Abstract

Optical second harmonic generation (SHG) from nanostructured graphene has been studied in the framework of classical electromagnetism using a surface integral equation method. Single disks and dimers are considered, demonstrating that the nonlinear conversion is enhanced when a localized surface plasmon resonance is excited at either the fundamental or second harmonic frequency. The proposed approach, beyond the electric dipole approximation used in the quantum description, reveals that SHG from graphene nanostructures with centrosymmetric shapes is possible when retardation effects and the excitation of high plasmonic modes at the second harmonic frequency are taken into account. Several SHG effects similar to those arising in metallic nanostructures, such as the silencing of the nonlinear emission and the design of double resonant nanostructures, are also reported. Finally, it is shown that the SHG from graphene disk dimers is very sensitive to a relative vertical displacement of the disks, opening new possibilities for the design of nonlinear plasmonic nanorulers.

© 2017 Optical Society of America

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2017 (3)

J. D. Cox, A. Marini, and F. J. G. de Abajo, “Plasmon-assisted high-harmonic generation in graphene,” Nature Comm. 8, 14380 (2017).
[Crossref]

J. D. Cox, R. Yu, and F. J. G. de Abajo, “Analytical description of the nonlinear plasmonic response in nanographene,” Phys. Rev. B 96(4), 045442 (2017).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Second order optical nonlinearity of graphene due to electric quadrupole and magnetic dipole effects,” Sci. Rep. 7, 43843 (2017).
[Crossref] [PubMed]

2016 (5)

T. V. Raziman and O. J. F. Martin, “Does the real part contain all the physical information,” J. Opt. 18(9), 95002 (2016).
[Crossref]

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing Graphene Chi(2) Using a Gold Photon Sieve,” Nano Lett. 16(1), 48–54 (2016).
[Crossref]

Z. Wang, T. Li, K. Almdal, N. Asger Mortensen, S. Xiao, and S. Ndoni, “Experimental demonstration of graphene plasmons working close to the near-infrared window,” Opt. Lett. 41(22), 5345 (2016).
[Crossref] [PubMed]

J. D. Cox, I. Silveiro, and F. J. G. de Abajo, “Quantum Effects in the Nonlinear Response of Graphene Plasmons,” ACS Nano,  10(2), 1995–2003 (2016).
[Crossref] [PubMed]

G. D. Bernasconi, J. Butet, and O. J. F. Martin, “Mode analysis of second-harmonic generation in plasmonic nanostructures,” J. Opt. Soc. Am. B 33(4), 768 (2016).
[Crossref]

2015 (9)

F. Ramirez, B. Liu, and S. Shen, “Extreme blueshift of surface plasmon resonance frequency in graphene nanodisk stacks,” J. Quant. Spectrosc. Radiat. Transf. 158, 27–35 (2015).
[Crossref]

G. Rosolen and B. Maes, “Asymmetric and connected graphene dimers for a tunable plasmonic response,” Phys. Rev. B,  92(20), 205405 (2015).
[Crossref]

J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano,  9(11), 10545–10562 (2015)
[Crossref] [PubMed]

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
[Crossref]

M. T. Manzoni, I. Silveiro, F. J. G. de Abajo, and D. E. Chang, “Second-order quantum nonlinear optical processes in single graphene nanostructures and arrays,” New J. Phys. 17(8), 83031 (2015).
[Crossref]

J. D. Cox and F. J. G. de Abajo, “Plasmon-Enhanced Nonlinear Wave Mixing in Nanostructured Graphene,” ACS Photonics,  2(2), 306–312 (2015).
[Crossref]

M. Jablan and D. E. Chang, “Multiplasmon Absorption in Graphene,” Phys. Rev. Lett. 114(23), 236801 (2015).
[Crossref] [PubMed]

T. O. Wehling, A. Huber, A. I. Lichtenstein, and M. I. Katsnelson, “Probing of valley polarization in graphene via optical second-harmonic generation,” Phys. Rev. B 91(4), 041404 (2015).
[Crossref]

T. V. Raziman, W. R. C. Somerville, O. J. F. Martin, and E. C. Le Ru, “Accuracy of surface integral equation matrix elements in plasmonic calculations,” J. Opt. Soc. Am. B 32(3), 485(2015).
[Crossref]

2014 (11)

Y. Q. An, J. E. Rowe, D. B. Dougherty, J. U. Lee, and A. C. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 115310 (2014).
[Crossref]

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: Role of edge states,” Phys. Rev. B 90(24), 241414 (2014).
[Crossref]

L. E. Golub and S. A. Tarasenko, “Valley polarization induced second harmonic generation in graphene,” Phys. Rev. B 90(20), 201402 (2014).
[Crossref]

D. A. Smirnova, I. V. Shadrivov, A. E. Miroshnichenko, A. I. Smirnov, and Y. S. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 035412 (2014).
[Crossref]

J. D. Cox and F. J. G. de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nature Comm. 5, 5725 (2014).
[Crossref]

B. Metzger, M. Hentschel, T. Schumacher, M. Lippitz, X. Ye, C. B. Murray, B. Knabe, K. Buse, and H. Giessen, “Doubling the Efficiency of Third Harmonic Generation by Positioning ITO Nanocrystals into the Hot-Spot of Plasmonic Gap-Antennas,” Nano Lett. 14(5), 2867–2872 (2014).
[Crossref] [PubMed]

H. Aouani, M. Rahmani, M. Navarro-Cia, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nature Nanotech. 9(4), 290–294 (2014).
[Crossref]

J. Butet and O. J. F. Martin, “Nonlinear plasmonic nanorulers,” ACS Nano 8(5), 4931–4939 (2014).
[Crossref] [PubMed]

J. Butet and O. J. F. Martin, “Fano resonances in the nonlinear optical response of coupled plasmonic nanostructures,” Opt. Express 22(24), 29693 (2014).
[Crossref]

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

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. De Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

2013 (11)

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano,  14(3), 2388–2395 (2013).
[Crossref]

S. Thongrattanasiri and F. J. G. de Abajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110(18), 187401 (2013).
[Crossref] [PubMed]

J. B. Khurgin and G. Sun, “Plasmonic enhancement of the third order nonlinear optical phenomena: Figures of merit,” Opt. Express 21(22), 27460–27480 (2013).
[Crossref] [PubMed]

E. Garmire, “Nonlinear optics in daily life,” Opt. Express 21(25), 30532–44 (2013).
[Crossref]

Z. Fang and X. Zhu, “Plasmonics in nanostructures,” Adv. Mater. 25(28), 3840–3856 (2013).
[Crossref] [PubMed]

J. D. Cox, M. R. Singh, M. a. Anton, and F. Carreño, “Plasmonic control of nonlinear two-photon absorption in graphene nanocomposites,” J. Phys. Condens. Matter 25(38), 385302 (2013).
[Crossref] [PubMed]

M. Gullans, D. E. Chang, F. H. L. Koppens, F. J. G. de Abajo, and M. D. Lukin, “Single-photon nonlinear optics with graphene plasmons,” Phys. Rev. Lett. 111(24), 247401 (2013).
[Crossref]

C. Forestiere, A. Capretti, and G. Miano, “Surface integral method for second harmonic generation in metal nanoparticles including both local-surface and nonlocal-bulk sources,” J. Opt. Soc. Am. B 30(9), 2355 (2013).
[Crossref]

Y. Q. An, F. Nelson, J. U. Lee, and A. C. Diebold, “Enhanced Optical Second-Harmonic Generation from the Current-Biased Graphene/SiO 2 /Si(001) Structure,” Nano Lett. 13(5), 2104–2109 (2013).
[Crossref] [PubMed]

V. a. Margulis, E. E. Muryumin, and E. a. Gaiduk, “Optical second-harmonic generation from two-dimensional hexagonal crystals with broken space inversion symmetry,” J. Phys. Condens. Matter 25(19), 195302 (2013).
[Crossref] [PubMed]

J. Butet, B. Gallinet, K. Thyagarajan, and O. J. F. Martin, “Second-harmonic generation from periodic arrays of arbitrary shape plasmonic nanostructures: a surface integral approach,” J. Opt. Soc. Am. B 30(11), 2970 (2013).
[Crossref]

2012 (9)

J. Berthelot, G. Bachelier, M. Song, P. Rai, G. Colas des Francs, A. Dereux, and A. Bouhelier, “Silencing and enhancement of second-harmonic generation in optical gap antennas,” Opt. Express 20(10), 10498 (2012).
[Crossref] [PubMed]

K. Thyagarajan, S. Rivier, A. Lovera, and O. J. F. Martin, “Enhanced second-harmonic generation from double resonant plasmonic antennae,” Opt. Express 20(12), 12860–12865 (2012).
[Crossref] [PubMed]

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano,  6(5), 3677–3694 (2012).
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M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photon. 3, 737–748 (2012).
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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)
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A. Manjavacas, P. Nordlander, and F. J. G. de Abajo, “Plasmon Blockade in Nanostructured Graphene,” ACS Nano 6(2), 1724–1731 (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)
[Crossref] [PubMed]

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 Nanotech.,  7(5), 330–334 (2012).
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H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14(12), 125001 (2012).
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2011 (6)

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).
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W. Wang, P. Apell, and J. Kinaret, “Edge plasmons in graphene nanostructures,” Phys. Rev. B 84(8), 085423 (2011).
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M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today 64(2), 39–44. (2011).
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N. J. Halas, S. Lal, W. S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chem. Rev. 111(6), 3913–3961 (2011).
[Crossref] [PubMed]

J. Karch, C. Drexler, P. Olbrich, M. Fehrenbacher, M. Hirmer, M. M. Glazov, S. A. Tarasenko, E. L. Ivchenko, B. Birkner, J. Eroms, D. Weiss, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Ostler, T. Seyller, and S. D. Ganichev, “Terahertz Radiation Driven Chiral Edge Currents in Graphene,” Phys. Rev. Lett. 107(27), 276601 (2011).
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S. A. Mikhailov, “Theory of the giant plasmon-enhanced second-harmonic generation in graphene and semiconductor two-dimensional electron systems,” Phys. Rev. B 84(4), 045432 (2011).
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2010 (3)

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105(9), 1–4. (2010).
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P. Pantazis, J. Maloney, D. Wu, S. E. Fraser, P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” PNAS 107(33), 14535–14540 (2010).
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F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene Photonics and Optoelectronics,” Nat. Photon. 3, 611–622 (2010).
[Crossref]

2009 (1)

2008 (1)

L. A. Falkovsky, “Optical properties of graphene,” J. Phys.: Conf. Ser. 129(11), 12004 (2008).

2007 (3)

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” EPJB,  56(4), 281–284 (2007).
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S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” EPL 79(2), 27002 (2007).
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E. H. Hwang and S. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
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2006 (1)

2005 (1)

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant Optical Antennas,” Science 308(5728), 1607–1609 (2005).
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2004 (2)

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon Hybridization in Nanoparticle Dimers,” Nano Lett. 4(5), 899–903 (2004).
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J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
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1999 (1)

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh Scattering from a Sphere of Centrosymmetric Material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
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Aizpurua, J.

Ajayan, P. M.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. De Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
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Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano,  14(3), 2388–2395 (2013).
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Almdal, K.

An, Y. Q.

Y. Q. An, J. E. Rowe, D. B. Dougherty, J. U. Lee, and A. C. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 115310 (2014).
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Y. Q. An, F. Nelson, J. U. Lee, and A. C. Diebold, “Enhanced Optical Second-Harmonic Generation from the Current-Biased Graphene/SiO 2 /Si(001) Structure,” Nano Lett. 13(5), 2104–2109 (2013).
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Anton, M. a.

J. D. Cox, M. R. Singh, M. a. Anton, and F. Carreño, “Plasmonic control of nonlinear two-photon absorption in graphene nanocomposites,” J. Phys. Condens. Matter 25(38), 385302 (2013).
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Aouani, H.

H. Aouani, M. Rahmani, M. Navarro-Cia, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nature Nanotech. 9(4), 290–294 (2014).
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Apell, P.

W. Wang, P. Apell, and J. Kinaret, “Edge plasmons in graphene nanostructures,” Phys. Rev. B 84(8), 085423 (2011).
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Asger Mortensen, N.

Avouris, P.

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14(12), 125001 (2012).
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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 Nanotech.,  7(5), 330–334 (2012).
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Bachelier, G.

Bao, Q.

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano,  6(5), 3677–3694 (2012).
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Baselli, M.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
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Bernasconi, G. D.

Berthelot, J.

Biagioni, P.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
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Birkner, B.

J. Karch, C. Drexler, P. Olbrich, M. Fehrenbacher, M. Hirmer, M. M. Glazov, S. A. Tarasenko, E. L. Ivchenko, B. Birkner, J. Eroms, D. Weiss, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Ostler, T. Seyller, and S. D. Ganichev, “Terahertz Radiation Driven Chiral Edge Currents in Graphene,” Phys. Rev. Lett. 107(27), 276601 (2011).
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Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene Photonics and Optoelectronics,” Nat. Photon. 3, 611–622 (2010).
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Boyd, R. W.

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J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano,  9(11), 10545–10562 (2015)
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Bryant, G. W.

Buse, K.

B. Metzger, M. Hentschel, T. Schumacher, M. Lippitz, X. Ye, C. B. Murray, B. Knabe, K. Buse, and H. Giessen, “Doubling the Efficiency of Third Harmonic Generation by Positioning ITO Nanocrystals into the Hot-Spot of Plasmonic Gap-Antennas,” Nano Lett. 14(5), 2867–2872 (2014).
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Butet, J.

Capretti, A.

Carreño, F.

J. D. Cox, M. R. Singh, M. a. Anton, and F. Carreño, “Plasmonic control of nonlinear two-photon absorption in graphene nanocomposites,” J. Phys. Condens. Matter 25(38), 385302 (2013).
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Cecchet, F.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing Graphene Chi(2) Using a Gold Photon Sieve,” Nano Lett. 16(1), 48–54 (2016).
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Celebrano, M.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
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Cerullo, G.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
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Chandra, B.

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 Nanotech.,  7(5), 330–334 (2012).
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Chang, D. E.

M. T. Manzoni, I. Silveiro, F. J. G. de Abajo, and D. E. Chang, “Second-order quantum nonlinear optical processes in single graphene nanostructures and arrays,” New J. Phys. 17(8), 83031 (2015).
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M. Jablan and D. E. Chang, “Multiplasmon Absorption in Graphene,” Phys. Rev. Lett. 114(23), 236801 (2015).
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M. Gullans, D. E. Chang, F. H. L. Koppens, F. J. G. de Abajo, and M. D. Lukin, “Single-photon nonlinear optics with graphene plasmons,” Phys. Rev. Lett. 111(24), 247401 (2013).
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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]

Chang, W. S.

N. J. Halas, S. Lal, W. S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chem. Rev. 111(6), 3913–3961 (2011).
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Cheng, J. L.

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Second order optical nonlinearity of graphene due to electric quadrupole and magnetic dipole effects,” Sci. Rep. 7, 43843 (2017).
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Christensen, J.

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)
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Christensen, T.

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: Role of edge states,” Phys. Rev. B 90(24), 241414 (2014).
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Ciccacci, F.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
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Colas des Francs, G.

Colomer, J. F.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing Graphene Chi(2) Using a Gold Photon Sieve,” Nano Lett. 16(1), 48–54 (2016).
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Cox, J. D.

J. D. Cox, A. Marini, and F. J. G. de Abajo, “Plasmon-assisted high-harmonic generation in graphene,” Nature Comm. 8, 14380 (2017).
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J. D. Cox, R. Yu, and F. J. G. de Abajo, “Analytical description of the nonlinear plasmonic response in nanographene,” Phys. Rev. B 96(4), 045442 (2017).
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J. D. Cox, I. Silveiro, and F. J. G. de Abajo, “Quantum Effects in the Nonlinear Response of Graphene Plasmons,” ACS Nano,  10(2), 1995–2003 (2016).
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J. D. Cox and F. J. G. de Abajo, “Plasmon-Enhanced Nonlinear Wave Mixing in Nanostructured Graphene,” ACS Photonics,  2(2), 306–312 (2015).
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J. D. Cox and F. J. G. de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nature Comm. 5, 5725 (2014).
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J. D. Cox, M. R. Singh, M. a. Anton, and F. Carreño, “Plasmonic control of nonlinear two-photon absorption in graphene nanocomposites,” J. Phys. Condens. Matter 25(38), 385302 (2013).
[Crossref] [PubMed]

Dadap, J. I.

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
[Crossref]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh Scattering from a Sphere of Centrosymmetric Material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[Crossref]

de Abajo, F. J. G.

J. D. Cox, R. Yu, and F. J. G. de Abajo, “Analytical description of the nonlinear plasmonic response in nanographene,” Phys. Rev. B 96(4), 045442 (2017).
[Crossref]

J. D. Cox, A. Marini, and F. J. G. de Abajo, “Plasmon-assisted high-harmonic generation in graphene,” Nature Comm. 8, 14380 (2017).
[Crossref]

J. D. Cox, I. Silveiro, and F. J. G. de Abajo, “Quantum Effects in the Nonlinear Response of Graphene Plasmons,” ACS Nano,  10(2), 1995–2003 (2016).
[Crossref] [PubMed]

J. D. Cox and F. J. G. de Abajo, “Plasmon-Enhanced Nonlinear Wave Mixing in Nanostructured Graphene,” ACS Photonics,  2(2), 306–312 (2015).
[Crossref]

M. T. Manzoni, I. Silveiro, F. J. G. de Abajo, and D. E. Chang, “Second-order quantum nonlinear optical processes in single graphene nanostructures and arrays,” New J. Phys. 17(8), 83031 (2015).
[Crossref]

J. D. Cox and F. J. G. de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nature Comm. 5, 5725 (2014).
[Crossref]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. De Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

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

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano,  14(3), 2388–2395 (2013).
[Crossref]

S. Thongrattanasiri and F. J. G. de Abajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110(18), 187401 (2013).
[Crossref] [PubMed]

M. Gullans, D. E. Chang, F. H. L. Koppens, F. J. G. de Abajo, and M. D. Lukin, “Single-photon nonlinear optics with graphene plasmons,” Phys. Rev. Lett. 111(24), 247401 (2013).
[Crossref]

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]

A. Manjavacas, P. Nordlander, and F. J. G. de Abajo, “Plasmon Blockade in Nanostructured Graphene,” ACS Nano 6(2), 1724–1731 (2012).
[Crossref] [PubMed]

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]

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]

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006).
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De Angelis, C.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
[Crossref]

Dereux, A.

Diebold, A. C.

Y. Q. An, J. E. Rowe, D. B. Dougherty, J. U. Lee, and A. C. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 115310 (2014).
[Crossref]

Y. Q. An, F. Nelson, J. U. Lee, and A. C. Diebold, “Enhanced Optical Second-Harmonic Generation from the Current-Biased Graphene/SiO 2 /Si(001) Structure,” Nano Lett. 13(5), 2104–2109 (2013).
[Crossref] [PubMed]

Dougherty, D. B.

Y. Q. An, J. E. Rowe, D. B. Dougherty, J. U. Lee, and A. C. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 115310 (2014).
[Crossref]

Drexler, C.

J. Karch, C. Drexler, P. Olbrich, M. Fehrenbacher, M. Hirmer, M. M. Glazov, S. A. Tarasenko, E. L. Ivchenko, B. Birkner, J. Eroms, D. Weiss, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Ostler, T. Seyller, and S. D. Ganichev, “Terahertz Radiation Driven Chiral Edge Currents in Graphene,” Phys. Rev. Lett. 107(27), 276601 (2011).
[Crossref]

Duo, L.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
[Crossref]

Eisenthal, K. B.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-Harmonic Rayleigh Scattering from a Sphere of Centrosymmetric Material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[Crossref]

Eisler, H.-J.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant Optical Antennas,” Science 308(5728), 1607–1609 (2005).
[Crossref] [PubMed]

Eroms, J.

J. Karch, C. Drexler, P. Olbrich, M. Fehrenbacher, M. Hirmer, M. M. Glazov, S. A. Tarasenko, E. L. Ivchenko, B. Birkner, J. Eroms, D. Weiss, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Ostler, T. Seyller, and S. D. Ganichev, “Terahertz Radiation Driven Chiral Edge Currents in Graphene,” Phys. Rev. Lett. 107(27), 276601 (2011).
[Crossref]

Falkovsky, L. A.

L. A. Falkovsky, “Optical properties of graphene,” J. Phys.: Conf. Ser. 129(11), 12004 (2008).

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

Fang, Z.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. De Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano,  14(3), 2388–2395 (2013).
[Crossref]

Z. Fang and X. Zhu, “Plasmonics in nanostructures,” Adv. Mater. 25(28), 3840–3856 (2013).
[Crossref] [PubMed]

Fehrenbacher, M.

J. Karch, C. Drexler, P. Olbrich, M. Fehrenbacher, M. Hirmer, M. M. Glazov, S. A. Tarasenko, E. L. Ivchenko, B. Birkner, J. Eroms, D. Weiss, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Ostler, T. Seyller, and S. D. Ganichev, “Terahertz Radiation Driven Chiral Edge Currents in Graphene,” Phys. Rev. Lett. 107(27), 276601 (2011).
[Crossref]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene Photonics and Optoelectronics,” Nat. Photon. 3, 611–622 (2010).
[Crossref]

Finazzi, M.

M. Celebrano, X. Wu, M. Baselli, S. Gromann, P. Biagioni, A. Locatelli, C. De Angelis, G. Cerullo, R. Osellame, B. Hecht, L. Duo, F. Ciccacci, and M. Finazzi, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nature Nanotech. 10(5), 412–417 (2015).
[Crossref]

Forestiere, C.

Fraser, S. E.

P. Pantazis, J. Maloney, D. Wu, S. E. Fraser, P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” PNAS 107(33), 14535–14540 (2010).
[Crossref] [PubMed]

P. Pantazis, J. Maloney, D. Wu, S. E. Fraser, P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” PNAS 107(33), 14535–14540 (2010).
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M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photon. 3, 737–748 (2012).
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Zhu, W.

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 Nanotech.,  7(5), 330–334 (2012).
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Zhu, X.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. De Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
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ACS Nano (8)

J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano,  9(11), 10545–10562 (2015)
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J. Butet and O. J. F. Martin, “Nonlinear plasmonic nanorulers,” ACS Nano 8(5), 4931–4939 (2014).
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Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano,  6(5), 3677–3694 (2012).
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J. D. Cox, I. Silveiro, and F. J. G. de Abajo, “Quantum Effects in the Nonlinear Response of Graphene Plasmons,” ACS Nano,  10(2), 1995–2003 (2016).
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Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated Tunability and Hybridization of Localized Plasmons in Nanostructured Graphene,” ACS Nano,  14(3), 2388–2395 (2013).
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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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A. Manjavacas, P. Nordlander, and F. J. G. de Abajo, “Plasmon Blockade in Nanostructured Graphene,” ACS Nano 6(2), 1724–1731 (2012).
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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)
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ACS Photonics (2)

F. J. G. de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics,  1(3), 135–152 (2014).
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J. D. Cox and F. J. G. de Abajo, “Plasmon-Enhanced Nonlinear Wave Mixing in Nanostructured Graphene,” ACS Photonics,  2(2), 306–312 (2015).
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Adv. Mater. (1)

Z. Fang and X. Zhu, “Plasmonics in nanostructures,” Adv. Mater. 25(28), 3840–3856 (2013).
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Chem. Rev. (1)

N. J. Halas, S. Lal, W. S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chem. Rev. 111(6), 3913–3961 (2011).
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EPJB (1)

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” EPJB,  56(4), 281–284 (2007).
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EPL (1)

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” EPL 79(2), 27002 (2007).
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J. Opt. (1)

T. V. Raziman and O. J. F. Martin, “Does the real part contain all the physical information,” J. Opt. 18(9), 95002 (2016).
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J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (5)

J. Phys. Condens. Matter (2)

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J. D. Cox, M. R. Singh, M. a. Anton, and F. Carreño, “Plasmonic control of nonlinear two-photon absorption in graphene nanocomposites,” J. Phys. Condens. Matter 25(38), 385302 (2013).
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J. Phys.: Conf. Ser. (1)

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J. Quant. Spectrosc. Radiat. Transf. (1)

F. Ramirez, B. Liu, and S. Shen, “Extreme blueshift of surface plasmon resonance frequency in graphene nanodisk stacks,” J. Quant. Spectrosc. Radiat. Transf. 158, 27–35 (2015).
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Nano Lett. (6)

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. De Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14(1), 299–304 (2014).
[Crossref]

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing Graphene Chi(2) Using a Gold Photon Sieve,” Nano Lett. 16(1), 48–54 (2016).
[Crossref]

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon Hybridization in Nanoparticle Dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

B. Metzger, M. Hentschel, T. Schumacher, M. Lippitz, X. Ye, C. B. Murray, B. Knabe, K. Buse, and H. Giessen, “Doubling the Efficiency of Third Harmonic Generation by Positioning ITO Nanocrystals into the Hot-Spot of Plasmonic Gap-Antennas,” Nano Lett. 14(5), 2867–2872 (2014).
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Y. Q. An, F. Nelson, J. U. Lee, and A. C. Diebold, “Enhanced Optical Second-Harmonic Generation from the Current-Biased Graphene/SiO 2 /Si(001) Structure,” Nano Lett. 13(5), 2104–2109 (2013).
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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).
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Nat. Photon. (2)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photon. 3, 737–748 (2012).
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F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene Photonics and Optoelectronics,” Nat. Photon. 3, 611–622 (2010).
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Nature Comm. (2)

J. D. Cox and F. J. G. de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nature Comm. 5, 5725 (2014).
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J. D. Cox, A. Marini, and F. J. G. de Abajo, “Plasmon-assisted high-harmonic generation in graphene,” Nature Comm. 8, 14380 (2017).
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Nature Nanotech. (3)

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 Nanotech.,  7(5), 330–334 (2012).
[Crossref]

H. Aouani, M. Rahmani, M. Navarro-Cia, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nature Nanotech. 9(4), 290–294 (2014).
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Figures (5)

Fig. 1
Fig. 1 (a) Top panel: Relative permittivity of a graphene nanodisk, obtained using the RPA with EF = 0.4 eV, ħτ−1 = 1.6 meV, T = 300 K, R = 50 nm, εb = 1 and t = 0.5 nm. Middle panel: Scattering by a 100 nm diameter graphene nanodisk (logarithmic scale) as a function of the incident electromagnetic wave energy. Bottom panel: SH intensity emitted by the same nanodisk (logarithmic scale) as a function of the incident electromagnetic wave energy. (b) The three eigenmodes found between 0.1 and 0.32 eV corresponding respectively to: a dipolar mode, a quadrupolar mode, and an octupolar mode. Red and blue colors correspond to positive and negative surface charges whereas white represent a neutral surface charges.
Fig. 2
Fig. 2 Schematic representation of the graphene dimers characterized by a gap distance g and vertical shift h.
Fig. 3
Fig. 3 (a) Dimer of graphene disks. Top panel: Scattered field intensity shown in logarithmic scale; inset: zoom at the resonance of the dipolar mode; Middle panel: Enhancement of the intensity of the field between the two disks shown in logarithmic scale; Bottom panel: Second harmonic intensity shown in linear scale ; inset: zoom at the resonance of the dipolar mode - as function of the incident wave energy for a dimer of 100 nm diameter graphene disks with gap distances ranging from g = 2 nm to 640 nm. (b) The first four eigenmodes of a dimer with a gap distance of 2 nm, revealed by the eigenmode analysis are shown. Red and blue colors correspond to positive and negative surface charges whereas white represent a neutral surface charges.
Fig. 4
Fig. 4 (a) Normalized imaginary part (five firsts) and real part (last) of the charge distribution at the graphene nanodimer surface evaluated at the energy of each peak observed in Fig. 3(b). Positive charges and negative charges are respectively represented in red and blue. (b) Real part of the longitudinal component of the SH electric field for a 100 nm diameter disk dimer with a gap distance of 2 nm at 0.136 eV.
Fig. 5
Fig. 5 (a) Response as a function of the incident energy for a dimer of two 100 nm diameter graphene disks with a constant lateral distance g = 2 nm and different vertical shifts ranging from h = 0 to 40 nm. Top panel: Scattered field intensity shown in logarithmic scale; inset: zoom at the resonance of the dipolar mode; Middle panel: Intensity enhancement in the gap shown in logarithmic scale; Bottom panel: SH intensity shown in linear scale; inset: zoom at the resonance of the dipolar mode. (b) Comparison between a dimer composed of two 100 nm graphene nanodisks with a gap distance g = 2 nm (left column) and the same dimer with an additional vertical shift (h = 20 nm; right column) at 0.135 eV. Top panel: Maps of the fundamental near-field intensity shown in logarithmic scale; Middle panels: SH near field intensity shown in logarithmic scale; Bottom panels: SH emission pattern. Maps are taken in the middle of the vertical shift. (c) SHG from a dimer composed of two 100 nm graphene nanodisks with a gap distance g = 2 nm as a function of the vertical shift h between the disks. Left panel: near-field intensity in the middle of the gap in the xy plane and next to the top or bottom disk; Right panel: far-field intensity.

Equations (5)

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σ k ( ω ) = 2 i e 2 k B T π 2 ( ω + i τ 1 ) ln ( 2 cosh ( E F 2 k B T ) ) + e 2 4 [ G ( ω / 2 ) 4 ω i π 0 + d Ω G ( Ω ) G ( ω / 2 ) ω 2 4 Ω 2 ]
G ( ω ) = sinh ( ω / k B T ) cosh ( E F / k B T ) + cosh ( ω / k B T ) .
σ E ( ω ) = 16 i e 2 π ω ω R l , n l max , n max l + 1 β ln 5 f ln , 0 1 ( ω + i η β ln ω R ) .
ε ( ω ) = ε b + i σ ( ω ) ε 0 ω t ,
ω d = e 2 π ( 12.5 E F ε 0 d ) .

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