Abstract

We investigate the nonlinear supermodes of surface plasmon polaritons in graphene multilayers with arbitrary number of graphene layers. Apart from the symmetric and anti-symmetric supermodes which exist in linear multilayer graphene waveguides, more asymmetric supermodes emerge in the nonlinear counterparts as the field symmetry is broken. The number of asymmetric supermodes relies largely on the layer number of graphene. There is a certain threshold of field intensity for the emergence of each individual asymmetric supermode. The threshold increases as the incident wavelength or chemical potential of graphene increases. The study may find applications in building all-optical mode converters and switches.

© 2017 Optical Society of America

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References

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2017 (2)

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 99(1), 1 (2017). doi:.
[Crossref]

2016 (3)

2015 (3)

2014 (2)

C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014).
[Crossref] [PubMed]

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photonics Rev. 8(2), 291–296 (2014).
[Crossref]

2013 (5)

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

A. Auditore, C. De Angelis, A. Locatelli, S. Boscolo, M. Midrio, M. Romagnoli, A.-D. Capobianco, and G. Nalesso, “Graphene sustained nonlinear modes in dielectric waveguides,” Opt. Lett. 38(5), 631–633 (2013).
[Crossref] [PubMed]

A. V. Gorbach, “Nonlinear graphene plasmonics: amplitude equation for surface plasmons,” Phys. Rev. A 87(1), 013830 (2013).
[Crossref]

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

2012 (3)

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

2011 (5)

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84(19), 195446 (2011).
[Crossref]

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

2010 (5)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82(20), 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(9), 097401 (2010).
[Crossref] [PubMed]

G. Xing, H. Guo, X. Zhang, T. C. Sum, and C. H. A. Huan, “The Physics of ultrafast saturable absorption in graphene,” Opt. Express 18(5), 4564–4573 (2010).
[Crossref] [PubMed]

N. M. R. Peres, “Colloquium: the transport properties of graphene: an introduction,” Rev. Mod. Phys. 82(3), 2673–2700 (2010).
[Crossref]

2008 (3)

S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20(38), 384204 (2008).
[Crossref] [PubMed]

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(5881), 1308 (2008).
[Crossref] [PubMed]

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

2007 (3)

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79(2), 27002 (2007).
[Crossref]

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[Crossref] [PubMed]

2003 (1)

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

1997 (1)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[Crossref]

1984 (1)

1978 (1)

Anderson, T.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

Auditore, A.

Bao, Q.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

Bartal, G.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[Crossref] [PubMed]

Bezares, F. J.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (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(5881), 1308 (2008).
[Crossref] [PubMed]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[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(5881), 1308 (2008).
[Crossref] [PubMed]

Boscolo, S.

Bravo-Abad, J.

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

Caldwell, J. D.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

Capobianco, A.-D.

Chang, D. E.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[Crossref]

Chun, H.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

Cohen, O.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

Coskun, T. H.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[Crossref]

Currie, M.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

Da, H.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

De Angelis, C.

Dong, Z.

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Feit, M. D.

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Fleck, J. A.

Fleischer, J. W.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

Fujiwara, M.

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

García de Abajo, F. J.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Garcia-Vidal, F. J.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84(19), 195446 (2011).
[Crossref]

García-Vidal, F. J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

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(5881), 1308 (2008).
[Crossref] [PubMed]

Genov, D. A.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[Crossref] [PubMed]

Gorbach, A. V.

A. V. Gorbach, “Nonlinear graphene plasmonics: amplitude equation for surface plasmons,” Phys. Rev. A 87(1), 013830 (2013).
[Crossref]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

Grigorenko, A. N.

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(5881), 1308 (2008).
[Crossref] [PubMed]

Guinea, F.

A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84(19), 195446 (2011).
[Crossref]

Günther, U.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Guo, H.

Hale, P. J.

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), 097401 (2010).
[Crossref] [PubMed]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Hendry, E.

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), 097401 (2010).
[Crossref] [PubMed]

Huan, C. H. A.

Huang, H.

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 99(1), 1 (2017). doi:.
[Crossref]

C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014).
[Crossref] [PubMed]

Iorsh, I. V.

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

Ishikawa, K. L.

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82(20), 201402 (2010).
[Crossref]

Kapon, E.

Katz, J.

Ke, S.

Kivshar, Y. S.

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photonics Rev. 8(2), 291–296 (2014).
[Crossref]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

Klaiman, S.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Koppens, F. H. L.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[Crossref] [PubMed]

Locatelli, A.

Loh, K. P.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

Long, H.

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 99(1), 1 (2017). doi:.
[Crossref]

Z. Wang, B. Wang, K. Wang, H. Long, and P. Lu, “Vector plasmonic lattice solitons in nonlinear graphene-pair arrays,” Opt. Lett. 41(15), 3619–3622 (2016).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Plasmonic lattice solitons in nonlinear graphene sheet arrays,” Opt. Express 23(25), 32679–32689 (2015).
[Crossref] [PubMed]

F. Wang, C. Qin, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Rabi oscillations of surface plasmon polaritons in graphene-pair arrays,” Opt. Express 23(24), 31136–31143 (2015).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014).
[Crossref] [PubMed]

Lu, P.

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 99(1), 1 (2017). doi:.
[Crossref]

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

Z. Wang, B. Wang, K. Wang, H. Long, and P. Lu, “Vector plasmonic lattice solitons in nonlinear graphene-pair arrays,” Opt. Lett. 41(15), 3619–3622 (2016).
[Crossref] [PubMed]

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Plasmonic lattice solitons in nonlinear graphene sheet arrays,” Opt. Express 23(25), 32679–32689 (2015).
[Crossref] [PubMed]

F. Wang, C. Qin, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Rabi oscillations of surface plasmon polaritons in graphene-pair arrays,” Opt. Express 23(24), 31136–31143 (2015).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014).
[Crossref] [PubMed]

Martin-Moreno, L.

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84(19), 195446 (2011).
[Crossref]

Midrio, M.

Mikhailov, S. A.

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), 097401 (2010).
[Crossref] [PubMed]

S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20(38), 384204 (2008).
[Crossref] [PubMed]

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79(2), 27002 (2007).
[Crossref]

Mitchell, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[Crossref]

Moger, J.

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), 097401 (2010).
[Crossref] [PubMed]

Moiseyev, N.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Nair, R. R.

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(5881), 1308 (2008).
[Crossref] [PubMed]

Nalesso, G.

Nesterov, M. L.

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

Nikitin, A. Y.

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

Nikitin, A. Yu.

A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84(19), 195446 (2011).
[Crossref]

Noda, T.

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

Novoselov, K. S.

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(5881), 1308 (2008).
[Crossref] [PubMed]

Peres, N. M. R.

N. M. R. Peres, “Colloquium: the transport properties of graphene: an introduction,” Rev. Mod. Phys. 82(3), 2673–2700 (2010).
[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(5881), 1308 (2008).
[Crossref] [PubMed]

Qin, C.

Qiu, C. W.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

Robinson, J.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

Romagnoli, M.

Sanaei, R.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

Savchenko, A. K.

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), 097401 (2010).
[Crossref] [PubMed]

Schwartz, T.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

Segev, M.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[Crossref]

Shadrivov, I. V.

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photonics Rev. 8(2), 291–296 (2014).
[Crossref]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

Smirnov, A. I.

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photonics Rev. 8(2), 291–296 (2014).
[Crossref]

Smirnova, D. A.

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photonics Rev. 8(2), 291–296 (2014).
[Crossref]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

Stauber, T.

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(5881), 1308 (2008).
[Crossref] [PubMed]

Sum, T. C.

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Tadjer, M.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

Takeuchi, S.

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

Tao, J.

Teng, J.

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Toubaru, K.

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Wang, B.

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 99(1), 1 (2017). doi:.
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

Z. Wang, B. Wang, K. Wang, H. Long, and P. Lu, “Vector plasmonic lattice solitons in nonlinear graphene-pair arrays,” Opt. Lett. 41(15), 3619–3622 (2016).
[Crossref] [PubMed]

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Plasmonic lattice solitons in nonlinear graphene sheet arrays,” Opt. Express 23(25), 32679–32689 (2015).
[Crossref] [PubMed]

F. Wang, C. Qin, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Rabi oscillations of surface plasmon polaritons in graphene-pair arrays,” Opt. Express 23(24), 31136–31143 (2015).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Wang, F.

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

F. Wang, C. Qin, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Rabi oscillations of surface plasmon polaritons in graphene-pair arrays,” Opt. Express 23(24), 31136–31143 (2015).
[Crossref] [PubMed]

Wang, K.

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 99(1), 1 (2017). doi:.
[Crossref]

Z. Wang, B. Wang, K. Wang, H. Long, and P. Lu, “Vector plasmonic lattice solitons in nonlinear graphene-pair arrays,” Opt. Lett. 41(15), 3619–3622 (2016).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

F. Wang, C. Qin, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Rabi oscillations of surface plasmon polaritons in graphene-pair arrays,” Opt. Express 23(24), 31136–31143 (2015).
[Crossref] [PubMed]

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Plasmonic lattice solitons in nonlinear graphene sheet arrays,” Opt. Express 23(25), 32679–32689 (2015).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Huang, H. Long, K. Wang, and P. Lu, “Low-loss plasmonic supermodes in graphene multilayers,” Opt. Express 22(21), 25324–25332 (2014).
[Crossref] [PubMed]

Wang, Q. J.

Wang, Z.

Xing, G.

Yang, J. K. W.

Yariv, A.

Yuan, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Zhang, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

G. Xing, H. Guo, X. Zhang, T. C. Sum, and C. H. A. Huan, “The Physics of ultrafast saturable absorption in graphene,” Opt. Express 18(5), 4564–4573 (2010).
[Crossref] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[Crossref] [PubMed]

Zhao, H. Q.

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

Ziegler, K.

S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20(38), 384204 (2008).
[Crossref] [PubMed]

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

ACS Nano (1)

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjer, “Quantifying pulsed laser induced damage to graphene,” Appl. Phys. Lett. 99(21), 211909 (2011).
[Crossref]

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

Europhys. Lett. (1)

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79(2), 27002 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Quantum Electron. 23(1), 4600105 (2017).

J. Lightwave Technol. (3)

J. Phys. Condens. Matter (1)

S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20(38), 384204 (2008).
[Crossref] [PubMed]

Laser Photonics Rev. (2)

D. A. Smirnova, I. V. Shadrivov, A. I. Smirnov, and Y. S. Kivshar, “Dissipative plasmon-solitons in multilayer graphene,” Laser Photonics Rev. 8(2), 291–296 (2014).
[Crossref]

M. L. Nesterov, J. Bravo-Abad, A. Y. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, “Graphene supports the propagation of subwavelength optical solitons,” Laser Photonics Rev. 7(2), L7–L11 (2013).
[Crossref]

Nano Lett. (2)

M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, “Highly efficient coupling of photons from nanoemitters into single-mode optical fibers,” Nano Lett. 11(10), 4362–4365 (2011).
[Crossref] [PubMed]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. A (1)

A. V. Gorbach, “Nonlinear graphene plasmonics: amplitude equation for surface plasmons,” Phys. Rev. A 87(1), 013830 (2013).
[Crossref]

Phys. Rev. B (4)

K. L. Ishikawa, “Nonlinear optical response of graphene in time domain,” Phys. Rev. B 82(20), 201402 (2010).
[Crossref]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear switching with a graphene coupler,” Phys. Rev. B 88(4), 045443 (2013).
[Crossref]

H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: Giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B 84(19), 195446 (2011).
[Crossref]

Phys. Rev. Lett. (7)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997).
[Crossref]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91(11), 113901 (2003).
[Crossref] [PubMed]

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99(1), 016803 (2007).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

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[Crossref]

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[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Schematic of the nonlinear graphene multilayer. The position of graphene is denoted by n with n ∈ [1, N] and N is the total number of graphene sheets. The interlayer space is denoted by d, σ g is the surface conductivity of graphene, and ε d is the relative permittivity of quartz medium between graphene.
Fig. 2
Fig. 2 Propagation constants of the supermodes for different number of graphene sheets. (a) real and (b) imaginary parts of the propagation constants as the peak intensity I0 = 2 V2/μm2. (c) real and (d) imaginary parts of the propagation constants as the peak intensity I0 = 40 V2/μm2. (e) real and (f) imaginary parts of the propagation constants of the asymmetric supermodes. The arrows represent the increasing order of the label s or s′.
Fig. 3
Fig. 3 Transverse electric field (Ex) distributions of the supermodes for the peak intensity I0 = 40 V2/μm2. (a) Mode profiles of four supermodes as N = 3. (b) Mode profiles of six supermodes as N = 4. The red dashed lines represent the position of graphene.
Fig. 4
Fig. 4 Electric field intensity (| E |2) distribution of the supermodes in the nonlinear graphene multilayer as N = 4. The field intensity is given by I0 = 40 V2/μm2.
Fig. 5
Fig. 5 (a) and (b) The nonlinear propagation constant β versus the peak intensity I0 in the nonlinear graphene multilayers of N = 3 and 4, respectively.
Fig. 6
Fig. 6 (a) and (b) The peak intensity threshold of the asymmetric modes versus different wavelength λ and chemical potential μ c as N = 4. (a) The threshold Ic for mode of s′ = 1. (b) The threshold Ic for mode of s′ = 2.

Equations (2)

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σ g , N L = i 3 8 e 2 π 2 ( e V F ) 2 μ c ω 3
( 0 η 0 k 0 x 1 ε r ( x ) x + k 0 η 0 k 0 ε r ( x ) η 0 0 ) ( H y E x ) = β ( H y E x )

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