Abstract

The highly unidirectional excitation of graphene plasmons (GPs) through near-field interference of orthogonally polarized dipoles is investigated. The preferred excitation direction of GPs by a circularly polarized dipole can be simply understood with the angular momentum conservation law. Moreover, the propagation direction of GPs can be switched not only by changing the phase difference between dipoles, but also by placing the z-polarized dipole to its image position, whereas the handedness of the background field remains the same. The unidirectional excitation of GPs can be extended into arc graphene surface as well. Furthermore, our proposal on directional generation of GPs can be realized in a semiconductor nanowire/graphene system, where a semiconductor nanowire can mimic a circularly polarized dipole when illuminated by two orthogonally polarized plane waves.

© 2016 Optical Society of America

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References

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2016 (1)

B. Wu, B. Zhu, G. Ren, and S. Jian, “Circular polarization-dependent wavefront control of plasmons on graphene,” IEEE Photonics Technol. Lett. 28, 1940–1943 (2016).
[Crossref]

2015 (5)

P. Arneberg, K. Storhaug, and L. Sandvik, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 609–613 (2015).

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, 165–168 (2015).
[Crossref] [PubMed]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

F. Liu, C. Qian, and Y. D. Chong, “Directional excitation of graphene surface plasmons,” Opt. Express 23, 2383–2391 (2015).
[Crossref] [PubMed]

B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23, 32420–32428 (2015).
[Crossref] [PubMed]

2014 (6)

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Opt. 16, 105002 (2014).
[Crossref]

D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
[Crossref]

J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346, 67–71 (2014).
[Crossref] [PubMed]

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8, 1086–1101 (2014).
[Crossref] [PubMed]

F. J. García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photonics 1, 135–152 (2014).
[Crossref]

2013 (4)

Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
[Crossref] [PubMed]

M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329–340 (2013).
[Crossref]

F. J. Rodríguez-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, 328–330 (2013).
[Crossref] [PubMed]

W. B. Lu, W. Zhu, H. J. Xu, Z. H. Ni, Z. G. Dong, and T. J. Cui, “Flexible transformation plasmonics using graphene,” Opt. Express 21, 10475–10482 (2013).
[Crossref] [PubMed]

2012 (4)

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

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

S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

2011 (6)

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

A. Y. Nikitin, F. Guinea, F. García-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84, 195446 (2011).
[Crossref]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (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, 251109 (2011).
[Crossref]

2009 (2)

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

H. Kim and B. Lee, “Unidirectional surface plasmon polariton excitation on single slit with oblique backside illumination,” Plasmonics 4, 153–159 (2009).
[Crossref]

2008 (1)

G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

2007 (2)

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

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref] [PubMed]

2006 (1)

V. Gusynin, S. Sharapov, and J. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

1994 (1)

S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497 (1994).
[Crossref]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref] [PubMed]

Alonso-González, P.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

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, 165–168 (2015).
[Crossref] [PubMed]

Andreev, G.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Arneberg, P.

P. Arneberg, K. Storhaug, and L. Sandvik, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 609–613 (2015).

Avouris, P.

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8, 1086–1101 (2014).
[Crossref] [PubMed]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

Bai, B.

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, 251109 (2011).
[Crossref]

Bao, W.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Britnell, L.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Buljan, H.

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

Capasso, F.

Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
[Crossref] [PubMed]

Carbotte, J.

V. Gusynin, S. Sharapov, and J. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

Casanova, F.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

Centeno, A.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

Chang, D. E.

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

Chen, J.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

Chong, Y. D.

Cui, T. J.

Dominguez, G.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Dong, Z. G.

Echtermeyer, T.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Engheta, N.

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

Enk, S. J. v.

S. J. v. Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” Europhys. Lett. 25, 497 (1994).
[Crossref]

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, 165–168 (2015).
[Crossref] [PubMed]

Falkovsky, L.

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

Fei, Z.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Ferrari, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref] [PubMed]

Gao, Y.

B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23, 32420–32428 (2015).
[Crossref] [PubMed]

B. Zhu, G. Ren, Y. Gao, B. Wu, Y. Lian, and S. Jian, “Creation of graphene plasmons vortex via cross shape nanoantennas under linearly polarized incidence,” Plasmonics, in press (2016).
[Crossref]

Garcia de Abajo, F. J.

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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
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T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
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B. Wu, B. Zhu, G. Ren, and S. Jian, “Circular polarization-dependent wavefront control of plasmons on graphene,” IEEE Photonics Technol. Lett. 28, 1940–1943 (2016).
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B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23, 32420–32428 (2015).
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Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
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F. H. Koppens, D. E. Chang, and F. J. Garcia de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011).
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S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
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S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
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S.-Y. Lee, I.-M. Lee, K.-Y. Kim, and B. Lee, “Comments on ’near-field interference for the unidirectional excitation of electromagnetic guided modes’,” arXiv:1306.5068 (2013).

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M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329–340 (2013).
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S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
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S.-Y. Lee, I.-M. Lee, K.-Y. Kim, and B. Lee, “Comments on ’near-field interference for the unidirectional excitation of electromagnetic guided modes’,” arXiv:1306.5068 (2013).

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S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
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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, 251109 (2011).
[Crossref]

Lian, Y.

B. Zhu, G. Ren, Y. Gao, B. Wu, Y. Lian, and S. Jian, “Creation of graphene plasmons vortex via cross shape nanoantennas under linearly polarized incidence,” Plasmonics, in press (2016).
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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, 165–168 (2015).
[Crossref] [PubMed]

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Liu, M.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
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T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
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M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329–340 (2013).
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[PubMed]

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Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Opt. 16, 105002 (2014).
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A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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A. Y. Nikitin, F. Guinea, F. García-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84, 195446 (2011).
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A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
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L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
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D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
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[Crossref] [PubMed]

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S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

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M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329–340 (2013).
[Crossref]

Park, J.

S.-Y. Lee, I.-M. Lee, J. Park, S. Oh, W. Lee, K.-Y. Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
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P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
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[Crossref] [PubMed]

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

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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, 165–168 (2015).
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Rauschenbeutel, A.

J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346, 67–71 (2014).
[Crossref] [PubMed]

Ren, G.

B. Wu, B. Zhu, G. Ren, and S. Jian, “Circular polarization-dependent wavefront control of plasmons on graphene,” IEEE Photonics Technol. Lett. 28, 1940–1943 (2016).
[Crossref]

B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23, 32420–32428 (2015).
[Crossref] [PubMed]

B. Zhu, G. Ren, Y. Gao, B. Wu, Y. Lian, and S. Jian, “Creation of graphene plasmons vortex via cross shape nanoantennas under linearly polarized incidence,” Plasmonics, in press (2016).
[Crossref]

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Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

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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, 165–168 (2015).
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K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
[Crossref]

F. J. Rodríguez-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, 328–330 (2013).
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P. Arneberg, K. Storhaug, and L. Sandvik, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 609–613 (2015).

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V. Gusynin, S. Sharapov, and J. Carbotte, “Unusual microwave response of dirac quasiparticles in graphene,” Phys. Rev. Lett. 96, 256802 (2006).
[Crossref] [PubMed]

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A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

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Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
[Crossref] [PubMed]

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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
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P. Arneberg, K. Storhaug, and L. Sandvik, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 609–613 (2015).

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

Tame, M.

M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329–340 (2013).
[Crossref]

Tan, Q.

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, 251109 (2011).
[Crossref]

Thiemens, M.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Thongrattanasiri, S.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

Vakil, A.

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

Varlamov, A.

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

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P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

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J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346, 67–71 (2014).
[Crossref] [PubMed]

Wagner, M.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Wan, C.

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

Wang, P.

Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Opt. 16, 105002 (2014).
[Crossref]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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B. Wu, B. Zhu, G. Ren, and S. Jian, “Circular polarization-dependent wavefront control of plasmons on graphene,” IEEE Photonics Technol. Lett. 28, 1940–1943 (2016).
[Crossref]

B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23, 32420–32428 (2015).
[Crossref] [PubMed]

B. Zhu, G. Ren, Y. Gao, B. Wu, Y. Lian, and S. Jian, “Creation of graphene plasmons vortex via cross shape nanoantennas under linearly polarized incidence,” Plasmonics, in press (2016).
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D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
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Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Opt. 16, 105002 (2014).
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Xu, H. J.

Yao, Y.

Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

Yu, N.

Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
[Crossref] [PubMed]

Yu, W.

Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Opt. 16, 105002 (2014).
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K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
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D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
[Crossref]

F. J. Rodríguez-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, 328–330 (2013).
[Crossref] [PubMed]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

Zhang, L.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Zhang, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

Zhao, Z.

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[PubMed]

Zhu, B.

B. Wu, B. Zhu, G. Ren, and S. Jian, “Circular polarization-dependent wavefront control of plasmons on graphene,” IEEE Photonics Technol. Lett. 28, 1940–1943 (2016).
[Crossref]

B. Zhu, G. Ren, Y. Gao, B. Wu, C. Wan, and S. Jian, “Graphene circular polarization analyzer based on unidirectional excitation of plasmons,” Opt. Express 23, 32420–32428 (2015).
[Crossref] [PubMed]

B. Zhu, G. Ren, Y. Gao, B. Wu, Y. Lian, and S. Jian, “Creation of graphene plasmons vortex via cross shape nanoantennas under linearly polarized incidence,” Plasmonics, in press (2016).
[Crossref]

Zhu, W.

Zurutuza, A.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
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ACS Nano (1)

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8, 1086–1101 (2014).
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ACS Photonics (2)

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Appl. Phys. Lett. (1)

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, 251109 (2011).
[Crossref]

Eur. Phys. J. B (1)

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

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IEEE Photonics Technol. Lett. (1)

B. Wu, B. Zhu, G. Ren, and S. Jian, “Circular polarization-dependent wavefront control of plasmons on graphene,” IEEE Photonics Technol. Lett. 28, 1940–1943 (2016).
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G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

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Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Opt. 16, 105002 (2014).
[Crossref]

Nano Lett. (2)

Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13, 1257–1264 (2013).
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Nat. Commun. (2)

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
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D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
[Crossref]

Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref] [PubMed]

Nat. Photonics (2)

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
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A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
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Nat. Phys. (1)

M. Tame, K. McEnery, Ş. Özdemir, J. Lee, S. Maier, and M. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329–340 (2013).
[Crossref]

Nature (3)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011).
[Crossref] [PubMed]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, and P. Godignon, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[PubMed]

Z. Fei, A. Rodin, G. Andreev, W. Bao, A. McLeod, M. Wagner, L. Zhang, Z. Zhao, M. Thiemens, and G. Dominguez, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
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Science (5)

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344, 1369–1373 (2014).
[Crossref] [PubMed]

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

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F. J. Rodríguez-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, 328–330 (2013).
[Crossref] [PubMed]

J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346, 67–71 (2014).
[Crossref] [PubMed]

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J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, Inc., 1999).

S.-Y. Lee, I.-M. Lee, K.-Y. Kim, and B. Lee, “Comments on ’near-field interference for the unidirectional excitation of electromagnetic guided modes’,” arXiv:1306.5068 (2013).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
[Crossref]

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

The even parity in angular spectrum originate from ρpz(−kx) = ρpz(kx), and the even parity in real space is result from ρpz(−x)=∫−∞∞ρpz(kx)exp(−ikxx)dkx=∫−∞∞ρpz(−kx)exp(ikxx)dkx=ρpz(x).

B. Zhu, G. Ren, Y. Gao, B. Wu, Y. Lian, and S. Jian, “Creation of graphene plasmons vortex via cross shape nanoantennas under linearly polarized incidence,” Plasmonics, in press (2016).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic of directional excitation of graphene plasmons by a dipole source. The dipole source can be mimicked by a semiconductor nanowire.
Fig. 2
Fig. 2 Directional excitation of GPs by a circularly polarized dipole. (a) Magnetic field distributions for GPs excited by a 2D circularly polarized dipole p2D = [1, a*]p0, where −iωp0 = 1A · m. (b) Angular momentum spectra of initial ( H y 0 ), reflective ( H y ref ) and total (Hy) magnetic field magnitude of the polarized dipole (solid lines) and an ideal circularly polarized dipole (dashed lines). The dotted lines indicate the wavenumbers of GPs. (c) The simulated (solid line) and analytically calculated (marked by red circle) spatial dependent charge density in graphene. (d) Spatial dependent near field ratio R x [ H y ref ] = | H y ref ( x ) / H y ref ( x ) | (colored in black), Rx[ρ] = |ρ(x)/ρ(−x)|(colored in red) and Rx[Px] = |Px(x)/Px(−x)|(colored in blue).
Fig. 3
Fig. 3 Directional excitation of GPs by two separated orthogonal polarized dipoles. (a) Magnetic field distributions for GPs excited by two orthogonal polarized dipoles px = [1, 0]p0 and pz = [0, a*]p0 located at (0, 0.01λ) and (0, −0.01λ), respectively. The insert figure shows the enlarged excitation region. (b) Angular momentum spectra of initial magnetic field magnitude H y 0 and charge density ρ (multiplied by c) in the graphene plane, the initial and induced quantities have opposite preferred direction for excitation of GPs. (c) The simulated (solid line) and analytically calculated (marked by red circle) spatial dependent charge density in graphene. (d) The spatial dependent extinction ratio for a single circularly polarized dipole (case I, colored in black), and two separated orthogonal polarized dipoles (case II, colored in red).
Fig. 4
Fig. 4 Directional excitation of GPs on a free-standing infinite long graphene cylinder. The distribution of magnetic field |Re{Hz}| for GPs excited by the configuration of case I(a) and two separated orthogonal polarized dipoles (c). Simulated and analytically calculated spatial dependent charge density for the case I(b) and case II(d). The phase in (c) is set as π/4 in order to show the two individual sources.
Fig. 5
Fig. 5 Directional excitation of GPs using In0.53Ga0.47As nanowires illuminated by two orthogonally polarized plane waves. (a) Schematics representation and electric field distributions |Ez| for excited GPs, the insert figure is normalized absorption cross section of the In0.53Ga0.47As nanowire with radii of 20 nm, 50 nm and 100 nm. (b) The dependence of the angular spectrum ratios Rk(kspp) on extinction parameter |pz/px|. The blue circle indicates the parameter of the considered nanowire. The insert figure is the polarized circle of the nanowire with (solid line) and without (broken line) graphene sheet. (c) The simulated and theoretically calculated spatial dependent charge density in graphene. The thick line indicates the charge distribution induced by ideal circular polarized dipole with dipole momentum as −iωpx = 76.72 pA · m, the thin line indicate the case of dipole momentum as −iωpx = 61.08 pA · m and pz/px = −0.0624 + 0.9953i, respectively. The dot marked line indicates the simulated result of semiconductor nanowire. (d) The energy flux ratio of ideal dipole (pz/px = i, solid line), actual dipole(pz/px = −0.0624 + 0.9953i, broken line) and simulated result of semiconductor nanowire (thick line).

Equations (11)

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H y 0 ( x , z ) = 1 μ 0 ( × A ) y = i ω 4 π [ p z k x k z p x ] exp ( i k z | z z dip | ) exp ( i k x x ) d k x
H y 0 ( k x , z ) = i ω p x 4 π [ p z p x k x k z 1 ] exp ( i k z | z z dip | ) .
r p ( k x ) = 1 2 k z ε s k z + k z + 2 α k z k z k 0
t p ( k x ) = 2 ε s k z ε s k z + k z + 2 α k z k z k 0
H y ref ( k x , z ) = r p ( k x ) i ω p x 4 π [ p z p x k x k z + 1 ] exp [ j k z ( z + z dip ) ] H y tr ( k x , z ) = t p ( k x ) i ω p x 4 π [ p z p x k x k z + 1 ] exp [ j k z ( z z dip ) ]
ρ s ind ( x ) = δ ( z ) i ω σ E x ( x , 0 ) x = δ ( z ) i ω x ( H y ( x , 0 ) H y ( x , 0 + ) ) .
ρ s ind ( k x ) = [ t p ( 1 + r p ) ] i k x 4 π [ p z k x k z + p x ] exp ( i k z z dip ) .
ρ ˜ p z ( k x ) = ( 1 + r p t p ) i p z 4 π k x 2 k z exp ( i k z z dip ) = ρ p z ( k x ) , ρ ˜ p x ( k x ) = ( 1 + r p t p ) i ( p x ) 4 π k x exp ( i k z z dip ) = ρ p x ( k x ) ,
P x ( x ) = 1 2 Re { E y H z * } d z .
ρ ( l ) ρ f ( x ) + ρ f ( x 2 π r ) , l ( π r , π r ]
E 1 = 1 2 ( x ^ + z ^ ) exp ( i k x x i k z z i ω t ) E 2 = 1 2 ( x ^ + z ^ ) exp ( i k x x i k z z i ω t i π / 2 ) .

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