Abstract

We analyze and model the nonlocal response of ultrathin hyperbolic metasurfaces (HMTSs) by applying an effective medium approach. We show that the intrinsic spatial dispersion in the materials employed to realize the metasurfaces imposes a wavenumber cutoff on the hyperbolic isofrequency contour, inversely proportional to the Fermi velocity, and we compare it with the cutoff arising from the structure granularity. In the particular case of HTMSs implemented by an array of graphene nanostrips, we find that graphene nonlocality can become the dominant mechanism that closes the hyperbolic contour – imposing a wavenumber cutoff at around 300k0 – in realistic configurations with periodicity L<π/(300k0), thus providing a practical design rule to implement HMTSs at THz and infrared frequencies. In contrast, more common plasmonic materials, such as noble metals, operate at much higher frequencies, and therefore their intrinsic nonlocal response is mainly relevant in hyperbolic metasurfaces and metamaterials with periodicity below a few nm, being very weak in practical scenarios. In addition, we investigate how spatial dispersion affects the spontaneous emission rate of emitters located close to HMTSs. Our results establish an upper bound set by nonlocality to the maximum field confinement and light-matter interactions achievable in practical HMTSs, and may find application in the practical development of hyperlenses, sensors and on-chip networks.

© 2015 Optical Society of America

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References

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  1. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
    [Crossref] [PubMed]
  2. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
    [Crossref]
  3. V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
    [Crossref] [PubMed]
  4. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
    [Crossref] [PubMed]
  5. A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
    [Crossref] [PubMed]
  6. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
    [Crossref]
  7. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
    [Crossref]
  8. O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
    [Crossref]
  9. I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
    [Crossref]
  10. F. H. L. 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]
  11. F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
    [Crossref]
  12. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013).
    [Crossref] [PubMed]
  13. J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
    [Crossref] [PubMed]
  14. P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
    [Crossref]
  15. D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
    [Crossref]
  16. S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
    [Crossref] [PubMed]
  17. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
    [Crossref]
  18. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
    [Crossref]
  19. G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
    [Crossref]
  20. D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).
  21. D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
    [Crossref]
  22. J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).
  23. A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
    [Crossref]
  24. M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
    [Crossref] [PubMed]
  25. O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).
  26. E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
    [Crossref]
  27. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
    [Crossref]
  28. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
    [Crossref] [PubMed]
  29. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
    [Crossref]
  30. W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
    [Crossref] [PubMed]
  31. M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).
  32. A. D. Boardman, Electromagnetic Surface Modes, John Wiley and Sons, Chichester (1982).
  33. H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
    [Crossref]
  34. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : the art of scientific computing, 3rd edition (Cambridge University Press, 2007).
  35. L. Chen, C. Zhang, J. Zhou, and L. J. Guo, “Probing the Ultrathin Limit of Hyperbolic Meta-material: Nonlocality Induced Topological Transitions,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FTu4C.6.
  36. D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
    [Crossref] [PubMed]
  37. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
    [Crossref]
  38. J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
    [Crossref] [PubMed]
  39. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
  40. N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).
  41. O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
    [Crossref] [PubMed]

2015 (7)

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

2014 (4)

F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

2013 (10)

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
[Crossref] [PubMed]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

2012 (5)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
[Crossref]

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

2011 (2)

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
[Crossref]

F. H. L. 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)

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

2009 (1)

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

2008 (1)

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

2007 (2)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

2006 (1)

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

2003 (1)

H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
[Crossref]

1987 (1)

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Akinwande, D.

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Altug, H.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Alù, A.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
[Crossref]

Alvarez-Melcon, A.

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

Araneo, R.

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Barrera, R. G.

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Bilow, H. J.

H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
[Crossref]

Bogdanov, A. A.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Botten, L. C.

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Buljan, H.

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

Burghignoli, P.

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Caloz, C.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Chang, D. E.

F. H. L. 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.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Chen, P.

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Correas-Serrano, D.

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

Crassee, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

de Leon, N. P.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

del CastilloMussot, M.

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Devlin, R. C.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Di Stefano, O.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Dibos, A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Drachev, V. P.

Esquius-Morote, M.

Etezadi, D.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Falkovsky, L. A.

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

Fallahi, A.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

Fina, N.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Forati, E.

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

Gaponenko, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

García de Abajo, F. J.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

F. H. L. 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]

Girlanda, R.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Gomez-Diaz, J. S.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

Gómez-Díaz, J. S.

Goussetis, G.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Grant, G.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Guermoune, A.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Ham, M. H.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Hanson, G. W.

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

High, A. A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Huang, H.

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Iorsh, I.

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Iorsh, I. V.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Jablan, M.

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

Jacob, Z.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Janner, D.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Kildishev, A. V.

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Kivshar, Yu. S.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Koppens, F. H. L.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

F. H. L. 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]

Kretzschmar, I.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Krishnamoorthy, H. N. S.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Kuzmenko, A. B.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Lee, B. H.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Limaj, O.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Lioubtchenko, D.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Lovat, G.

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Low, T.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

Lukin, M. D.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Luukkonen, O.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Martín-Moreno, L.

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

McPhedran, R. C.

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Menon, V. M.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Mochan, W. L.

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Moon, K. J.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Mortensen, N. A.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Mosig, J.

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

Myoung, J. M.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Narimanov, E.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Nguyen, H. V.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Nicorovici, N. A. P.

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Nikitin, A. Y.

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Orlita, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Ostler, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Ovcharenko, A. I.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Park, H.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Perczel, J.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Perruisseau-Carrier, J.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
[Crossref]

Pieruccini, M.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Podolskiy, V. A.

Polking, M.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Potemski, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Pruneri, V.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Raisanen, A.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Rodrigo, D.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Ross, C. A.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Savasta, S.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Seyller, T.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Siaj, M.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

Simovski, C.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Skulason, H. S.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Soljacic, M.

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

Son, J. G.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Son, M.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Song, M.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Sounas, D. L.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Strano, M. S.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Szkopek, T.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Tamagnone, M.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

Thongrattanasiri, S.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

Tretyakov, S.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Trushkov, I.

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

Tymchenko, M.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Varlamov, A. A.

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

Walter, A. L.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Wild, D. S.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Wubs, M.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Yakovlev, A. B.

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

Yan, W.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Yermakov, O. Y.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

ACS Nano (1)

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

ACS Photonics (2)

F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Adv. Mater. (1)

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Eur. Phys. J. B (1)

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

IEEE Anten. Wirel. Propag. Lett. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

IEEE Trans. Antenn. Propag. (4)

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
[Crossref]

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

IEEE Trans. Microw. Theory Tech. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. Appl. Phys. (1)

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
[Crossref]

J. Phys. Condens. Matter (2)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Nano Lett. (2)

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

F. H. L. 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]

Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Nature (1)

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Mater. Express (1)

Phys. B Condens. Matter (1)

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Phys. Rev. B (7)

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
[Crossref]

Phys. Rev. B Condens. Matter (2)

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

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Phys. Rev. Lett. (2)

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

Science (2)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Other (4)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : the art of scientific computing, 3rd edition (Cambridge University Press, 2007).

L. Chen, C. Zhang, J. Zhou, and L. J. Guo, “Probing the Ultrathin Limit of Hyperbolic Meta-material: Nonlocality Induced Topological Transitions,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FTu4C.6.

A. D. Boardman, Electromagnetic Surface Modes, John Wiley and Sons, Chichester (1982).

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Figures (8)

Fig. 1
Fig. 1 Schematic of a hyperbolic metasurface composed of 2D strips with widths W and unit cell period L. The inset shows an equivalent representation of one unit-cell modelled by two strips described by effective conductivity tensors σ ¯ ¯ and σ C ¯ ¯ .
Fig. 2
Fig. 2 Nonlocal conductivity tensor elements of graphene in Cartesian coordinates, normalized versus the material local response. Results are given as a function of real k x / k 0 and k y / k 0 in-plane wavenumbers. Other parameters are f=2.5 THz,   μ c =0.02 eV , and τ=0.3 ps.
Fig. 3
Fig. 3 Imaginary parts of the (a) longitudinal and (b) transverse nonlocal conductivity components of graphene normalized versus the local response. Results are given as a function of real k x / k 0 and k y / k 0 in-plane wavenumbers. Other parameters are f=2.5 THz,  μ c =0.02 eV , and τ=0.3 ps
Fig. 4
Fig. 4 Effective conductivity tensor of a nonlocal graphene-based HMTS. The structure comprises an array of spatially-dispersive graphene strips with width W=50 nm and period L=100 nm. Other parameters are f=2.5 THz, μ c =0.2 eV, and τ=0.3 ps.
Fig. 5
Fig. 5 Isofrequency contours of the hyperbolic plasmons supported by an array of graphene strips with W=0.5 L for (a) L=30 nm, (b) L=100 nm, (c) L=300 nm, and (d) L=700 nm. Solid blue (green) lines computed with EMA considering local (nonlocal) graphene conductivity σ g ¯ ¯ . Dashed red lines computed using a full-wave mode matching approach with local σ g ¯ ¯ . Markers computed with FEM eigenfrequency solver COMSOL Multyphysics using nonlocal σ g ¯ ¯ .
Fig. 6
Fig. 6 Isofrequency contours of plasmons supported by an array of graphene strips versus (a) strip width W ( L=100 nm, f=4 THz,   μ c =0.02 eV, τ = 0.3 ps) and (b) electron relaxation time τ ( L=100 nm, W=30 nm, f=2.5 THz, μ c =0.3 eV). (c)-(d) shows similar results versus chemical potential μ c with (c) L=100 nm, W=25 nm, and f=2.5 THz and (d) L=120 nm, W=110 nm, and f=4.0 THz. Solid (dashed) lines computed with EMA considering local (nonlocal) graphene conductivity σ g ¯ ¯ .
Fig. 7
Fig. 7 SER (in logarithm scale) of a z-oriented dipole as a function of its distance from an array of graphene strips, computed with EMA derived in Section II. (a)-(b) Results versus strip width W for local and nonlocal graphene responses, respectively. Parameters are L=50 nm, f=2.5 THz, μ c =0.2 eV, and τ = 0.3 ps.
Fig. 8
Fig. 8 Fundamental limits imposed by nonlocality to the SER of a z-oriented emitter. (a) SER (in logarithm scale) versus the dipole distance to the metasurface. The width of the graphene strips is set to W=15 nm and μ c =0.2 eV. (b) SER (in logarithm scale) of an emitter located exactly on the HMTSs versus the structure physical dimensions and chemical potential. Parameters are L=100 nm, f=2.5 THz, and τ = 0.3 ps.

Equations (21)

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σ m ¯ ¯ =( σ xx m σ xy m σ yx m σ yy m )
σ c ¯ ¯ =( σ xx c 0 0 0 )    with   σ xx c =i ω ε 0 ε eff L π ln[ csc( πG 2L ) ]
E x ( x )= J x 1 σ xx ( x ) E y σ xy σ xx ( x ) , J y ( x )= J x σ yx ( x ) σ xx ( x ) + E y [ σ yy ( x ) σ yx ( x ) σ xy ( x ) σ xx ( x ) ],
1 σ xx eff = 1 L L 1 σ xx (x) dx , σ xy eff = σ xx eff L L σ xy (x) σ xx (x) dx , σ xy eff = σ xx eff L L σ xy (x) σ xx (x) dx ,
σ yy eff = 1 L L σ yy (x)dx 1 L L σ xy (x) σ yx (x) σ xx (x) dx + σ xy eff σ yx eff σ xx eff ,
σ xx eff = L σ xx m σ xx c W σ xx c +(LW) σ xx g , σ xy,eff = σ xy,eff = σ xx,eff W L σ xy m σ xx m .
σ yy eff = W L σ yy m W L σ yx m σ xy m σ xx m σ xx0 + σ yx eff σ xy eff σ xx eff .
σ xx g ( k x , k y )=γ I ϕ xx + γ D Δ k y ( I ϕ xx k y I ϕ yx k x ) D σ ,
σ xy g ( k x , k y )=γ I ϕ xy + γ D Δ k y ( I ϕ xy k y I ϕ yy k x ) D σ ,
σ yx g ( k x , k y )=γ I ϕ yx + γ D Δ k x ( I ϕ yx k x I ϕ xx k y ) D σ ,
σ yy g ( k x , k y )=γ I ϕ yy + γ D Δ k x ( I ϕ yy k x I ϕ xy k y ) D σ ,
I ϕ xx ( k x , k y )=2π v F 2 k y 2 k ρ 2 Rα v F k x k q 2 α 2 k q 2 (1R) v F 2 (α+ v F k x ) k ρ 4 ,
I ϕ xy ( k x , k y )= I ϕ yx ( k x , k y )=2π k x k y v F 2 k ρ 2 R+2α v F k x + α 2 k q 2 (1R) v F 2 (α+ v F k x ) k ρ 4 ,
I ϕ yy ( k x , k y )=2π v F 2 k x 2 k ρ 2 R+α v F k x k q 2 + α 2 k q 2 (1R) v F 2 (α+ v F k x ) k ρ 4 ,
γ=i q e 2 k B T π 2 2 log{ 2[ 1+cosh( μ c k B T ) ] }, γ D =i v F 2πωτ , D σ =1+ γ D Δ k ρ 2 ,
Δ= 2π v F k ρ 2 ( 1 α α 2 v F 2 k ρ 2 ),R( k x , k y )= α+ v F k x α 2 v F 2 k ρ 2 ,α=ω+i/τ,
k ρ 2 = k x 2 + k y 2 , k q 2 = k x 2 k y 2
σ d ¯ ¯ ( k ρ )=( σ ρ ( k ρ ) 0 0 σ ϕ ( k ρ ) ),
ε m ρ ( k ρ )=1 ω p 2 ω 2 +iωγ β 2 k ρ 2 , ε m ϕ ( k ρ )=1 ω p 2 ω 2 +iωγ ,
σ eff ¯ ¯ ( k x , k y )=( σ xx eff ( k x , k y ) σ xy eff ( k x , k y ) σ yx eff ( k x , k y ) σ yy eff ( k x , k y ) ).
k 0 k z [ 4+ η 0 2 { σ xx eff ( k x , k y ) σ yy eff ( k x , k y ) σ xy eff ( k x , k y ) σ yx eff ( k x , k y ) } ]+2 k 0 2 η 0 [ σ xx eff ( k x , k y )+ σ yy eff ( k x , k y ) ] 2 η 0 [ σ xx eff ( k x , k y ) k x 2 + σ yy eff ( k x , k y ) k y 2 + k x k y ( σ xy eff ( k x , k y )+ σ xy eff ( k x , k y ) ) ]=0,

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