Abstract

Graphene plasmons have received significant attention recently due to its attractive properties such as high spatial confinement and tunability. However, exciting plasmons on graphene effectively still remains a challenge owing to the large wave-vector mismatch between the optical beam in air and graphene plasmon. In this paper, we present a novel scheme capable of exciting graphene surface plasmons (GSPs) on a flat suspended graphene by using only s-polarized optical beams through four-wave mixing (FWM) process, where the GSPs fields were derived analytically based on the Green's function analysis, under the basis of momentum conservation. By incorporating the merits of nonlinear optics, the presented scheme avoids any patterning of either graphene or substrate. We believe that the proposed scheme potentially paves the way towards an efficient pure optical excitation, switching and modulation of GSPs for realizing graphene-based nano-photonic and optoelectronic integrated circuits.

© 2015 Optical Society of America

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2015 (1)

2014 (3)

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(6190), 1369–1373 (2014).
[Crossref] [PubMed]

J. Tao, X. C. Yu, B. Hu, A. Dubrovkin, and Q. J. Wang, “Graphene-based tunable plasmonic Bragg reflector with a broad bandwidth,” Opt. Lett. 39(2), 271–274 (2014).
[Crossref] [PubMed]

X. H. Yao, M. Tokman, and A. Belyanin, “Efficient Nonlinear Generation of THz Plasmons in Graphene and Topological Insulators,” Phys. Rev. Lett. 112(5), 055501 (2014).
[Crossref] [PubMed]

2013 (10)

W. L. Gao, G. Shi, Z. H. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. F. Xu, “Excitation and Active Control of Propagating Surface Plasmon Polaritons in Graphene,” Nano Lett. 13(8), 3698–3702 (2013).
[Crossref] [PubMed]

X. L. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102(13), 131101 (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] [PubMed]

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

Y. Yao, M. A. Kats, P. Genevet, N. F. Yu, Y. Song, J. Kong, and F. Capasso, “Broad Electrical Tuning of Graphene-Loaded Plasmonic Antennas,” Nano Lett. 13(3), 1257–1264 (2013).
[Crossref] [PubMed]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref] [PubMed]

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

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

Z. Y. Li and N. F. Yu, “Modulation of mid-infrared light using graphene-metal plasmonic antennas,” Appl. Phys. Lett. 102(13), 131108 (2013).
[Crossref]

X. L. Zhu, W. Yan, N. A. Mortensen, and S. S. Xiao, “Bends and splitters in graphene nanoribbon waveguides,” Opt. Express 21(3), 3486–3491 (2013).
[Crossref] [PubMed]

2012 (6)

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. Castro Neto, 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]

J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene Plasmon Waveguiding and Hybridization in Individual and Paired Nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

W. L. Gao, J. Shu, C. Y. Qiu, and Q. F. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012).
[Crossref] [PubMed]

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

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
[Crossref]

2011 (2)

M. Liu, X. B. Yin, E. Ulin-Avila, B. S. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

2010 (6)

F. Schwierz, “Graphene transistors,” Nat. Nanotechnol. 5(7), 487–496 (2010).
[Crossref] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

K. F. MacDonald and N. I. Zheludev, “Active plasmonics: current status,” Laser Photonics Rev. 4(4), 562–567 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref] [PubMed]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

M. Topsakal and S. Ciraci, “Elastic and plastic deformation of graphene, silicene, and boron nitride honeycomb nanoribbons under uniaxial tension: A first-principles density-functional theory study,” Phys. Rev. B 81(2), 024107 (2010).
[Crossref]

2009 (2)

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-Space Excitation of Propagating Surface Plasmon Polaritons by Nonlinear Four-Wave Mixing,” Phys. Rev. Lett. 103(26), 266802 (2009).
[Crossref] [PubMed]

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

2008 (3)

S. Palomba and L. Novotny, “Nonlinear excitation of surface plasmon polaritons by four-wave mixing,” Phys. Rev. Lett. 101(5), 056802 (2008).
[Crossref] [PubMed]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33(23), 2874–2876 (2008).
[Crossref] [PubMed]

2007 (2)

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[Crossref] [PubMed]

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).
[Crossref]

Ajayan, P. M.

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref] [PubMed]

W. L. Gao, G. Shi, Z. H. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. F. Xu, “Excitation and Active Control of Propagating Surface Plasmon Polaritons in Graphene,” Nano Lett. 13(8), 3698–3702 (2013).
[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(6190), 1369–1373 (2014).
[Crossref] [PubMed]

J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

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. Castro Neto, 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]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Atwater, H. A.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref] [PubMed]

Avouris, P.

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

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

Badioli, M.

J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Bagci, H.

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

Bao, W.

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. Castro Neto, 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]

Basov, D. N.

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. Castro Neto, 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]

Belyanin, A.

X. H. Yao, M. Tokman, and A. Belyanin, “Efficient Nonlinear Generation of THz Plasmons in Graphene and Topological Insulators,” Phys. Rev. Lett. 112(5), 055501 (2014).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

Brar, V. W.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref] [PubMed]

Buljan, H.

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

Calle, F.

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

Camara, N.

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Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano 7(3), 2388–2395 (2013).
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C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
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J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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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. Castro Neto, 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).
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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(6190), 1369–1373 (2014).
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T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
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H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, M. Freitag, X. S. Li, F. Guinea, P. Avouris, and F. N. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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Liu, M.

M. Liu, X. B. Yin, E. Ulin-Avila, B. S. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano 7(3), 2388–2395 (2013).
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T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6(8), 554–559 (2012).
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V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly Confined Tunable Mid-Infrared Plasmonics in Graphene Nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
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H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, M. Freitag, X. S. Li, F. Guinea, P. Avouris, and F. N. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
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ACS Nano (3)

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X. L. Zhu, W. Yan, P. U. Jepsen, O. Hansen, N. A. Mortensen, and S. S. Xiao, “Experimental observation of plasmons in a graphene monolayer resting on a two-dimensional subwavelength silicon grating,” Appl. Phys. Lett. 102(13), 131101 (2013).
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Laser Photonics Rev. (1)

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Nature (3)

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S. Palomba and L. Novotny, “Nonlinear excitation of surface plasmon polaritons by four-wave mixing,” Phys. Rev. Lett. 101(5), 056802 (2008).
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[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Scheme for exciting GSPs by using FWM process. The incident optical beams from free-space are having the oscillation frequencies of ω1 and ω2, respectively, where the incidence angles are denoted as θ1and θ2. The figure illustrates the resonant incident angles (blue arrows) and the direction of GSPs propagation (purple arrow).
Fig. 2
Fig. 2 Dispersion curve of GSPs with the Fermi energy level of E f =0.4 eV (solid red line). Dashed line represents the light line in free-space, where k f = (πn) 1/2 is the Fermi wave vector. The carrier mobility used is µ = 10000 cm2/(V·s). The GSP dispersion curve is lying beyond the light line, which prohibits the plasmon to be excited by a single incident beam. The vectorial sum of three incident photons, as shown solid line in pink and purple, make it possible to couple light into GSPs in the red line.
Fig. 3
Fig. 3 Relationship between the incident angles of θ 1 and θ 2 for exciting GSPs using FWM with different Fermi energy levels of graphene. The incident wavelengths are λ 1 =40 μm and λ 2 =25 μm.
Fig. 4
Fig. 4 The relationship between the two components of the χ (3),xyyy and χ (3),zyyy with different monolayer graphene Fermi energy levels of E f =0.2 eV, 0.4 eV, 0.6 eV and 0.8 eV .
Fig. 5
Fig. 5 (a)-(b) Electric field wavefront distribution of Ey for the two pump lasers with incident wavelength of 40 µm and 25 µm, and incident angles of 50° and 25.2°, respectively. (c) Electric field distribution of Ez for the excited GSPs on a monolayer graphene sheet with a Fermi energy level of E f =0.4 eV . The ratio of χ (3),xyyy / χ (3),zyyy is 0.014.
Fig. 6
Fig. 6 Relationship between the two components of χ (3),xyyy and χ (3),zyyy for monolayer (N = 1), bilayer (N = 2), and four-layer graphene (N = 4) with a Fermi energy level of E f =0.4 eV .

Equations (26)

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ω spp =2 ω 1 ω 2
k spp ( ω spp )= ε 0 ε 1 + ε 2 2 2i ω spp σ( ω spp ) .
Re{ k spp ( ω spp ) }=2 k 1 sin θ 1 k 2 sin θ 2
E 1,y = E 1 e i k x1 x e i k z1 z e i ω 1 t E 2,y = E 2 e i k x2 x e i k z2 z e i ω 2 t ,
P i ( r )= χ (3),iyyy E 1,y E 1,y E 2,y ( r ),
E ( r )=iωμ μ 0 V G ( r , r' ) j ( r' )dV',
j ( r' )=iω P ( r' ).
E ( r' )=iωμ μ 0 xyz G ( r , r' ) j ( r' )dx'dy'dz' =iωμ μ 0 + + + G 0 ( r , r' )[ a ^ x χ (3),xyyy + a ^ y χ (3),yyyy + a ^ z χ (3),zyyy ] E 1 2 E 2 × e i[2 k x1 k x2 ]x' e i[2 k z1 k z2 ]z' e i ω spp t δ(z' z 0 )dx'dy'dz',
E ( r ) = ωμ μ 0 2 [ M xx ( k x , k y =0) χ (3),xyyy + M xz ( k x , k y =0) χ (3),zyyy M yy ( k x , k y =0) χ (3),yyyy M zx ( k x , k y =0) χ (3),xyyy + M zz ( k x , k y =0) χ (3),zyyy ] × e i(2 k x1 k x2 )x e i k zspp ( k x , k y =0)z × E 1 2 E 2 e i ω spp t ,
E x =A e i k x,spp x e i k z,spp z , E y =0, E z =B e i k x,spp x e i k z,spp z ,
M xx ( k x , k y =0) χ (3),xyyy + M xz ( k x , k y =0) χ (3),zyyy M zx ( k x , k y =0) χ (3),xyyy + M zz ( k x , k y =0) χ (3),zyyy = A B =D.
[ k z1,spp 2 +D k x k z1,spp ] χ (3),xyyy =[D k x 2 + k x k z1,spp ] χ (3),zyyy ,
σ( ω spp )= 2i e 2 k B T π 2 ( ω spp +i τ 1 ) ln[ 2cosh( E f 2 k B T ) ] + e 2 4 { 1 2 + 1 π arctan( ω spp 2 E f 2 k B T ) i 2π ln[ ( ω spp +2 E f ) 2 ( ω spp 2 E f ) 2 + (2 k B T) 2 ] },
j ( r' )=iω ε 0 [ε( r' ) ε ref ( r' )] E ( r' ) =iω ε 0 Δε( r' ) E ( r' ) =iω P ( r' ).
E 1,y E 1,y E 2,y = E 1 2 e i2 k x1 x e i2 k z1 z e i2 ω 1 t × E 2 e i k x2 x e i k z2 z e i ω 2 t = E 1 2 E 2 e i[2 k x1 k x2 ]x e i[2 k z1 k z2 ]z e i(2 ω 1 ω 2 )t = E 1 2 E 2 e i[2 k x1 k x2 ]x e i[2 k z1 k z2 ]z e i ω spp t .
P i ( r )= χ (3),iyyy E 1 2 E 2 e i[2 k x1 k x2 ]x e i[2 k z1 k z2 ]z e i ω spp t ,
P x ( r ')= χ (3),xyyy E 1 2 E 2 e i[2 k x1 k x2 ]x' e i[2 k z1 k z2 ]z' e i ω spp t ,
P y ( r' )= χ (3),yyyy E 1 2 E 2 e i[2 k x1 k x2 ]x' e i[2 k z1 k z2 ]z' e i ω spp t ,
P z ( r' )= χ (3),zyyy E 1 2 E 2 e i[2 k x1 k x2 ]x' e i[2 k z1 k z2 ]z' e i ω spp t .
G 0 ( r , r' )= i 8 π 2 + + M ( k x , k y ) e i k x (xx') e i k y (yy') e i k z1 spp (zz') d k x d k y ,
M ( k x , k y =0)= 1 k 1,spp 2 k z1,spp ( k x , k y =0) [ k 1,spp 2 k x 2 0 k x k z1,spp 0 k 1,spp 2 0 k x k z1,spp 0 k x 2 ] =[ M xx 0 M xz 0 M yy 0 M zx 0 M zz ].
k x 2 ε x + k z 2 ε z = ( ω c ) 2 ε x ε z .
E ( r )=iωμ μ 0 + + + + i 8 π 2 + + M ( k x , k y ) e i k x (xx') e i k y (yy') e i k z1 spp z ×[ a ^ x χ (3),xyyy + a ^ y χ (3),yyyy + a ^ z χ (3),zyyy ] E 1 2 E 2 e i[2 k x1 k x2 ]x' e i ω spp t d k x d k y dx'dy' =iωμ μ 0 + + + i 8 π 2 M ( k x , k y ) e i k x x e i k x x' e i k y y e i k z1 spp z ×[ a ^ x χ (3),xyyy + a ^ y χ (3),yyyy + a ^ z χ (3),zyyy ] E 1 2 E 2 e i[2 k x1 k x2 ]x' e i ω spp1 t 2πδ( k y )d k x d k y dx' = ωμ μ 0 4π + + M ( k x , k y =0) e i k x x e i k x x' e i k z1 spp ( k x , k y =0)z ×[ a ^ x χ (3),xyyy + a ^ y χ (3),yyyy + a ^ z χ (3),zyyy ] E 1 2 E 2 e i[2 k x1 k x2 ]x' e i ω spp t d k x dx',
+ e i[2 k x1 k x2 ]x' e i k x x' dx'=2πδ[2 k x1 k x2 k x ] .
E ( r )= ωμ μ 0 4π + M ( k x , k y =0) e i k x x e i k z spp ( k x , k y =0)z [ a ^ x χ (3),xyyy + a ^ y χ (3),yyyy + a ^ z χ (3),zyyy ] × E 1 2 E 2 2πδ[2 k x1 k x2 k x ] e i ω spp t d k x = ωμμ 2 M ( k x =2 k x1 k x2 , k y =0) e i(2 k x1 k x2 )x e i k z spp ( k x =2 k x1 k x2 , k y =0)z ×[ a ^ x χ (3),xyyy + a ^ y χ (3),yyyy + a ^ z χ (3),zyyy ]× E 1 2 E 2 e i ω spp t .
E ( r )= ωμ μ 0 2 [ M xx ( k x , k y =0) 0 M xz ( k x , k y =0) 0 M yy ( k x , k y =0) 0 M zx ( k x , k y =0) 0 M zz ( k x , k y =0) ] × e i(2 k x1 k x2 ) e i k zspp1 ( k x =2 k x1 k x2 , k y =0)z ×[ χ (3),xyyy χ (3),yyyy χ (3),zyyy ]× E 1 2 E 2 e i ω spp t E ( r ) = ωμ μ 0 2 [ M xx ( k x , k y =0) χ (3),xyyy + M xz ( k x , k y =0) χ (3),zyyy M yy ( k x , k y =0) χ (3),yyyy M zx ( k x , k y =0) χ (3),zyyy + M zz ( k x , k y =0) χ (3),zyyy ] × e i(2 k x1 k x2 ) e i k z,spp ( k x , k y =0)z × E 1 2 E 2 e i ω spp t .

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