Abstract

We study the discrete soliton formation in one- and two-dimensional arrays of nanowires coated with graphene monolayers. Highly confined solitons, including the fundamental and the higher-order modes, are found to be supported by the proposed structure with a low level of power flow. Numerical analysis reveals that, by tuning the input intensity and Fermi energy, the beam diffraction, soliton dimension and propagation loss can be fully controlled in a broad range, indicating potential values of the graphene-based solitons in nonlinear/active nanophotonic systems.

© 2016 Optical Society of America

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    [Crossref]
  2. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104(10), 106802 (2010).
    [Crossref] [PubMed]
  3. 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]
  4. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Self-focusing and spatial plasmon-polariton solitons,” Opt. Express 17(24), 21732–21737 (2009).
    [Crossref] [PubMed]
  5. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
    [Crossref]
  6. F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
    [Crossref]
  7. F. H. 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]
  8. A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
    [Crossref] [PubMed]
  9. K. Kim, J.-Y. Choi, T. Kim, S.-H. Cho, and H.-J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature 479(7373), 338–344 (2011).
    [Crossref] [PubMed]
  10. J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
    [Crossref] [PubMed]
  11. A. V. Gorbach, A. Marini, and D. V. Skryabin, “Graphene-clad tapered fiber: effective nonlinearity and propagation losses,” Opt. Lett. 38(24), 5244–5247 (2013).
    [Crossref] [PubMed]
  12. X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
    [Crossref] [PubMed]
  13. Y. Wu, B. Yao, A. Zhang, Y. Rao, Z. Wang, Y. Cheng, Y. Gong, W. Zhang, Y. Chen, and K. S. Chiang, “Graphene-coated microfiber Bragg grating for high-sensitivity gas sensing,” Opt. Lett. 39(5), 1235–1237 (2014).
    [Crossref] [PubMed]
  14. Y. Gao, G. Ren, B. Zhu, J. Wang, and S. Jian, “Single-mode graphene-coated nanowire plasmonic waveguide,” Opt. Lett. 39(20), 5909–5912 (2014).
    [Crossref] [PubMed]
  15. Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express 22(20), 24322–24331 (2014).
    [Crossref] [PubMed]
  16. H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
    [Crossref] [PubMed]
  17. D. A. Smirnova, I. V. Shadrivov, A. E. Miroshnichenko, A. I. Smirnov, and Y. S. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 035412 (2014).
    [Crossref]
  18. R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
    [Crossref] [PubMed]
  19. A. Y. Nikitin, F. Guinea, F. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85(8), 081405 (2012).
    [Crossref]
  20. M. A. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013).
    [Crossref] [PubMed]
  21. T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
    [Crossref] [PubMed]
  22. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
    [Crossref] [PubMed]
  23. 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(19), 196405 (2008).
    [Crossref] [PubMed]
  24. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
    [Crossref] [PubMed]
  25. P. Avouris, “Graphene: electronic and photonic properties and devices,” Nano Lett. 10(11), 4285–4294 (2010).
    [Crossref] [PubMed]
  26. Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
    [Crossref] [PubMed]
  27. 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]
  28. 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(2), 021014 (2013).
    [Crossref]
  29. 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]
  30. M. I. Molina and Y. S. Kivshar, “Discrete and surface solitons in photonic graphene nanoribbons,” Opt. Lett. 35(17), 2895–2897 (2010).
    [Crossref] [PubMed]
  31. Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
    [Crossref]
  32. D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015).
    [Crossref]
  33. 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]
  34. C. Huang, F. Ye, Z. Sun, and X. Chen, “Tunable subwavelength photonic lattices and solitons in periodically patterned graphene monolayer,” Opt. Express 22(24), 30108–30117 (2014).
    [Crossref] [PubMed]
  35. A. Savin and Y. S. Kivshar, “Surface solitons at the edges of graphene nanoribbons,” Europhysics Lett. 89(4), 46001 (2010).
    [Crossref]
  36. 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]
  37. L. M. Zhao, D. Y. Tang, H. Zhang, X. Wu, Q. Bao, and K. P. Loh, “Dissipative soliton operation of an ytterbium-doped fiber laser mode locked with atomic multilayer graphene,” Opt. Lett. 35(21), 3622–3624 (2010).
    [Crossref] [PubMed]
  38. S. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhysics Lett. 79(2), 27002 (2007).
    [Crossref]
  39. R. Wang, Y. Hao, Z. Wang, H. Gong, and J. T. L. Thong, “Large-diameter graphene nanotubes synthesized using Ni nanowire templates,” Nano Lett. 10(12), 4844–4850 (2010).
    [Crossref] [PubMed]
  40. Y. Wu, B. Yao, A. Zhang, Y. Rao, Z. Wang, Y. Cheng, Y. Gong, W. Zhang, Y. Chen, and K. S. Chiang, “Graphene-coated microfiber Bragg grating for high-sensitivity gas sensing,” Opt. Lett. 39(5), 1235–1237 (2014).
    [Crossref] [PubMed]
  41. A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
    [Crossref]
  42. I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
    [Crossref] [PubMed]
  43. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
  44. P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, “How perfect can graphene be?” Phys. Rev. Lett. 103(13), 136403 (2009).
    [Crossref] [PubMed]

2015 (2)

Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
[Crossref]

D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015).
[Crossref]

2014 (8)

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]

C. Huang, F. Ye, Z. Sun, and X. Chen, “Tunable subwavelength photonic lattices and solitons in periodically patterned graphene monolayer,” Opt. Express 22(24), 30108–30117 (2014).
[Crossref] [PubMed]

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref] [PubMed]

Y. Wu, B. Yao, A. Zhang, Y. Rao, Z. Wang, Y. Cheng, Y. Gong, W. Zhang, Y. Chen, and K. S. Chiang, “Graphene-coated microfiber Bragg grating for high-sensitivity gas sensing,” Opt. Lett. 39(5), 1235–1237 (2014).
[Crossref] [PubMed]

Y. Gao, G. Ren, B. Zhu, J. Wang, and S. Jian, “Single-mode graphene-coated nanowire plasmonic waveguide,” Opt. Lett. 39(20), 5909–5912 (2014).
[Crossref] [PubMed]

Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express 22(20), 24322–24331 (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(3), 035412 (2014).
[Crossref]

Y. Wu, B. Yao, A. Zhang, Y. Rao, Z. Wang, Y. Cheng, Y. Gong, W. Zhang, Y. Chen, and K. S. Chiang, “Graphene-coated microfiber Bragg grating for high-sensitivity gas sensing,” Opt. Lett. 39(5), 1235–1237 (2014).
[Crossref] [PubMed]

2013 (7)

J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
[Crossref] [PubMed]

A. V. Gorbach, A. Marini, and D. V. Skryabin, “Graphene-clad tapered fiber: effective nonlinearity and propagation losses,” Opt. Lett. 38(24), 5244–5247 (2013).
[Crossref] [PubMed]

X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
[Crossref] [PubMed]

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(2), 021014 (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]

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]

M. A. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013).
[Crossref] [PubMed]

2012 (7)

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

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[Crossref] [PubMed]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20(27), 28017–28024 (2012).
[Crossref] [PubMed]

A. Y. Nikitin, F. Guinea, F. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85(8), 081405 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
[Crossref] [PubMed]

2011 (4)

A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
[Crossref]

F. H. 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]

K. Kim, J.-Y. Choi, T. Kim, S.-H. Cho, and H.-J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature 479(7373), 338–344 (2011).
[Crossref] [PubMed]

H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
[Crossref] [PubMed]

2010 (8)

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

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104(10), 106802 (2010).
[Crossref] [PubMed]

P. Avouris, “Graphene: electronic and photonic properties and devices,” Nano Lett. 10(11), 4285–4294 (2010).
[Crossref] [PubMed]

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]

M. I. Molina and Y. S. Kivshar, “Discrete and surface solitons in photonic graphene nanoribbons,” Opt. Lett. 35(17), 2895–2897 (2010).
[Crossref] [PubMed]

R. Wang, Y. Hao, Z. Wang, H. Gong, and J. T. L. Thong, “Large-diameter graphene nanotubes synthesized using Ni nanowire templates,” Nano Lett. 10(12), 4844–4850 (2010).
[Crossref] [PubMed]

L. M. Zhao, D. Y. Tang, H. Zhang, X. Wu, Q. Bao, and K. P. Loh, “Dissipative soliton operation of an ytterbium-doped fiber laser mode locked with atomic multilayer graphene,” Opt. Lett. 35(21), 3622–3624 (2010).
[Crossref] [PubMed]

A. Savin and Y. S. Kivshar, “Surface solitons at the edges of graphene nanoribbons,” Europhysics Lett. 89(4), 46001 (2010).
[Crossref]

2009 (4)

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Self-focusing and spatial plasmon-polariton solitons,” Opt. Express 17(24), 21732–21737 (2009).
[Crossref] [PubMed]

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

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[Crossref] [PubMed]

P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, “How perfect can graphene be?” Phys. Rev. Lett. 103(13), 136403 (2009).
[Crossref] [PubMed]

2008 (2)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).

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(19), 196405 (2008).
[Crossref] [PubMed]

2007 (2)

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

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]

Alaee, R.

Avouris, P.

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref] [PubMed]

P. Avouris, “Graphene: electronic and photonic properties and devices,” Nano Lett. 10(11), 4285–4294 (2010).
[Crossref] [PubMed]

Bao, Q.

Barnes, W. L.

I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
[Crossref] [PubMed]

Barra, A.-L.

P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, “How perfect can graphene be?” Phys. Rev. Lett. 103(13), 136403 (2009).
[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]

Bludov, Y. V.

Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
[Crossref]

Bointon, T. H.

I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
[Crossref] [PubMed]

Bolotin, K. I.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).

Bonaccorso, F.

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

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]

Buljan, H.

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

Capolino, F.

Chang, D. E.

F. H. 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]

Chen, X.

Chen, Y.

Cheng, Y.

Chiang, K. S.

Cho, S.-H.

K. Kim, J.-Y. Choi, T. Kim, S.-H. Cho, and H.-J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature 479(7373), 338–344 (2011).
[Crossref] [PubMed]

Choi, J.-Y.

K. Kim, J.-Y. Choi, T. Kim, S.-H. Cho, and H.-J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature 479(7373), 338–344 (2011).
[Crossref] [PubMed]

Christensen, J.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Chung, H.-J.

K. Kim, J.-Y. Choi, T. Kim, S.-H. Cho, and H.-J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature 479(7373), 338–344 (2011).
[Crossref] [PubMed]

Cormode, D.

A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
[Crossref]

Craciun, M. F.

I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
[Crossref] [PubMed]

Cui, Y.

H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
[Crossref] [PubMed]

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(2), 021014 (2013).
[Crossref]

Dai, H.

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D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015).
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P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, “How perfect can graphene be?” Phys. Rev. Lett. 103(13), 136403 (2009).
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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(2), 021014 (2013).
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Panoiu, N. C.

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Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
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P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, “How perfect can graphene be?” Phys. Rev. Lett. 103(13), 136403 (2009).
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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
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Ren, G.

Reynolds, C.

A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
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A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
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H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
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Russo, S.

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A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
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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]

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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(19), 196405 (2008).
[Crossref] [PubMed]

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[Crossref]

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]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Self-focusing and spatial plasmon-polariton solitons,” Opt. Express 17(24), 21732–21737 (2009).
[Crossref] [PubMed]

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X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
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K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).

Skryabin, D. V.

Smirnov, A. I.

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

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).
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Smirnov, L. A.

D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015).
[Crossref]

Smirnova, D. A.

Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
[Crossref]

D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015).
[Crossref]

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

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]

Soljacic, M.

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

Stormer, H. L.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).

Sun, Z.

Tan, D. T.

J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
[Crossref] [PubMed]

Tang, D. Y.

Thong, J. T. L.

R. Wang, Y. Hao, Z. Wang, H. Gong, and J. T. L. Thong, “Large-diameter graphene nanotubes synthesized using Ni nanowire templates,” Nano Lett. 10(12), 4844–4850 (2010).
[Crossref] [PubMed]

Thongrattanasiri, S.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

Vasilevskiy, M. I.

Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
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Wang, H.

H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
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Wang, J.

Wang, R.

R. Wang, Y. Hao, Z. Wang, H. Gong, and J. T. L. Thong, “Large-diameter graphene nanotubes synthesized using Ni nanowire templates,” Nano Lett. 10(12), 4844–4850 (2010).
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Wang, Z.

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I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
[Crossref] [PubMed]

Wu, X.

Wu, Y.

Xu, Q.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[Crossref] [PubMed]

Yang, X.

X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
[Crossref] [PubMed]

Yang, Y.

H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
[Crossref] [PubMed]

Yao, B.

Ye, F.

C. Huang, F. Ye, Z. Sun, and X. Chen, “Tunable subwavelength photonic lattices and solitons in periodically patterned graphene monolayer,” Opt. Express 22(24), 30108–30117 (2014).
[Crossref] [PubMed]

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104(10), 106802 (2010).
[Crossref] [PubMed]

Yeh, P.-C.

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(2), 021014 (2013).
[Crossref]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Zhang, A.

Zhang, H.

Zhang, W.

Zhang, X.

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, L. M.

Zhao, P.

X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
[Crossref] [PubMed]

Zhu, B.

Zhu, Y.

X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
[Crossref] [PubMed]

ACS Appl. Mater. Interfaces (1)

X. Yang, K. Fan, Y. Zhu, J. Shen, X. Jiang, P. Zhao, S. Luan, and C. Li, “Electric papers of graphene-coated Co₃O₄ fibers for high-performance lithium-ion batteries,” ACS Appl. Mater. Interfaces 5(3), 997–1002 (2013).
[Crossref] [PubMed]

ACS Nano (4)

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref] [PubMed]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (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]

Adv. Mater. (1)

I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, and M. F. Craciun, “Novel highly conductive and transparent graphene-based conductors,” Adv. Mater. 24(21), 2844–2849 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
[Crossref]

Europhysics Lett. (2)

A. Savin and Y. S. Kivshar, “Surface solitons at the edges of graphene nanoribbons,” Europhysics Lett. 89(4), 46001 (2010).
[Crossref]

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

Laser Photonics Rev. (2)

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, 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]

Nano Lett. (4)

R. Wang, Y. Hao, Z. Wang, H. Gong, and J. T. L. Thong, “Large-diameter graphene nanotubes synthesized using Ni nanowire templates,” Nano Lett. 10(12), 4844–4850 (2010).
[Crossref] [PubMed]

P. Avouris, “Graphene: electronic and photonic properties and devices,” Nano Lett. 10(11), 4285–4294 (2010).
[Crossref] [PubMed]

H. Wang, Y. Yang, Y. Liang, J. T. Robinson, Y. Li, A. Jackson, Y. Cui, and H. Dai, “Graphene-wrapped sulfur particles as a rechargeable lithium-sulfur battery cathode material with high capacity and cycling stability,” Nano Lett. 11(7), 2644–2647 (2011).
[Crossref] [PubMed]

F. H. 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]

Nanotechnology (1)

J. Gosciniak and D. T. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24(18), 185202 (2013).
[Crossref] [PubMed]

Nat. Photonics (2)

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

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Nature (1)

K. Kim, J.-Y. Choi, T. Kim, S.-H. Cho, and H.-J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature 479(7373), 338–344 (2011).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. B (6)

A. Y. Nikitin, F. Guinea, F. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85(8), 081405 (2012).
[Crossref]

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

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

Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. M. Peres, and M. I. Vasilevskiy, “Discrete solitons in graphene metamaterials,” Phys. Rev. B 91(4), 045424 (2015).
[Crossref]

D. A. Smirnova, R. E. Noskov, L. A. Smirnov, and Y. S. Kivshar, “Dissipative plasmon solitons in graphene nanodisk arrays,” Phys. Rev. B 91(7), 075409 (2015).
[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]

Phys. Rev. Lett. (5)

P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, “How perfect can graphene be?” Phys. Rev. Lett. 103(13), 136403 (2009).
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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(19), 196405 (2008).
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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]

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. 104(10), 106802 (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]

Phys. Rev. X (1)

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(2), 021014 (2013).
[Crossref]

Science (1)

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[Crossref] [PubMed]

Solid State Commun. (1)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).

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Figures (3)

Fig. 1
Fig. 1

(a) Schematics of an array of dielectric nanowires coated by graphene monolayer. (b) Influence of the wavelength and Fermi energy on the coupling coefficient for neighboring nanowires. (c), (d) Diffraction relation of the fundamental transmission band for (c) E F =0.5eV , (d) E F =1.1eV , with λ 0 =10 μm . In (b)-(d), a=100 nm , s=4a .

Fig. 2
Fig. 2

Profiles of the normalized tangential electric field Ez and light intensity |E|2 for 1D graphene-based solitons at different Fermi energy and beam intensity: (a) E F =0.5eV , I max =3.6× 10 14 V 2 /m 2 ; (b) E F =1.1eV , I max =7.2× 10 15 V 2 /m 2 . (c) The normalized soliton width, [inset of (c)] soliton mode index and (d) soliton power vs. beam intensity. In (c) and (d), the blue, red and green curve represents E F =0.5eV , E F =0.8eV and E F =1.1eV , respectively. (e) The soliton propagation length, [inset of (e)] soliton mode index and (f) normalized soliton width vs. Fermi energy, for I max =7.2× 10 15 V 2 /m 2 . The inset of (f) shows the |E|2 soliton profiles at various EF. In (a)-(f), a=100 nm , s=4a , λ 0 =10 μm .

Fig. 3
Fig. 3

(a) Diffraction relation of the fundamental transmission band for the 2D array, with λ 0 =10 μm . (b) The normalized soliton width vs. beam intensity, for the fundamental and quadrupole modes. (c)-(f) Ez and |E|2 profiles of the (c), (d) fundamental mode, and (e), (f) quadrupole mode, for I max =2.2× 10 16 V 2 /m 2 . In (a)-(f), a=100 nm , s=4a , E F =1.1eV , λ 0 =10 μm .

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

σ L (ω)= e 2 E F π 2 i ω+i τ 1 ,
σ NL (ω)=i 3 8 e 2 π 2 ( e V F E F ω ) 2 E F ω ,
W= ( 2 | E | 2 (x x 0 ) 2 dx | E | 2 d x ) 1/2 ,

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