Abstract

Plasmons in highly doped graphene offer the means to dramatically enhance light absorption in the atomically thin material. Ultimately the absorbed light energy induces an increase in electron temperature, accompanied by large shifts in the chemical potential. This intrinsically incoherent effect leads to strong intensity-dependent modifications of the optical response, complementing the remarkable coherent nonlinearities arising in graphene due to interband transitions and anharmonic intraband electron motion. Through rigorous time-domain quantum-mechanical simulations of graphene nanoribbons, we show that the incoherent mechanism dominates over the coherent response for the high levels of intensity required to trigger nonperturbative optical phenomena such as saturable absorption. We anticipate that these findings will elucidate the role of coherent and incoherent nonlinearities for future studies and applications of plasmon-assisted nonlinear optics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  11. J. D. Cox and F. J. García de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nat. Commun. 5, 5725 (2014).
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  14. T. Christensen, W. Yan, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
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    [Crossref]
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  27. Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6, 3677–3694 (2012).
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  28. A. Marini, J. D. Cox, and F. J. García de Abajo, “Theory of graphene saturable absorption,” Phys. Rev. B 95, 125408 (2017).
    [Crossref]
  29. F. J. García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photon. 1, 135–152 (2014).
    [Crossref]
  30. M. M. Jadidi, J. C. König-Otto, S. Winnerl, A. B. Sushkov, H. D. Drew, T. E. Murphy, and M. Mittendorff, “Nonlinear terahertz absorption of graphene plasmons,” Nano Lett. 16, 2734–2738 (2016).
    [Crossref]
  31. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
    [Crossref]
  32. 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, 2541–2547 (2013).
    [Crossref]
  33. M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, and H. A. Atwater, “Tunable large resonant absorption in a midinfrared graphene Salisbury screen,” Phys. Rev. B 90, 165409 (2014).
    [Crossref]
  34. 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]

2017 (3)

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

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

A. Marini, J. D. Cox, and F. J. García de Abajo, “Theory of graphene saturable absorption,” Phys. Rev. B 95, 125408 (2017).
[Crossref]

2016 (3)

M. M. Jadidi, J. C. König-Otto, S. Winnerl, A. B. Sushkov, H. D. Drew, T. E. Murphy, and M. Mittendorff, “Nonlinear terahertz absorption of graphene plasmons,” Nano Lett. 16, 2734–2738 (2016).
[Crossref]

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]

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]

2015 (4)

M. Jablan and D. E. Chang, “Multiplasmon absorption in graphene,” Phys. Rev. Lett. 114, 236801 (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]

T. Christensen, W. Yan, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (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, 10545–10562 (2015).
[Crossref]

2014 (5)

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

M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, and H. A. Atwater, “Tunable large resonant absorption in a midinfrared graphene Salisbury screen,” Phys. Rev. B 90, 165409 (2014).
[Crossref]

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

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, 115310 (2014).
[Crossref]

P. Bowlan, E. Martinez-Moreno, K. Reimann, T. Elsaesser, and M. Woerner, “Ultrafast terahertz response of multilayer graphene in the nonperturbative regime,” Phys. Rev. B 89, 041408 (2014).
[Crossref]

2013 (3)

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).
[Crossref]

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, 2541–2547 (2013).
[Crossref]

2012 (5)

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]

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]

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6, 3677–3694 (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]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[Crossref]

2011 (3)

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10, 911–921 (2011).
[Crossref]

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
[Crossref]

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

2010 (3)

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

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

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

2008 (3)

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, 442–453 (2008).
[Crossref]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref]

2007 (1)

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

1997 (1)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[Crossref]

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, 115310 (2014).
[Crossref]

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, 442–453 (2008).
[Crossref]

Atwater, H. A.

M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, and H. A. Atwater, “Tunable large resonant absorption in a midinfrared graphene Salisbury screen,” Phys. Rev. B 90, 165409 (2014).
[Crossref]

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, 2541–2547 (2013).
[Crossref]

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

Bai, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
[Crossref]

Bao, Q.

Bechtel, H. A.

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

Bian, F.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
[Crossref]

Blake, P.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref]

Booth, T. J.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref]

Bowlan, P.

P. Bowlan, E. Martinez-Moreno, K. Reimann, T. Elsaesser, and M. Woerner, “Ultrafast terahertz response of multilayer graphene in the nonperturbative regime,” Phys. Rev. B 89, 041408 (2014).
[Crossref]

Brar, V. W.

M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, and H. A. Atwater, “Tunable large resonant absorption in a midinfrared graphene Salisbury screen,” Phys. Rev. B 90, 165409 (2014).
[Crossref]

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, 2541–2547 (2013).
[Crossref]

Brevet, P.-F.

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, 10545–10562 (2015).
[Crossref]

Butet, J.

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, 10545–10562 (2015).
[Crossref]

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]

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]

M. Jablan and D. E. Chang, “Multiplasmon absorption in graphene,” Phys. Rev. Lett. 114, 236801 (2015).
[Crossref]

Cheng, J. L.

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]

Chiu, H.-Y.

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]

Choi, M.

M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim, S. Kim, M. Choi, and H. A. Atwater, “Tunable large resonant absorption in a midinfrared graphene Salisbury screen,” Phys. Rev. B 90, 165409 (2014).
[Crossref]

Christensen, T.

T. Christensen, W. Yan, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
[Crossref]

Christopher, P.

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10, 911–921 (2011).
[Crossref]

Constant, T. J.

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]

Cox, J. D.

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

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

A. Marini, J. D. Cox, and F. J. García de Abajo, “Theory of graphene saturable absorption,” Phys. Rev. B 95, 125408 (2017).
[Crossref]

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

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).
[Crossref]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[Crossref]

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, 115310 (2014).
[Crossref]

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, 115310 (2014).
[Crossref]

Drew, H. D.

M. M. Jadidi, J. C. König-Otto, S. Winnerl, A. B. Sushkov, H. D. Drew, T. E. Murphy, and M. Mittendorff, “Nonlinear terahertz absorption of graphene plasmons,” Nano Lett. 16, 2734–2738 (2016).
[Crossref]

Elsaesser, T.

P. Bowlan, E. Martinez-Moreno, K. Reimann, T. Elsaesser, and M. Woerner, “Ultrafast terahertz response of multilayer graphene in the nonperturbative regime,” Phys. Rev. B 89, 041408 (2014).
[Crossref]

Feld, M. S.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[Crossref]

García de Abajo, F. J.

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

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

A. Marini, J. D. Cox, and F. J. García de Abajo, “Theory of graphene saturable absorption,” Phys. Rev. B 95, 125408 (2017).
[Crossref]

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

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

Geim, A. K.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref]

Geng, B.

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

Gerstenkorn, C.

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]

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T. Christensen, W. Yan, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407 (2015).
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Supplementary Material (1)

NameDescription
» Supplement 1       We provide here additional simulations similar to those of Figs. 2 and 3 of the main text, but under excitation by pulses of different fluence.

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Figures (3)

Fig. 1.
Fig. 1. Enhancement of saturable absorption and third-harmonic response by graphene plasmons under intense cw illumination. (a) Incident light excites plasmons in a graphene nanoribbon. Plasmon dissipation is usually described through direct inelastic decay to the initial electronic distribution at a rate γ0 (static model, left). A more realistic description incorporates dynamic heating of the electrons at a rate γT before further relaxation to phonons at a rate γph (dynamic model, right). (b) Normal-incidence absorption cross section of a highly doped (Fermi energy EF=0.5  eV) graphene nanoribbon (20 nm width) for different cw incident light intensities (upper-right legend), normalized to the graphene area. Light is polarized across the ribbon, which exhibits a transverse plasmon resonance of ωp=0.382  eV energy (low intensity limit). The cross section is plotted as a function of detuning, defined as the photon energy relative to ωp. The results are obtained from self-consistent-field tight-binding simulations incorporating either the static (γ0=25  meV, dashed curves) or dynamic (γT=20  meV, γph=5  meV, solid curves) relaxation scheme. (c) Intensity dependence of the absorption cross section for three different photon energies, indicated by the color-coded arrows in (b). (d), (e) Third-harmonic susceptibility χ3ω(3) in electrostatic units, assuming a 3.3×108 cm graphene thickness, under the same conditions as (b), (c).
Fig. 2.
Fig. 2. Transient absorption of ultrashort light pulses tuned to a graphene plasmon. (a)–(d) Time-domain simulations showing the response of the nanoribbon considered in Fig. 1 to normally impinging Gaussian light pulses of varying FWHM duration Δ. The pulse central photon energy is tuned to the ribbon transverse plasmon (ωp=0.382  eV), and the fluence is fixed to 1  J/m2. We show (a) the induced dipole moment, (b) the electronic heat Q, (c) the chemical potential μ, and (d) the electronic temperature T. Results are obtained within the dynamic relaxation scheme [see Fig. 1(a)], assuming γT=20  meV and γph=5  meV. (e)–(j) Spectral decomposition of the induced dipole moments under excitation by the pulses considered in (a)–(d), calculated in the dynamic (solid curves) and static (γ0=25  meV, dashed curves) relaxation schemes and represented as a function of detuning (photon energy relative to ωp). The incident pulse spectra are shown for comparison (dotted curves). (k), (l) Fraction of energy absorbed from the directly impinging light (fluence times graphene area) as a function of either (k) pulse duration for the indicated pulse fluence or (l) peak intensity for different durations.
Fig. 3.
Fig. 3. All-optical modulation by transient graphene plasmons in a pump–probe configuration. (a) Induced dipole moment (left axis) upon pump–probe excitation (50 fs delay) of the ribbon in Fig. 1 using pulses tuned to the ribbon transverse plasmon, as calculated within the dynamic relaxation scheme. The incident light electric field is shown for comparison (right axis). (b) Fraction of energy absorbed from the directly impinging probe pulse calculated in the static (dashed curve) and dynamic (solid curve) relaxation schemes. (c) Spectral decomposition of the cross section for absorption of the probe pulse as a function of pulse delay and detuning. The pulse duration and fluence are 10 fs and 1  J/m2 in all cases.

Equations (3)

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ρ˙=i[H,ρ]Γ(ρ),
jϵj[fj(μ,T)fj]=0
2jfj(μ,T)=Ne,

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