Abstract

Graphene is a nonlinear material which can be used as a saturable absorber, frequency mixer and frequency multiplier. We theoretically study the third harmonic generation from graphene lying on different dielectric (dispersionless or polar) substrates, metalized or non-metalized on the back side. We show that the third harmonic intensity emitted from graphene lying on a substrate, can be increased by orders of magnitude as compared to the isolated graphene, due the LO-phonon resonances in a polar dielectric or due to the interference effects in the substrates metalized on the back side. In some frequency intervals, the presence of the polar dielectric substrate compensates the strongly decreasing with ω frequency dependence of the third-order conductivity of graphene making the response almost frequency independent. Our results can be used for the development of graphene based frequency multipliers operating in microwave through infrared frequencies.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]

2016 (9)

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12, 124–127 (2016).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Erratum: Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature [phys. rev. b 91, 235320 (2015)],” Phys. Rev. B 93, 039904 (2016).
[Crossref]

S. A. Mikhailov, “Quantum theory of the third-order nonlinear electrodynamic effects in graphene,” Phys. Rev. B 93, 085403 (2016).
[Crossref]

S. A. Mikhailov, N. A. Savostianova, and A. S. Moskalenko, “Negative dynamic conductivity of a current-driven array of graphene nanoribbons,” Phys. Rev. B 94, 035439 (2016).
[Crossref]

H. Rostami and M. Polini, “Theory of third-harmonic generation in graphene: A diagrammatic approach,” Phys. Rev. B 93, 161411 (2016).
[Crossref]

J. D. Cox, I. Silviero, and F. J. G. de Abajo, “Quantum effects in the nonlinear response of graphene plasmons,” ACS Nano 10, 1995–2003 (2016).
[Crossref] [PubMed]

M. A. Sharif, M. H. M. Ara, B. Ghafary, S. Salmani, and S. Mohajer, “Experimental observation of low threshold optical bistability in exfoliated graphene with low oxidation degree,” Opt. Mater. 53, 80–86 (2016).
[Crossref]

M. Tokman, Y. Wang, I. Oladyshkin, A. R. Kutayiah, and A. Belyanin, “Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene,” Phys. Rev. B 93, 235422 (2016).
[Crossref]

Y. Wang, M. Tokman, and A. Belyanin, “Second-order nonlinear optical response of graphene,” Phys. Rev. B 94, 195442 (2016).
[Crossref]

2015 (3)

J. D. Cox and F. J. G. de Abajo, “Plasmon-enhanced nonlinear wave mixing in nanostructured graphene,” ACS Photonics 2, 306–312 (2015).
[Crossref]

N. A. Savostianova and S. A. Mikhailov, “Giant enhancement of the third harmonic in graphene integrated in a layered structure,” Appl. Phys. Lett. 107, 181104 (2015).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature,” Phys. Rev. B 91, 235320 (2015).
[Crossref]

2014 (10)

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Dc current induced second order optical nonlinearity in graphene,” Optics Express 22, 15868–15876 (2014).
[Crossref] [PubMed]

X. Yao, M. Tokman, and A. Belyanin, “Efficient nonlinear generation of THz plasmons in graphene and topological insulators,” Phys. Rev. Lett. 112, 055501 (2014).
[Crossref] [PubMed]

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, 035412 (2014).
[Crossref]

N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
[Crossref]

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

M. M. Glazov and S. Ganichev, “High frequency electric field induced nonlinear effects in graphene,” Phys. Rep. 535, 101–138 (2014).
[Crossref]

R. R. Hartmann, J. Kono, and M. E. Portnoi, “Terahertz science and technology of carbon nanomaterials,” Nanotechnology 25, 322001 (2014).
[Crossref] [PubMed]

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

2013 (4)

H. K. Avetissian, G. F. Mkrtchian, K. G. Batrakov, S. A. Maksimenko, and A. Hoffmann, “Multiphoton resonant excitations and high-harmonic generation in bilayer graphene,” Phys. Rev. B 88, 165411 (2013).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Nonlinear control of absorption in one-dimensional photonic crystal with graphene-based defect,” Opt. Lett. 38, 3550–3553 (2013).
[Crossref] [PubMed]

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
[Crossref]

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

2012 (6)

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, 554–559 (2012).
[Crossref]

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
[Crossref]

S. A. Jafari, “Nonlinear optical response in gapped graphene,” J. Phys. Condens. Matter 24, 205802 (2012).
[Crossref] [PubMed]

S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New J. Phys. 14, 115024 (2012).
[Crossref]

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37, 1856–1858 (2012).
[Crossref] [PubMed]

2011 (3)

G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
[Crossref]

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98, 073106 (2011).
[Crossref]

S. A. Mikhailov, “Theory of the giant plasmon-enhanced second-harmonic generation in graphene and semiconductor two-dimensional electron systems,” Phys. Rev. B 84, 045432 (2011).
[Crossref]

2010 (5)

J. J. Dean and H. M. van Driel, “Graphene and few-layer graphite probed by second-harmonic generation: Theory and experiment,” Phys. Rev. B 82, 125411 (2010).
[Crossref]

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82, 201402 (2010).
[Crossref]

D. Popa, Z. Sun, F. Torrisi, T. Hasan, F. Wang, and A. C. Ferrari, “Sub 200 fs pulse generation from a graphene mode-locked fiber laser,” Appl. Phys. Lett. 97, 203106 (2010).
[Crossref]

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
[Crossref]

E. Hendry, P. J. Hale, J. J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
[Crossref] [PubMed]

2009 (3)

S. A. Mikhailov, “Non-linear graphene optics for terahertz applications,” Microelectron. J. 40, 712–715 (2009).
[Crossref]

S. A. Mikhailov, “Nonlinear cyclotron resonance of a massless quasiparticle in graphene,” Phys. Rev. B 79, 241309 (2009).
[Crossref]

J. J. Dean and H. M. van Driel, “Second harmonic generation from graphene and graphitic film,” Appl. Phys. Lett. 95, 261910 (2009).
[Crossref]

2008 (1)

S. A. Mikhailov and K. Ziegler, “Non-linear electromagnetic response of graphene: Frequency multiplication and the self-consistent field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
[Crossref]

2007 (2)

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
[Crossref]

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007).
[Crossref] [PubMed]

2000 (1)

M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61, 8187–8201 (2000).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Álvarez, P.

G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
[Crossref]

Ara, M. H. M.

M. A. Sharif, M. H. M. Ara, B. Ghafary, S. Salmani, and S. Mohajer, “Experimental observation of low threshold optical bistability in exfoliated graphene with low oxidation degree,” Opt. Mater. 53, 80–86 (2016).
[Crossref]

Avetissian, H. K.

H. K. Avetissian, G. F. Mkrtchian, K. G. Batrakov, S. A. Maksimenko, and A. Hoffmann, “Multiphoton resonant excitations and high-harmonic generation in bilayer graphene,” Phys. Rev. B 88, 165411 (2013).
[Crossref]

Avouris, P.

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

Bao, Q.

Batrakov, K. G.

H. K. Avetissian, G. F. Mkrtchian, K. G. Batrakov, S. A. Maksimenko, and A. Hoffmann, “Multiphoton resonant excitations and high-harmonic generation in bilayer graphene,” Phys. Rev. B 88, 165411 (2013).
[Crossref]

Beba, D.

S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New J. Phys. 14, 115024 (2012).
[Crossref]

Belyanin, A.

M. Tokman, Y. Wang, I. Oladyshkin, A. R. Kutayiah, and A. Belyanin, “Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene,” Phys. Rev. B 93, 235422 (2016).
[Crossref]

Y. Wang, M. Tokman, and A. Belyanin, “Second-order nonlinear optical response of graphene,” Phys. Rev. B 94, 195442 (2016).
[Crossref]

X. Yao, M. Tokman, and A. Belyanin, “Efficient nonlinear generation of THz plasmons in graphene and topological insulators,” Phys. Rev. Lett. 112, 055501 (2014).
[Crossref] [PubMed]

Bludov, Y. V.

N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
[Crossref]

Boreman, G. D.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980), 6th ed.

Bykov, A. Y.

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
[Crossref]

Camblor, R.

G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
[Crossref]

Chang, D. E.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12, 124–127 (2016).
[Crossref]

Cheng, J. L.

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Erratum: Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature [phys. rev. b 91, 235320 (2015)],” Phys. Rev. B 93, 039904 (2016).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature,” Phys. Rev. B 91, 235320 (2015).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Dc current induced second order optical nonlinearity in graphene,” Optics Express 22, 15868–15876 (2014).
[Crossref] [PubMed]

Chiu, H.-Y.

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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
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M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Nonlinear control of absorption in one-dimensional photonic crystal with graphene-based defect,” Opt. Lett. 38, 3550–3553 (2013).
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Dadap, J. I.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

de Abajo, F. J. G.

J. D. Cox, I. Silviero, and F. J. G. de Abajo, “Quantum effects in the nonlinear response of graphene plasmons,” ACS Nano 10, 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, 306–312 (2015).
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J. D. Cox and F. J. G. de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nat. Commun. 5, 5725 (2014).
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M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
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M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Nonlinear control of absorption in one-dimensional photonic crystal with graphene-based defect,” Opt. Lett. 38, 3550–3553 (2013).
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M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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Dragoman, D.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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Dragoman, M.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
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D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98, 073106 (2011).
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D. Popa, Z. Sun, F. Torrisi, T. Hasan, F. Wang, and A. C. Ferrari, “Sub 200 fs pulse generation from a graphene mode-locked fiber laser,” Appl. Phys. Lett. 97, 203106 (2010).
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M. M. Glazov and S. Ganichev, “High frequency electric field induced nonlinear effects in graphene,” Phys. Rep. 535, 101–138 (2014).
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N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
[Crossref]

Ghafary, B.

M. A. Sharif, M. H. M. Ara, B. Ghafary, S. Salmani, and S. Mohajer, “Experimental observation of low threshold optical bistability in exfoliated graphene with low oxidation degree,” Opt. Mater. 53, 80–86 (2016).
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M. M. Glazov and S. Ganichev, “High frequency electric field induced nonlinear effects in graphene,” Phys. Rep. 535, 101–138 (2014).
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Grande, M.

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Nonlinear control of absorption in one-dimensional photonic crystal with graphene-based defect,” Opt. Lett. 38, 3550–3553 (2013).
[Crossref] [PubMed]

Gu, T.

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, 554–559 (2012).
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Hale, P. J.

E. Hendry, P. J. Hale, J. J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
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R. R. Hartmann, J. Kono, and M. E. Portnoi, “Terahertz science and technology of carbon nanomaterials,” Nanotechnology 25, 322001 (2014).
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D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98, 073106 (2011).
[Crossref]

D. Popa, Z. Sun, F. Torrisi, T. Hasan, F. Wang, and A. C. Ferrari, “Sub 200 fs pulse generation from a graphene mode-locked fiber laser,” Appl. Phys. Lett. 97, 203106 (2010).
[Crossref]

Hendry, E.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12, 124–127 (2016).
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E. Hendry, P. J. Hale, J. J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
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Hoffmann, A.

H. K. Avetissian, G. F. Mkrtchian, K. G. Batrakov, S. A. Maksimenko, and A. Hoffmann, “Multiphoton resonant excitations and high-harmonic generation in bilayer graphene,” Phys. Rev. B 88, 165411 (2013).
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S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

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, 554–559 (2012).
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S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

Hornett, S. M.

T. J. Constant, S. M. Hornett, D. E. Chang, and E. Hendry, “All-optical generation of surface plasmons in graphene,” Nat. Phys. 12, 124–127 (2016).
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G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
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N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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Johnson, T. W.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
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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, 035412 (2014).
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Kockaert, P.

Kono, J.

R. R. Hartmann, J. Kono, and M. E. Portnoi, “Terahertz science and technology of carbon nanomaterials,” Nanotechnology 25, 322001 (2014).
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M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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Kumar, J.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
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Kumar, N.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
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M. Tokman, Y. Wang, I. Oladyshkin, A. R. Kutayiah, and A. Belyanin, “Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene,” Phys. Rev. B 93, 235422 (2016).
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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, 554–559 (2012).
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G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
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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, 554–559 (2012).
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Massar, S.

McMillan, J. F.

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, 554–559 (2012).
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G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
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S. A. Mikhailov, N. A. Savostianova, and A. S. Moskalenko, “Negative dynamic conductivity of a current-driven array of graphene nanoribbons,” Phys. Rev. B 94, 035439 (2016).
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N. A. Savostianova and S. A. Mikhailov, “Giant enhancement of the third harmonic in graphene integrated in a layered structure,” Appl. Phys. Lett. 107, 181104 (2015).
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S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New J. Phys. 14, 115024 (2012).
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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, 045432 (2011).
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E. Hendry, P. J. Hale, J. J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
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S. A. Mikhailov, “Non-linear graphene optics for terahertz applications,” Microelectron. J. 40, 712–715 (2009).
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S. A. Mikhailov, “Nonlinear cyclotron resonance of a massless quasiparticle in graphene,” Phys. Rev. B 79, 241309 (2009).
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S. A. Mikhailov and K. Ziegler, “Non-linear electromagnetic response of graphene: Frequency multiplication and the self-consistent field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
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S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
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S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007).
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S. A. Mikhailov, “Electromagnetic nonlinearities in graphene,” in “Carbon nanotubes and graphene for photonic applications,” S. Yamashita, Y. Saito, and J. H. Choi, eds. (Woodhead Publishing Limited, 2013), chap. 7, pp. 171–219.
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Miroshnichenko, A. E.

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, 035412 (2014).
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Mkrtchian, G. F.

H. K. Avetissian, G. F. Mkrtchian, K. G. Batrakov, S. A. Maksimenko, and A. Hoffmann, “Multiphoton resonant excitations and high-harmonic generation in bilayer graphene,” Phys. Rev. B 88, 165411 (2013).
[Crossref]

Moger, J. J.

E. Hendry, P. J. Hale, J. J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
[Crossref] [PubMed]

Mohajer, S.

M. A. Sharif, M. H. M. Ara, B. Ghafary, S. Salmani, and S. Mohajer, “Experimental observation of low threshold optical bistability in exfoliated graphene with low oxidation degree,” Opt. Mater. 53, 80–86 (2016).
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Moskalenko, A. S.

S. A. Mikhailov, N. A. Savostianova, and A. S. Moskalenko, “Negative dynamic conductivity of a current-driven array of graphene nanoribbons,” Phys. Rev. B 94, 035439 (2016).
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Muller, A. A.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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Murzina, T. V.

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
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Neculoiu, D.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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Obraztsova, E. D.

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
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Oh, S.-H.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
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Oladyshkin, I.

M. Tokman, Y. Wang, I. Oladyshkin, A. R. Kutayiah, and A. Belyanin, “Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene,” Phys. Rev. B 93, 235422 (2016).
[Crossref]

Olmon, R. L.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
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Osgood, R. M.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

Peres, N. M. R.

N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
[Crossref]

Petrone, N.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

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, 554–559 (2012).
[Crossref]

Ping, L. K.

Plana, R.

M. Dragoman, D. Neculoiu, G. Deligeorgis, G. Konstantinidis, D. Dragoman, A. Cismaru, A. A. Muller, and R. Plana, “Millimeter-wave generation via frequency multiplication in graphene,” Appl. Phys. Lett. 97, 093101 (2010).
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H. Rostami and M. Polini, “Theory of third-harmonic generation in graphene: A diagrammatic approach,” Phys. Rev. B 93, 161411 (2016).
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Popa, D.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98, 073106 (2011).
[Crossref]

D. Popa, Z. Sun, F. Torrisi, T. Hasan, F. Wang, and A. C. Ferrari, “Sub 200 fs pulse generation from a graphene mode-locked fiber laser,” Appl. Phys. Lett. 97, 203106 (2010).
[Crossref]

Portnoi, M. E.

R. R. Hartmann, J. Kono, and M. E. Portnoi, “Terahertz science and technology of carbon nanomaterials,” Nanotechnology 25, 322001 (2014).
[Crossref] [PubMed]

Raschke, M. B.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
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Rostami, H.

H. Rostami and M. Polini, “Theory of third-harmonic generation in graphene: A diagrammatic approach,” Phys. Rev. B 93, 161411 (2016).
[Crossref]

Rybin, M. G.

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
[Crossref]

Salmani, S.

M. A. Sharif, M. H. M. Ara, B. Ghafary, S. Salmani, and S. Mohajer, “Experimental observation of low threshold optical bistability in exfoliated graphene with low oxidation degree,” Opt. Mater. 53, 80–86 (2016).
[Crossref]

Santos, J. E.

N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
[Crossref]

Savchenko, A. K.

E. Hendry, P. J. Hale, J. J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
[Crossref] [PubMed]

Savostianova, N. A.

S. A. Mikhailov, N. A. Savostianova, and A. S. Moskalenko, “Negative dynamic conductivity of a current-driven array of graphene nanoribbons,” Phys. Rev. B 94, 035439 (2016).
[Crossref]

N. A. Savostianova and S. A. Mikhailov, “Giant enhancement of the third harmonic in graphene integrated in a layered structure,” Appl. Phys. Lett. 107, 181104 (2015).
[Crossref]

Scalora, M.

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Nonlinear control of absorption in one-dimensional photonic crystal with graphene-based defect,” Opt. Lett. 38, 3550–3553 (2013).
[Crossref] [PubMed]

Schubert, M.

M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61, 8187–8201 (2000).
[Crossref]

Shadrivov, I. V.

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, 035412 (2014).
[Crossref]

Sharif, M. A.

M. A. Sharif, M. H. M. Ara, B. Ghafary, S. Salmani, and S. Mohajer, “Experimental observation of low threshold optical bistability in exfoliated graphene with low oxidation degree,” Opt. Mater. 53, 80–86 (2016).
[Crossref]

Shelton, D.

R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86, 235147 (2012).
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Silviero, I.

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J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Erratum: Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature [phys. rev. b 91, 235320 (2015)],” Phys. Rev. B 93, 039904 (2016).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature,” Phys. Rev. B 91, 235320 (2015).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Dc current induced second order optical nonlinearity in graphene,” Optics Express 22, 15868–15876 (2014).
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J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
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N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
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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, 035412 (2014).
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Nat. Photonics (1)

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J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
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Optics Express (1)

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Dc current induced second order optical nonlinearity in graphene,” Optics Express 22, 15868–15876 (2014).
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Phys. Rev. B (20)

M. Tokman, Y. Wang, I. Oladyshkin, A. R. Kutayiah, and A. Belyanin, “Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene,” Phys. Rev. B 93, 235422 (2016).
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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, 035412 (2014).
[Crossref]

N. M. R. Peres, Y. V. Bludov, J. E. Santos, A.-P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90, 125425 (2014).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
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Phys. Rev. Lett. (3)

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X. Yao, M. Tokman, and A. Belyanin, “Efficient nonlinear generation of THz plasmons in graphene and topological insulators,” Phys. Rev. Lett. 112, 055501 (2014).
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S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007).
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S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

Prog. Electromag. Res. (1)

G. Hotopan, S. Ver Hoeye, C. Vazquez, R. Camblor, M. Fernández, F. Las Heras, P. Álvarez, and R. Menéndez, “Millimeter wave microstrip mixer based on graphene,” Prog. Electromag. Res. 118, 57–69 (2011).
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Other (5)

S. A. Mikhailov, “Electromagnetic nonlinearities in graphene,” in “Carbon nanotubes and graphene for photonic applications,” S. Yamashita, Y. Saito, and J. H. Choi, eds. (Woodhead Publishing Limited, 2013), chap. 7, pp. 171–219.
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Figures (13)

Fig. 1
Fig. 1 The geometry of the considered structure. A wave with the frequency ω is incident on the structure graphene (G) – layer 1 (L1) – layer 2 (L2) in the z direction. The system emits the third harmonic 3ω in the backward and forward directions. The layer L1 is assumed to be a dielectric with a dispersionless dielectric constant or a polar dielectric with the dielectric function (4). The back side of the layer L1 can be uncovered or covered by a metallic layer L2.
Fig. 2
Fig. 2 The parameter η(3) of the AGA structure (a) as a function of the electron density at different input-wave frequencies (f = 30 THz, λ = 10 μm and f = 10 THz, λ = 30 μm) and different values of the relaxation time τ in graphene and (b) as a function of the input-wave frequency f at τ = 1 ps and at different values of the electron density, measured in units 1012 cm−2. The resonances correspond to ħω/EF = 2/3, 1, and 2; in (a) from right to left, in (b) from left to right.
Fig. 3
Fig. 3 The parameter η(3) of the AGDA structure (a) as a function of the input-wave frequency f at the dielectric thickness d = 12.5 μm and (b) as a function of d at f = 6 THz (λ = 50 μm). Other parameters are τ = 1 ps and ns = 1011 cm−2; the refractive index of the dielectric is 2. The black curve in (a) shows for comparison the result for the isolated graphene layer (AGA) corresponding to the black solid curve in Fig. 2(b). Red dotted and green dashed curves show the third-harmonic wave intensity emitted in the backward and forward direction respectively.
Fig. 4
Fig. 4 The parameter η(3) of the AGDMA structure as a function of the input-wave frequency f at the dielectric thickness of (a) d = 12.5 μm and (b) d = 6.25 μm. Other parameters are τ = 1 ps, ns = 1011 cm−2; the refractive index of the dielectric is 2, the metal (Au) thickness is 0.2 μm. The black curve in (a) shows for comparison the result for the isolated graphene layer (AGA) corresponding to the black solid curve in Fig. 2(b). Red dotted and green dashed curves show the third-harmonic wave intensity (at the frequency 3 f) emitted in the backward and forward direction respectively.
Fig. 5
Fig. 5 (a) The real and imaginary parts of the dielectric function P(ω), Eq. (4), as well as (b) the transmission, (c) the reflection and (d) the absorption coefficients of an AGPA structure as a function of the input-wave frequency f at the dielectric thickness d = 12.5 μm. Parameters of the polar dielectric = 1, ωTO/2π = 15 THz, ωLO/2π = 30 THz, γTO/2π = 0.2 THz. Parameters of graphene are τ = 1 ps, ns = 1011 cm−2.
Fig. 6
Fig. 6 The parameter η(3) of the AGPA structure as a function of the input-wave frequency f at the polar dielectric thickness d = 14.3 μm in (a) a broad range 0–40 THz and (b) in the narrow frequency range around the resonance ω = ωres, Eq. (11). Parameters of graphene are τ = 1 ps and ns = 0.707 × 1011 cm−2, parameters of the polar dielectric: = 1, fTO = 15 THz, fLO = 30 THz, γTO/2π = 0.2 THz. The black curves show for comparison the result for the isolated graphene layer (AGA). The red dotted and green dashed curves show the third-harmonic wave intensity (at the frequency 3 f) emitted in the backward and forward direction respectively.
Fig. 7
Fig. 7 The parameter η(3) of the AGPA structure as a function of the dielectric thickness d at the input-wave frequency of (a) f = 5.02 THz (corresponds to the maximum of η(3) at the ħω/EF = 2/3 resonance) and (b) f = 30 THz, away from the graphene resonance at the upper boundary of the Reststrahlen-Band. Parameters of graphene are τ = 1 ps and ns = 0.707 × 1011 cm−2, parameters of the polar dielectric: = 1, fTO = 15 THz, fLO = 30 THz, γTO/2π = 0.2 THz. Red solid and green dashed curves show the third-harmonic wave intensity emitted in the backward and forward direction respectively. The value of the efficiency η AGPA ( 3 ) at d → 0 in (a), which is not clearly seen in the figure, is ≃ 2.18 × 10−14 (cm2/W)2 both for the forward and backward emitted radiation.
Fig. 8
Fig. 8 The absolute value of the ω-component of the electric field at the plane z = 0, |Ex (z = 0)|, and its cube as a function of the input-wave frequency f in the vicinity of the Reststrahlen-Band 15 − 30 THz. Parameters of the AGPA structure are the same as in Fig. 6.
Fig. 9
Fig. 9 The parameter η(3) of the AGPA structure as a function of the input-wave frequency f at the polar dielectric thickness d = 10.54 μm in the case when the resonance frequency (11) is close to the TO-phonon frequency fTO = 15 THz. Parameters of graphene are τ = 1 ps and ns = 0.636 × 1012 cm−2, parameters of the polar dielectric: = 1, fTO = 15 THz, fLO = 30 THz, γTO/2π = 0.2 THz. The black curves show for comparison the result for the isolated graphene layer (AGA). The red dotted and green dashed curves show the third-harmonic wave intensity emitted in the backward and forward direction respectively.
Fig. 10
Fig. 10 (a,b) The parameter η(3) of the AGPA structure as a function of the input-wave frequency f at the polar dielectric thickness d = 5.95 μm in the case when the triple resonance frequency (11) is close to the LO-phonon frequency fLO = 30 THz; (a) – in a broad frequency range up to 50 THz, (b) in the vicinity of the graphene resonance (11) at 10 THz. (c) The efficiency η(3) as a function of d at f = 10 THz. Parameters of graphene are τ = 1 ps and ns = 0.2827 × 1012 cm−2, parameters of the polar dielectric: = 1, fTO = 15 THz, fLO = 30 THz, γTO/2π = 0.2 THz. The black curves in (a) and (b) show for comparison the result for the isolated graphene layer (AGA). The red dotted and green dashed curves show the third-harmonic wave intensity emitted in the backward and forward direction respectively.
Fig. 11
Fig. 11 (a,b) The parameter η(3) of the AGPA structure as a function of the input-wave frequency f at the polar dielectric thickness d = 14.82 μm in the case when the graphene resonance frequency (11) is close to the LO-phonon frequency of the substrate fLO = 30 THz; (a) – in a broad frequency range up to 50 THz, (b) in the vicinity of the graphene resonance at 30 THz. (c) The same parameter as a function of the dielectric thickness at f = 30 THz. Parameters of graphene in (a)–(c) are τ = 1 ps and ns = 2.54469×1012 cm−2. (d) The efficiency η(3) as a function of the electron density at d = 14.82 μm and f = 30 THz. Parameters of the polar dielectric in all panels: = 1, fTO = 15 THz, fLO = 30 THz, γTO/2π = 0.2 THz. The black curves in (a) and (b) show for comparison the result for the isolated graphene layer (AGA). The red dotted and green dashed curves show the third-harmonic wave intensity emitted in the backward and forward direction respectively.
Fig. 12
Fig. 12 The parameter η(3) of the AGPMA structure (a) as a function of the input-wave frequency f at the polar dielectric thickness d = 11.31 μm in the case when the graphene resonance frequency (11) is close to the LO-phonon frequency of the substrate fLO = 30 THz and (b) as a function of d at f = 30 THz. Parameters of graphene are τ = 1 ps and ns = 2.54469 × 1012 cm−2, parameters of the polar dielectric: = 1, fTO = 15 THz, fLO = 30 THz, γTO/2π = 0.2 THz, the metal thickness is 0.2 μm. The black curve in (a) shows for comparison the result for the isolated graphene layer (AGA). The red dashed curve shows the third-harmonic wave intensity emitted in the backward direction.
Fig. 13
Fig. 13 The parameter η(3) of the AGDMA structure as a function of the input-wave frequency f at different values of the dielectric thickness d and at the relaxation times (a) τ = 1 ps and (b) τ = 0.1 ps. The density of electrons in graphene is ns = 0.3 × 1012 cm−2, the refractive index of the dielectric substrate is nω = n3ω = 2, the metal (Au) thickness is 0.4 μm. The black curves show for comparison the result for the isolated graphene layer (AGA). Only the emission in the backward direction is shown for all curves (the emission in the forward direction is negligibly small).

Equations (14)

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rot E = 1 c H t ,
rot H = ( ω , z ) c E t + 4 π c j δ ( z ) ,
H y ( + 0 ) H y ( 0 ) = 4 π c j x ( 0 ) .
P ( ω ) = ( 1 + ω LO 2 ω TO 2 ω TO 2 ω 2 i ω γ TO ) ,
M ( ω ) = 1 ω p 2 ω ( ω + i γ m ) ,
j x ( t ) = σ x x ( 1 ) ( ω ) E ω x ( 0 ) e i ω t + σ x x x x ( 3 ) ( ω , ω , ω ) [ E ω x ( 0 ) ] 3 e i 3 ω t + c . c .
σ α β ( 1 ) ( ω ) = δ α β e 2 π ( i Ω + i Γ + i 4 ln 2 ( Ω + i Γ ) 2 + ( Ω + i Γ ) ) ,
I 3 ω = η ( 3 ) I ω 3 .
d = λ ω 2 n ω m 1 , m 1 = 0 , 1 , 2 , ,
d = λ 3 ω 2 n 3 ω m 2 , m 2 = 0 , 1 , 2 , ,
ω = ω res = 2 E F 3 = 2 3 v F π n s ;
ω = 2 3 v F π n s = π c n d m 1 ,
d = λ ω 2 n ω ( m 1 + 1 2 ) , m 1 = 0 , 1 , 2 , ,
0 = ω LO 2 ω TO 2

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