Abstract

We investigate the supermodes in arbitrary layers of graphene sheets, which are collective guided modes formed by coupling of surface plasmon polaritons (SPPs) in each graphene sheet. In terms of the dispersion relation, we analyse the effective indexes and mode profiles of the supermodes. Numerical simulations reveal that the supermodes can be well approximated by linear superposition of SPPs in individual graphene sheets. Among all the possible supermodes, there is an interesting one possessing both lowest propagation loss and shortest mode wavelength. The loss of the supermode decreases as the number of layers increases and saturates at about 5 layers. The graphene multilayers may find potential applications in low-loss plasmonic waveguides and so constructed optical devices.

© 2014 Optical Society of America

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References

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  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).
  2. X. Zhang, Z. Li, J. Chen, S. Yue, and Q. Gong, “A dichroic surface-plasmon-polariton splitter based on an asymmetric T-shape nanoslit,” Opt. Express 21(12), 14548–14554 (2013).
    [Crossref] [PubMed]
  3. P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
    [Crossref] [PubMed]
  4. S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
    [Crossref] [PubMed]
  5. V. M. Shalaev and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier, 2007).
  6. M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer, 2007).
  7. Y. G. Chen, F. Y. Yang, J. Liu, and Z. Y. Li, “Broadband focusing and demultiplexing of surface plasmon polaritons on metal surface by holographic groove patterns,” Opt. Express 22(12), 14727–14737 (2014).
    [Crossref] [PubMed]
  8. M. Aramesh, J. Cervenka, A. Roberts, A. Djalalian-Assl, R. Rajasekharan, J. Fang, K. Ostrikov, and S. Prawer, “Coupling of a single-photon emitter in nanodiamond to surface plasmons of a nanochannel-enclosed silver nanowire,” Opt. Express 22(13), 15530–15541 (2014).
    [Crossref] [PubMed]
  9. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Low-scattering surface plasmon refraction with isotropic materials,” Opt. Express 22(11), 13547–13554 (2014).
    [Crossref] [PubMed]
  10. Z. Ruan, H. Wu, M. Qiu, and S. Fan, “Spatial control of surface plasmon polariton excitation at planar metal surface,” Opt. Lett. 39(12), 3587–3590 (2014).
    [Crossref] [PubMed]
  11. J. Chen and X. Wang, “Plasmon mode characteristics of metallic nanowire in uniaxial anisotropic dielectric,” Opt. Lett. 39(14), 4088–4091 (2014).
    [Crossref] [PubMed]
  12. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [Crossref] [PubMed]
  13. E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
    [Crossref]
  14. N. M. R. Peres, “Colloquium: The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82(3), 2673–2700 (2010).
    [Crossref]
  15. V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
    [Crossref]
  16. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
    [Crossref]
  17. P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
    [Crossref] [PubMed]
  18. J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
    [Crossref] [PubMed]
  19. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
    [Crossref]
  20. M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
    [Crossref] [PubMed]
  21. J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
    [Crossref] [PubMed]
  22. F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
    [Crossref] [PubMed]
  23. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
    [Crossref]
  24. B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
    [Crossref]
  25. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
    [Crossref] [PubMed]
  26. A. Auditore, C. de Angelis, A. Locatelli, and A. B. Aceves, “Tuning of surface plasmon polaritons beat length in graphene directional couplers,” Opt. Lett. 38(20), 4228–4231 (2013).
    [Crossref] [PubMed]
  27. B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
    [Crossref] [PubMed]
  28. R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
    [Crossref] [PubMed]
  29. M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
    [Crossref] [PubMed]
  30. B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
    [Crossref]
  31. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984).
    [Crossref] [PubMed]
  32. P. Tassin, T. Koschny, and C. M. Soukoulis, “Applied physics. Graphene for terahertz applications,” Science 341(6146), 620–621 (2013).
    [Crossref] [PubMed]
  33. R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
    [Crossref]
  34. M. L. Cooper and S. Mookherjea, “Numerically-assisted coupled-mode theory for silicon waveguide couplers and arrayed waveguides,” Opt. Express 17(3), 1583–1599 (2009).
    [Crossref] [PubMed]
  35. A. Yariv and P. Yeh, Optical Electronics in Modern Communications (Oxford University Press, Inc, 2006).
  36. Y. Fan, B. Wang, K. Wang, H. Long, and P. Lu, “Talbot effect in weakly coupled monolayer graphene sheet arrays,” Opt. Lett. 39(12), 3371–3373 (2014).
    [Crossref] [PubMed]
  37. B. Wang, H. Huang, K. Wang, H. Long, and P. Lu, “Plasmonic routing in aperiodic graphene sheet arrays,” Opt. Lett. 39(16), 4867–4870 (2014).
    [Crossref] [PubMed]
  38. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006).
    [Crossref] [PubMed]
  39. R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15(19), 12174–12182 (2007).
    [Crossref] [PubMed]
  40. Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
    [Crossref]

2014 (10)

S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
[Crossref] [PubMed]

R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
[Crossref]

B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
[Crossref]

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Low-scattering surface plasmon refraction with isotropic materials,” Opt. Express 22(11), 13547–13554 (2014).
[Crossref] [PubMed]

Y. Fan, B. Wang, K. Wang, H. Long, and P. Lu, “Talbot effect in weakly coupled monolayer graphene sheet arrays,” Opt. Lett. 39(12), 3371–3373 (2014).
[Crossref] [PubMed]

Y. G. Chen, F. Y. Yang, J. Liu, and Z. Y. Li, “Broadband focusing and demultiplexing of surface plasmon polaritons on metal surface by holographic groove patterns,” Opt. Express 22(12), 14727–14737 (2014).
[Crossref] [PubMed]

Z. Ruan, H. Wu, M. Qiu, and S. Fan, “Spatial control of surface plasmon polariton excitation at planar metal surface,” Opt. Lett. 39(12), 3587–3590 (2014).
[Crossref] [PubMed]

M. Aramesh, J. Cervenka, A. Roberts, A. Djalalian-Assl, R. Rajasekharan, J. Fang, K. Ostrikov, and S. Prawer, “Coupling of a single-photon emitter in nanodiamond to surface plasmons of a nanochannel-enclosed silver nanowire,” Opt. Express 22(13), 15530–15541 (2014).
[Crossref] [PubMed]

J. Chen and X. Wang, “Plasmon mode characteristics of metallic nanowire in uniaxial anisotropic dielectric,” Opt. Lett. 39(14), 4088–4091 (2014).
[Crossref] [PubMed]

B. Wang, H. Huang, K. Wang, H. Long, and P. Lu, “Plasmonic routing in aperiodic graphene sheet arrays,” Opt. Lett. 39(16), 4867–4870 (2014).
[Crossref] [PubMed]

2013 (7)

X. Zhang, Z. Li, J. Chen, S. Yue, and Q. Gong, “A dichroic surface-plasmon-polariton splitter based on an asymmetric T-shape nanoslit,” Opt. Express 21(12), 14548–14554 (2013).
[Crossref] [PubMed]

A. Auditore, C. de Angelis, A. Locatelli, and A. B. Aceves, “Tuning of surface plasmon polaritons beat length in graphene directional couplers,” Opt. Lett. 38(20), 4228–4231 (2013).
[Crossref] [PubMed]

M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
[Crossref] [PubMed]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

P. Tassin, T. Koschny, and C. M. Soukoulis, “Applied physics. Graphene for terahertz applications,” Science 341(6146), 620–621 (2013).
[Crossref] [PubMed]

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
[Crossref] [PubMed]

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

2012 (6)

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

2011 (3)

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

2010 (2)

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[Crossref] [PubMed]

N. M. R. Peres, “Colloquium: The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82(3), 2673–2700 (2010).
[Crossref]

2009 (2)

2008 (1)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

2007 (3)

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15(19), 12174–12182 (2007).
[Crossref] [PubMed]

2006 (1)

1984 (1)

Aceves, A. B.

Alaee, R.

R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
[Crossref]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

Alù, A.

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Amin, M.

Aramesh, M.

Auditore, A.

Bagci, H.

M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
[Crossref] [PubMed]

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
[Crossref] [PubMed]

Baillargeat, D.

S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
[Crossref] [PubMed]

Berini, P.

Bezus, E. A.

Buckley, R.

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

Bunch, J. S.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Calle, F.

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

Castro Neto, A. H.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

Cervenka, J.

Chang, D. E.

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Chen, J.

Chen, P. Y.

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Chen, Y. G.

Cheng, J.

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Christensen, J.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Cooper, M. L.

Craighead, H. G.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Das Sarma, S.

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

de Abajo, F. J.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

de Angelis, C.

Djalalian-Assl, A.

Doskolovich, L. L.

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Fan, S.

Fan, Y.

Fang, J.

Farhat, M.

Frank, I. W.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

García de Abajo, F. J.

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

García-Vidal, F. J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[Crossref] [PubMed]

Gong, Q.

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Guenneau, S.

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
[Crossref] [PubMed]

Guinea, F.

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

Hanson, G. W.

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

Ho, H. P.

S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
[Crossref] [PubMed]

Huang, H.

Huidobro, P. A.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[Crossref] [PubMed]

Hwang, E. H.

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

Jablan, M.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

Kapon, E.

Katz, J.

Kazanskiy, N. L.

Koppens, F. H.

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Koppens, F. H. L.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Koschny, T.

P. Tassin, T. Koschny, and C. M. Soukoulis, “Applied physics. Graphene for terahertz applications,” Science 341(6146), 620–621 (2013).
[Crossref] [PubMed]

Kotov, V. N.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

Lederer, F.

R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
[Crossref]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

Li, S.

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Li, Z.

Li, Z. Y.

Liu, J.

Y. G. Chen, F. Y. Yang, J. Liu, and Z. Y. Li, “Broadband focusing and demultiplexing of surface plasmon polaritons on metal surface by holographic groove patterns,” Opt. Express 22(12), 14727–14737 (2014).
[Crossref] [PubMed]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Locatelli, A.

Long, H.

Lu, P.

Manjavacas, A.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Martín-Moreno, L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[Crossref] [PubMed]

McEuen, P. L.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Mookherjea, S.

Nesterov, M. L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[Crossref] [PubMed]

Novoselov, K. S.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Ostrikov, K.

Parpia, J. M.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Pedrós, J.

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

Pereira, V. M.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

Peres, N. M. R.

N. M. R. Peres, “Colloquium: The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82(3), 2673–2700 (2010).
[Crossref]

Ping Loh, K.

B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
[Crossref]

Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Prawer, S.

Qiu, M.

Rajasekharan, R.

Roberts, A.

Rockstuhl, C.

R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
[Crossref]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

Ruan, Z.

Schiefele, J.

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

Soljacic, M.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

Sols, F.

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

Soukoulis, C. M.

P. Tassin, T. Koschny, and C. M. Soukoulis, “Applied physics. Graphene for terahertz applications,” Science 341(6146), 620–621 (2013).
[Crossref] [PubMed]

Sun, Y.

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Tanenbaum, D. M.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Tassin, P.

P. Tassin, T. Koschny, and C. M. Soukoulis, “Applied physics. Graphene for terahertz applications,” Science 341(6146), 620–621 (2013).
[Crossref] [PubMed]

Teng, J.

B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
[Crossref]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Thongrattanasiri, S.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Uchoa, B.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

van der Zande, A. M.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Verbridge, S. S.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

Wang, B.

B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
[Crossref]

Y. Fan, B. Wang, K. Wang, H. Long, and P. Lu, “Talbot effect in weakly coupled monolayer graphene sheet arrays,” Opt. Lett. 39(12), 3371–3373 (2014).
[Crossref] [PubMed]

B. Wang, H. Huang, K. Wang, H. Long, and P. Lu, “Plasmonic routing in aperiodic graphene sheet arrays,” Opt. Lett. 39(16), 4867–4870 (2014).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Wang, K.

Wang, X.

Wu, H.

Yang, F. Y.

Yariv, A.

Yong, K. T.

S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
[Crossref] [PubMed]

Yu, R.

R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
[Crossref]

Yuan, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Yue, S.

Zeng, S.

S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
[Crossref] [PubMed]

Zhang, X.

B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
[Crossref]

X. Zhang, Z. Li, J. Chen, S. Yue, and Q. Gong, “A dichroic surface-plasmon-polariton splitter based on an asymmetric T-shape nanoslit,” Opt. Express 21(12), 14548–14554 (2013).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Zheng, Z.

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

ACS Nano (2)

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Chem. Soc. Rev. (1)

S. Zeng, D. Baillargeat, H. P. Ho, and K. T. Yong, “Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications,” Chem. Soc. Rev. 43(10), 3426–3452 (2014).
[Crossref] [PubMed]

J. Appl. Phys. (2)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

B. Wang, X. Zhang, K. Ping Loh, and J. Teng, “Tunable broadband transmission and phase modulation of light through graphene multilayers,” J. Appl. Phys. 115(21), 213102 (2014).
[Crossref]

Nano Lett. (2)

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Opt. Express (9)

X. Zhang, Z. Li, J. Chen, S. Yue, and Q. Gong, “A dichroic surface-plasmon-polariton splitter based on an asymmetric T-shape nanoslit,” Opt. Express 21(12), 14548–14554 (2013).
[Crossref] [PubMed]

Y. G. Chen, F. Y. Yang, J. Liu, and Z. Y. Li, “Broadband focusing and demultiplexing of surface plasmon polaritons on metal surface by holographic groove patterns,” Opt. Express 22(12), 14727–14737 (2014).
[Crossref] [PubMed]

M. Aramesh, J. Cervenka, A. Roberts, A. Djalalian-Assl, R. Rajasekharan, J. Fang, K. Ostrikov, and S. Prawer, “Coupling of a single-photon emitter in nanodiamond to surface plasmons of a nanochannel-enclosed silver nanowire,” Opt. Express 22(13), 15530–15541 (2014).
[Crossref] [PubMed]

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Low-scattering surface plasmon refraction with isotropic materials,” Opt. Express 22(11), 13547–13554 (2014).
[Crossref] [PubMed]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
[Crossref] [PubMed]

M. L. Cooper and S. Mookherjea, “Numerically-assisted coupled-mode theory for silicon waveguide couplers and arrayed waveguides,” Opt. Express 17(3), 1583–1599 (2009).
[Crossref] [PubMed]

P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006).
[Crossref] [PubMed]

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15(19), 12174–12182 (2007).
[Crossref] [PubMed]

Opt. Lett. (6)

Phys. Rev. B (3)

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

R. Yu, R. Alaee, F. Lederer, and C. Rockstuhl, “Manipulating the interaction between localized and delocalized surface plasmon-polaritons in graphene,” Phys. Rev. B 90(8), 085409 (2014).
[Crossref]

Phys. Rev. Lett. (3)

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
[Crossref] [PubMed]

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

N. M. R. Peres, “Colloquium: The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82(3), 2673–2700 (2010).
[Crossref]

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84(3), 1067–1125 (2012).
[Crossref]

Science (3)

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315(5811), 490–493 (2007).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

P. Tassin, T. Koschny, and C. M. Soukoulis, “Applied physics. Graphene for terahertz applications,” Science 341(6146), 620–621 (2013).
[Crossref] [PubMed]

Other (4)

A. Yariv and P. Yeh, Optical Electronics in Modern Communications (Oxford University Press, Inc, 2006).

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

V. M. Shalaev and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier, 2007).

M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer, 2007).

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Figures (7)

Fig. 1
Fig. 1 Schematic of the graphene multilayer. The position of graphene is denoted by xn with n∈[1, N] and N is the total number of graphene sheets. The interlayer space is denoted by d, σ g is the surface conductivity of graphene, and ε d the relative permittivity of the dielectric medium between graphene. An+, An and An+1+, An+1 are field amplitudes of SPPs in different regions.
Fig. 2
Fig. 2 Effective indexes of the supermodes in the graphene multilayer with different number of graphene sheets. (a) real and (b) imaginary parts of the effective indexes. The red circles denote the lowest loss supermodes. The green arrows represent the increasing order of the label s.
Fig. 3
Fig. 3 Normalized transverse magnetic field distributions of the supermodes in the graphene multilayers. (a)-(d) Mode profiles of the supermodes as N = 2, 3, 4, 5 respectively. The red dashed lines represent the position of graphene. The green circles indicate the numerical results based on the FDFD computation.
Fig. 4
Fig. 4 Normalized magnetic field distributions of the supermodes calculated by FDFD simulations based on coupled-mode theory. (a)-(e) The supermodes s = 1, 2, 3, 4, 5 as N = 5. The weight factor Cn is on the top of the corresponding figure.
Fig. 5
Fig. 5 Normalized magnetic field distributions of the lowest loss supermodes (s = N) in the graphene multilayer as Ν = 1, 2, 3, 4, 5, respectively.
Fig. 6
Fig. 6 Mode wavelength and propagation length of lowest loss supermode (s = N) as Ν = 1, 2, 3, 4, 5 versus: interlayer space d, chemical potential μ c and excitation wavelength λ. (a) and (b) fixed λ = 10 μm and μ c = 0.15 eV. (c) and (d) fixed λ = 10 μm and d = 50 nm. (e) and (f) fixed d = 50 nm and μ c = 0.15 eV.
Fig. 7
Fig. 7 Influence of the parameters: d, λ, N, μ c on the FOM of lowest loss supermode (s = N). (a) The function FOM(d, λ, N) at fixed μ c = 0.15 eV. (b) The function FOM(d, λ, μ c ) for fixed number of graphene sheets N = 5. (a) and (b) The dielectric medium is air (ε d = 1). (c) and (d) The dielectric medium is KCl (ε d = 2.13). In (b) and (d), the magnitudes of FOM for μ c = 0.15 eV (the lowest slice) are doubled.

Equations (5)

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H y ( x ) = { A n + exp [ i k x ( x x n 1 ) ] + A n exp [ i k x ( x x n ) ] x n 1 < x < x n A n + 1 + exp [ i k x ( x x n ) ] + A n + 1 exp [ i k x ( x x n + 1 ) ] x n < x < x n + 1
E z ( x ) = { η 0 k x ε d k 0 { A n + exp [ i k x ( x x n 1 ) ] A n exp [ i k x ( x x n ) ] } x n 1 < x < x n η 0 k x ε d k 0 { A n + 1 + exp [ i k x ( x x n ) ] A n + 1 exp [ i k x ( x x n + 1 ) ] } x n < x < x n + 1
[ A n + 1 + A n + 1 ] = M n [ A n + A n ] = [ 1 / 2 1 / ( 2 κ ) 1 / ( 2 u ) 1 / ( 2 κ u ) ] [ ( 1 i ξ κ ) u 1 + i ξ κ κ u κ ] [ A n + A n ]
[ A N + 1 + A N + 1 ] = M N M N 1 M 2 M 1 [ A 1 + A 1 ] = [ m 11 m 12 m 21 m 22 ] [ A 1 + A 1 ]
H y ( x , z ) = ( n = 1 N C n H y , n ( x ) ) exp ( i β z )

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