Abstract

We propose a scheme to directionally couple light into graphene plasmons by placing a graphene sheet on a magneto-optical substrate. When a magnetic field is applied parallel to the surface, the graphene plasmon dispersion relation becomes asymmetric in the forward and backward directions. It is possible to achieve unidirectional excitation of graphene plasmons with normally incident illumination by applying a grating to the substrate. The directionality can be actively controlled by electrically gating the graphene, or by varying the magnetic bias. This scheme may have applications in graphene-based opto-electronics and sensing.

© 2015 Optical Society of America

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References

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  4. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100(11), 117401 (2008).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. R. Roldán, J.-N. Fuchs, and M. O. Goerbig, “Collective modes of doped graphene and a standard two-dimensional electron gas in a strong magnetic field: linear magnetoplasmons versus magnetoexcitons,” Phys. Rev. B 80(8), 085408 (2009).
    [Crossref]
  28. N. Chamanara, S. Dimitrios, and C. Christophe, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. Exp. 21(9), 11248–11256 (2013).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  33. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. Garcia de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2011).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  37. J. G. Rivas, C. Janke, P. H. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Ex. 13(3), 847–859 (2005).
    [Crossref]
  38. B. Hu, Q. J. Wang, and Y. Zhang, “Slowing down terahertz waves with tunable group velocities in a broad frequency range by surface magneto plasmons,” Opt. Ex. 20(9), 10071–11076 (2012).
    [Crossref]
  39. B. Hu, Q. J. Wang, and Y. Zhang, “Broadly tunable one-way terahertz plasmonic waveguide based on nonreciprocal surface magneto plasmons,” Opt. Lett. 37(11), 1895–1897 (2012).
    [Crossref] [PubMed]
  40. S. Thongrattanasiri, A. Manjavacas, and F. J. G. de Abajo, “Quantum finite-size effects in graphene plasmons,” ACS Nano 6(2), 1766–1775 (2012).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  43. H. Yan, Z. Li, X. Li, W. Zhu, P. Avouris, and F. Xia, “Infrared spectroscopy of tunable dirac terahertz magneto-plasmons in graphene,” Nano Lett. 12(7), 3766–3771 (2012).
    [Crossref] [PubMed]
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    [Crossref]
  45. C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nature Nano. 5(10), 722–726 (2010).
    [Crossref]
  46. J. C. Jang, S. Adam, J.-H. Chen, E. D. Williams, S. Das Sarma, and M. S. Fuhrer, “Tuning the effective fine structure constant in graphene: opposing effects of dielectric screening on short- and long-range potential scattering,” Phys. Rev. Lett. 101(14), 146805 (2008).
    [Crossref] [PubMed]
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    [Crossref]
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2014 (1)

F. Liu, Y. D. Chong, S. Adam, and M. Polini, “Gate-tunable coherent perfect absorption of terahertz radiation in graphene,” 2D Mat. 1, 031001 (2014).
[Crossref]

2013 (12)

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nature Phot. 7(5), 394–399 (2013).
[Crossref]

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]

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]

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

F. J. Rodrguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light: Sci. & Appl. 2(3), e70 (2013).
[Crossref]

N. Chamanara, S. Dimitrios, and C. Christophe, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. Exp. 21(9), 11248–11256 (2013).
[Crossref]

J. S. Gomez-Diaz, J. R. Mosig, and J. Perruisseau-Carrier, “Effect of spatial dispersion on surface waves propagating along graphene sheets,” Antennas and Prop., IEEE Trans. 61(7), 3589–3596 (2013).
[Crossref]

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Yu Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 114008 (2013).
[Crossref]

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27(10), 1341001 (2013).
[Crossref]

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” N. J. Phys. 15(11), 113003 (2013).
[Crossref]

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, H. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3(4), 041019 (2013).

2012 (12)

B. Hu, Q. J. Wang, and Y. Zhang, “Broadly tunable one-way terahertz plasmonic waveguide based on nonreciprocal surface magneto plasmons,” Opt. Lett. 37(11), 1895–1897 (2012).
[Crossref] [PubMed]

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

C. J. Docherty and M. B. Johnston, “Terahertz properties of graphene,” J. Infrared, Millimeter, and Terahertz Waves,  33(8), 797–815 (2012)
[Crossref]

H. Yan, Z. Li, X. Li, W. Zhu, P. Avouris, and F. Xia, “Infrared spectroscopy of tunable dirac terahertz magneto-plasmons in graphene,” Nano Lett. 12(7), 3766–3771 (2012).
[Crossref] [PubMed]

B. Hu, Q. J. Wang, and Y. Zhang, “Slowing down terahertz waves with tunable group velocities in a broad frequency range by surface magneto plasmons,” Opt. Ex. 20(9), 10071–11076 (2012).
[Crossref]

S. Thongrattanasiri, A. Manjavacas, and F. J. G. de Abajo, “Quantum finite-size effects in graphene plasmons,” ACS Nano 6(2), 1766–1775 (2012).
[Crossref] [PubMed]

A. Ferreira, N. M. R. Peres, and A. H. Castro Neto, “Confined magneto-optical waves in graphene,” Phys. Rev. B 85(20), 205426 (2012).
[Crossref]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. CastroNeto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

A. Ferreira and N. M. R. Peres, “Complete light absorption in graphene-metamaterial corrugated structures,” Phys. Rev. B 86(20), 205401 (2012).
[Crossref]

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nature Nano. 7(5), 330–334 (2012).
[Crossref]

M. Engel, M. Steiner, A. Lombardo, A. C. Ferrari, H. v. Loehneysen, P. Avouris, and R. Krupke, “Light-matter interaction in a microcavity-controlled graphene transistor,” Nature Commun. 3, 906–911 (2012).
[Crossref]

M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
[Crossref] [PubMed]

2011 (7)

A. A. Balandin, “Thermal properties of graphene and nanostructured carbon materials,” Nature Mat. 10(8), 569–581 (2011).
[Crossref]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nano. 6(10), 630–634 (2011).
[Crossref]

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

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

B. Guo, L. Fang, B. Zhang, and J. R. Gong, “Graphene doping: a review,” Insciences J. 1(2), 80–89 (2011).
[Crossref]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. Garcia de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2011).
[Crossref] [PubMed]

Z.-Y. Ong and E. Pop, “Effect of substrate modes on thermal transport in supported graphene,” Phys. Rev. B 84(7), 075471 (2011).
[Crossref]

2010 (2)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nature Nano. 5(10), 722–726 (2010).
[Crossref]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref] [PubMed]

2009 (3)

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

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109 (2009).
[Crossref]

R. Roldán, J.-N. Fuchs, and M. O. Goerbig, “Collective modes of doped graphene and a standard two-dimensional electron gas in a strong magnetic field: linear magnetoplasmons versus magnetoexcitons,” Phys. Rev. B 80(8), 085408 (2009).
[Crossref]

2008 (6)

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008).
[Crossref] [PubMed]

C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of mono-layer graphene,” Science 321(5887), 385–388 (2008).
[Crossref] [PubMed]

A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100(11), 117401 (2008).
[Crossref] [PubMed]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Ex. 16(14), 10858–10866 (2008).
[Crossref]

T. Xu, Y. Zhao, D. Gan, C. Wang, C. Du, and X. Luo, “Directional excitation of surface plasmons with subwave-length slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

J. C. Jang, S. Adam, J.-H. Chen, E. D. Williams, S. Das Sarma, and M. S. Fuhrer, “Tuning the effective fine structure constant in graphene: opposing effects of dielectric screening on short- and long-range potential scattering,” Phys. Rev. Lett. 101(14), 146805 (2008).
[Crossref] [PubMed]

2007 (1)

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

2006 (1)

Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W. Tan, M. Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L. Stormer, and P. Kim, “Landau-level splitting in graphene in high magnetic fields,” Phys. Rev. Lett. 96(13), 136806 (2006).
[Crossref] [PubMed]

2005 (1)

J. G. Rivas, C. Janke, P. H. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Ex. 13(3), 847–859 (2005).
[Crossref]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Adam, S.

F. Liu, Y. D. Chong, S. Adam, and M. Polini, “Gate-tunable coherent perfect absorption of terahertz radiation in graphene,” 2D Mat. 1, 031001 (2014).
[Crossref]

J. C. Jang, S. Adam, J.-H. Chen, E. D. Williams, S. Das Sarma, and M. S. Fuhrer, “Tuning the effective fine structure constant in graphene: opposing effects of dielectric screening on short- and long-range potential scattering,” Phys. Rev. Lett. 101(14), 146805 (2008).
[Crossref] [PubMed]

Akimov, I. A.

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, H. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3(4), 041019 (2013).

Alaee, R.

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

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. CastroNeto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Andrews, A. M.

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J. G. Rivas, C. Janke, P. H. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Ex. 13(3), 847–859 (2005).
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C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nature Nano. 5(10), 722–726 (2010).
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T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Yu Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 114008 (2013).
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Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W. Tan, M. Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L. Stormer, and P. Kim, “Landau-level splitting in graphene in high magnetic fields,” Phys. Rev. Lett. 96(13), 136806 (2006).
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M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
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H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nature Phot. 7(5), 394–399 (2013).
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H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nature Nano. 7(5), 330–334 (2012).
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H. Yan, Z. Li, X. Li, W. Zhu, P. Avouris, and F. Xia, “Infrared spectroscopy of tunable dirac terahertz magneto-plasmons in graphene,” Nano Lett. 12(7), 3766–3771 (2012).
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S. Thongrattanasiri, A. Manjavacas, and F. J. G. de Abajo, “Quantum finite-size effects in graphene plasmons,” ACS Nano 6(2), 1766–1775 (2012).
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Antennas and Prop., IEEE Trans. (1)

J. S. Gomez-Diaz, J. R. Mosig, and J. Perruisseau-Carrier, “Effect of spatial dispersion on surface waves propagating along graphene sheets,” Antennas and Prop., IEEE Trans. 61(7), 3589–3596 (2013).
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Appl. Phys. Lett. (2)

T. Xu, Y. Zhao, D. Gan, C. Wang, C. Du, and X. Luo, “Directional excitation of surface plasmons with subwave-length slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
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X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
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Insciences J. (1)

B. Guo, L. Fang, B. Zhang, and J. R. Gong, “Graphene doping: a review,” Insciences J. 1(2), 80–89 (2011).
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Int. J. Mod. Phys. B (1)

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27(10), 1341001 (2013).
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N. J. Phys. (1)

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” N. J. Phys. 15(11), 113003 (2013).
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Nano Lett. (3)

H. Yan, Z. Li, X. Li, W. Zhu, P. Avouris, and F. Xia, “Infrared spectroscopy of tunable dirac terahertz magneto-plasmons in graphene,” Nano Lett. 12(7), 3766–3771 (2012).
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Nature (1)

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H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nature Nano. 7(5), 330–334 (2012).
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C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nature Nano. 5(10), 722–726 (2010).
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Nature Phot. (1)

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nature Phot. 7(5), 394–399 (2013).
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Opt. Ex. (3)

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Ex. 16(14), 10858–10866 (2008).
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Figures (3)

Fig. 1
Fig. 1 (a) Schematic of the device under investigation. A graphene sheet lies on a grated magneto-optical substrate which has period Λ. A magnetic field is applied along the ŷ direction, parallel to the surface and perpendicular to the normal of the grating. Normally incident monochromatic light excites surface plasmons propagating in a single direction. (b) Dispersion relations for graphene surface plasmons on magneto-optical substrates, calculated numerically from Eqs. (4)(7). The wavenumber and frequency are respectively scaled relative to kF and ωF, the Fermi wavenumber and frequency in the graphene sheet (which are both proportional to the Fermi level). We assume a Fermi velocity of 106 ms−1 and τ = 0 in the graphene sheet. The relative dielectric constants ε1 = 3 and ε2 = 1, and the magneto-optical parameter α, are taken to be frequency-independent. For α = 0, the dispersion relation is symmetric; setting α ≠ 0 makes it asymmetric.
Fig. 2
Fig. 2 (a) Absorption spectrum of graphene surface plasmon resonances, obtained by full-wave simulations of the system shown in Fig. 1(a). With zero applied magnetic field (blue triangles), there is a single absorption peak, corresponding to bi-directional excitation of surface plasmons. With a 1T applied magnetic field (red circles), there are two distinct absorption peaks, corresponding to plasmons propagating in the ± directions. N labels the grating index; the plasmon wavevectors are ±2πN/Λ, where Λ is the grating period. (b)–(d) Plots of Hz for three resonances; black arrows indicate the local Poynting vector.
Fig. 3
Fig. 3 Active control of plasmon directionality. (a) Plasmon resonance peaks, corresponding to left-moving (L) and right-moving (R) plasmons, can be shifted by tuning the graphene Fermi level EF. At a fixed operating frequency (vertical dashes), we excite right-moving plasmons at EF = 1 eV, or right-moving plasmons at EF = 1.28 eV. (b) Power flux through one unit cell, in the x direction (calculated by integrating the Poynting vector) versus EF, for a fixed 56.6 THz illumination frequency. These results were obtained from finite-element simulations, with all other parameters the same as in Fig. 2.

Equations (12)

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ɛ = [ ɛ 1 0 i α 0 ɛ 0 - i α 0 ɛ 1 ] .
E j ( x , z ) = ( E jx , 0 , E jz ) e iqx e - k j | z |
H j ( x , z ) = ( 0 , H jy , 0 ) e iqx e - k j | z | ,
ɛ 2 k 2 + ɛ 1 2 - α 2 k 1 ɛ 1 + q α + i σ θ ɛ 0 = 0 ,
k 1 = q 2 - ( ɛ 1 2 - α 2 ɛ 1 ) ω 2 c 2
k 2 = q 2 - ɛ 2 ω 2 c 2 .
σ ( ω ) = e 2 E F π 2 i ω + i / τ ,
ω e 2 E F π 2 ɛ 0 | q | ɛ 1 + ɛ 2 α ,
δ ω ω 0 ~ α ɛ 1 + ɛ 2 ,
ɛ 1 ( ω ) = ( 1 - ω p 2 ω 2 - ω c 2 ) ω
α ( ω ) = ω p 2 ω c ω ( ω 2 - ω c 2 ) ɛ ,
δ ω ~ ω H c ω p 2 / ω 0 2 ɛ - 1 + 1 - ω p 2 / ω 0 2 .

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