variation of the absorption (A) spectra as a function of the groove width (g) in the range of 5 nm (bottom) to 50 nm (top). The M/D/M cavity length (w) is fixed at 180 nm, and the groove height (h) at 40 nm, with t = 5 nm. The circles and the triangles represent the high and low energy modes inside the vertical groove. The absorption peaks labeled by short black lines, on the left, are related to F-P like SPP resonances in the horizontal M/D/M resonator satisfying the resonance condition βMDM · w ≈ 5π. (b) A plot of the high and low energy modes, from (a) indicates an energy gap.
Fig. 5.
Fig. 5. (a) The horizontal electric field (Ex) profiles along the dotted yellow line, are plotted for (b) g = 10 nm, (c) g = 20 nm, and (d) g = 40 nm, for the respective lower and higher energy modes, taken from Fig. 4(a). The electric fields may be related to the surface current at the bottom of the groove. The black dashed lines in (b-d) indicate the position of the groove walls.
Fig. 6.
Fig. 6. The generation of hybrid modes mediated by the interaction of the M/A/M with the M/D/M energy levels. The groove M/A/M mode is coupled to the M/D/M resonance mode, yielding local surface charges and currents (depicted by the black arrows) and related to the electric field profiles of Fig. 5.
Fig. 7.
Fig. 7. The coupling of the M/A/M modes [characterized by Eq. (1)] with the M/D/M modes [characterized by Eq. (3)] leads to energy level interaction and gap formation as seen in the absorption spectra as a function of the grating height (h). The numbers after the MDM refer to the m in Eq. (3). Here, the g = 20 nm, while p = 200 nm, w = 180 nm, and t = 5 nm. The incident photon wavelength is varied from 400 nm to 1200 nm. The magnitude of the absorption coefficient is indicated on the right.
Fig. 8.
Fig. 8. (a) An overall summary of the horizontal M/D/M and vertical M/A/M modes with related SPPs). The magnitude of the out-of-plane magnetic field are plotted for peaks (b) E ∼ 2.33 eV, (c) E ∼ 2.44 eV – from Fig. 4(a), as well as for the lower energy modes, i.e., (d) E ∼ 2.01 eV. The current flow directions are related to induced magnetic moments and their related interactions. A higher degree of interaction leads to a larger energy gap and broader energy gap. The magnitude of the magnetic field is indicated at the right.

Equations (10)

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β M A M h + ϕ / 2 n π / 2
tan ( ζ g / 2 ) i η / ε m ζ
ζ = ( k 0 2 β M A M 2 ) 1 / 2
η = ( ε m k 0 2 β M A M 2 ) 1 / 2
β M D M w + δ m π
tan ( κ d t ) = 2 ε d ε m κ d κ m / ( ε m 2 κ d 2 ε d 2 κ m 2 )
κ d = ( ε d k 0 2 β M D M 2 ) 1 / 2
κ m = ( β M D M 2 ε m k 0 2 ) 1 / 2
H = ( E M D M V V E M A M )
E + / = 1 2 [ ( E M A M + E M D M ) ± ( E M A M + E M D M ) 2 + 4 V 2 ]