Abstract

Optical bistability of graphene surface plasmon is investigated numerically, using grating coupling method at normal light incidence. The linear surface plasmon resonance is strongly dependent on Femi-level of graphene, hence it can be tuned in a large wavelength range. Due to the field enhancement of graphene surface plasmon resonance and large third-order nonlinear response of graphene, a low-threshold optical hysteresis has been observed. The threshold value with 20MW/cm2 and response time with 1.7ps have been verified. Especially, it is found that this optical bistability phenomenon is angular insensitivity for near 15° incident angle. The threshold of optical bistability can be further lowered to 0.5MW/cm2 by using graphene nanoribbons, and the response time is also shorten to 800fs. We believe that our results will find potential applications in bistable devices and all-optical switching from mid-IR to THz range.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Graphene Tamm plasmon-induced low-threshold optical bistability at terahertz frequencies

Leyong Jiang, Jiao Tang, Jiao Xu, Zhiwei Zheng, Jun Dong, Jun Guo, Shengyou Qian, Xiaoyu Dai, and Yuanjiang Xiang
Opt. Mater. Express 9(1) 139-150 (2019)

Tunable optical bistability of dielectric/nonlinear graphene/dielectric heterostructures

Xiaoyu Dai, Leyong Jiang, and Yuanjiang Xiang
Opt. Express 23(5) 6497-6508 (2015)

Graphene-based absorber exploiting guided mode resonances in one-dimensional gratings

M. Grande, M. A. Vincenti, T. Stomeo, G. V. Bianco, D. de Ceglia, N. Aközbek, V. Petruzzelli, G. Bruno, M. De Vittorio, M. Scalora, and A. D’Orazio
Opt. Express 22(25) 31511-31519 (2014)

References

  • View by:
  • |
  • |
  • |

  1. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  2. D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
    [Crossref] [PubMed]
  3. G. Assanto, Z. Wang, D. J. Hagan, and E. W. VanStryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67(15), 2120–2122 (1995).
    [Crossref]
  4. H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
    [Crossref]
  5. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics 4(8), 535–544 (2010).
    [Crossref]
  6. D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
    [Crossref]
  7. V. G. Taeed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
    [Crossref]
  8. F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
    [Crossref]
  9. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
    [Crossref]
  10. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystalwaveguides,” J. Opt. Soc. Am. B 19(9), 2241 (2002).
    [Crossref]
  11. Q. M. Ngo, S. Kim, S. H. Song, and R. Magnusson, “Optical bistable devices based on guided-mode resonance in slab waveguide gratings,” Opt. Express 17(26), 23459–23467 (2009).
    [Crossref]
  12. P. Vincent, N. Paraire, M. Neviere, A. Koster, and R. Reinisch, “Grating in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2(7), 1106–1116 (1985)
    [Crossref]
  13. G. M. Wysin, H. J. Simon, and R. T. Deck, “Optical bistability with surface plasmons,” Opt. Lett. 6, 30–32 (1981).
    [Crossref] [PubMed]
  14. C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. 33(8), 869–871 (2008).
    [Crossref] [PubMed]
  15. S. Tang, B. Zhu, S. Xiao, J. Shen, and L. Zhou, “Low-threshold optical bistabilities in ultrathin nonlinear metamaterials,” Opt. Lett. 39(11), 3212–3215 (2014).
    [Crossref] [PubMed]
  16. G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
    [Crossref] [PubMed]
  17. F. H. L. Koppens, D. E. Chang, and F. J. Garcia de Abajo, “Graphene plasmonics: a platform for strong light matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
    [Crossref] [PubMed]
  18. Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS nano 6(5), 3677–3694 (2012).
    [Crossref] [PubMed]
  19. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
    [Crossref]
  20. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
    [Crossref] [PubMed]
  21. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
    [Crossref]
  22. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
    [Crossref] [PubMed]
  23. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
    [Crossref]
  24. C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
    [Crossref]
  25. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
    [Crossref]
  26. H. Zhang, S. Virally, Q. Bao, K. P. Loh, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
    [Crossref] [PubMed]
  27. N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
    [Crossref]
  28. 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]
  29. A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
    [Crossref]
  30. X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmon,” Sci. Rep. 5, 12271 (2015).
    [Crossref]
  31. T. Christensen, W. Yan, A. Jauho, M. Wubs, and N. A. Mortensen, “Kerr nonlinearity and plasmonic bistability in graphene nanoribbons,” Phys. Rev. B 92, 121407(R) (2015).
    [Crossref]
  32. T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
    [Crossref]
  33. C. Horvath, D. Bachman, R. Indoe, and V. Van, “Photothermal nonlinearity and optical bistability in a graphene–silicon waveguide resonator,” Opt. Lett. 38(23), 5036–5039 (2013).
    [Crossref] [PubMed]
  34. Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
    [Crossref]
  35. 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]
  36. T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
    [Crossref]
  37. P. Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
    [Crossref] [PubMed]
  38. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
    [Crossref]
  39. C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
    [Crossref] [PubMed]
  40. K. Bolotin, K. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9), 351–355 (2008).
    [Crossref]
  41. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995).
    [Crossref]
  42. M. G. Moharam, D. A. Pommet, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995).
    [Crossref]

2016 (1)

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

2015 (3)

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmon,” Sci. Rep. 5, 12271 (2015).
[Crossref]

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

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

2014 (3)

C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
[Crossref]

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[Crossref]

S. Tang, B. Zhu, S. Xiao, J. Shen, and L. Zhou, “Low-threshold optical bistabilities in ultrathin nonlinear metamaterials,” Opt. Lett. 39(11), 3212–3215 (2014).
[Crossref] [PubMed]

2013 (2)

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
[Crossref]

C. Horvath, D. Bachman, R. Indoe, and V. Van, “Photothermal nonlinearity and optical bistability in a graphene–silicon waveguide resonator,” Opt. Lett. 38(23), 5036–5039 (2013).
[Crossref] [PubMed]

2012 (6)

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]

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

H. Zhang, S. Virally, Q. Bao, K. P. Loh, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (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]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

2011 (4)

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

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

P. Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

2010 (4)

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

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics 4(8), 535–544 (2010).
[Crossref]

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

2009 (2)

Q. M. Ngo, S. Kim, S. H. Song, and R. Magnusson, “Optical bistable devices based on guided-mode resonance in slab waveguide gratings,” Opt. Express 17(26), 23459–23467 (2009).
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

2008 (3)

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. 33(8), 869–871 (2008).
[Crossref] [PubMed]

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

2007 (1)

2006 (1)

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

2003 (1)

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

2002 (2)

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystalwaveguides,” J. Opt. Soc. Am. B 19(9), 2241 (2002).
[Crossref]

2001 (1)

H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
[Crossref]

1995 (3)

1985 (1)

1981 (1)

1976 (1)

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[Crossref]

Akimov, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Alu, A.

P. Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Angelis, F. D.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Assanto, G.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. VanStryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67(15), 2120–2122 (1995).
[Crossref]

Atmatzakis, E.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Bachman, D.

Baker, N. J.

Bao, Q.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

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

H. Zhang, S. Virally, Q. Bao, K. P. Loh, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Bludov, Yu. V.

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[Crossref]

Boden, S. A.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Bolotin, K.

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

Bonaccorso, F.

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

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Chang, D. E.

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

Chang, Y. C.

C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
[Crossref]

Chen, C.

Chen, J.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Chen, P. Y.

P. Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Choi, D. Y.

Christensen, T.

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

Christopoulos, T.

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Dai, X.

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmon,” Sci. Rep. 5, 12271 (2015).
[Crossref]

Dean, C. R.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Deck, R. T.

Deng, Y.

Dijkhuis, J. I.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Eggleton, B. J.

Fabrizio, E. D.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Felber, F. S.

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[Crossref]

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]

Fink, Y.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

Finsterbusch, K.

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics 4(8), 535–544 (2010).
[Crossref]

Fu, L.

Fudenberg, G.

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

Gaeta, A. L.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
[Crossref]

Gao, W.

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]

Garcia de Abajo, F. J.

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

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Geng, B.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Godbout, N.

Golubev, V. G.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Grann, E. B.

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Grivas, N.

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Hagan, D. J.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. VanStryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67(15), 2120–2122 (1995).
[Crossref]

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]

Hanson, G. W.

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

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]

Hone, J.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

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

Horvath, C.

Hu, X. H.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

Ibanescu, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

Indoe, R.

Jauho, A.

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

Jauho, A. P.

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[Crossref]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Jiang, L.

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmon,” Sci. Rep. 5, 12271 (2015).
[Crossref]

Jiang, X.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Jiang, Z.

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

Joannopoulos, J. D.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

Johnson, S. G.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Kerst, R.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Kim, P.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

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

Kim, S.

Kivshar, Y. S.

Klima, M.

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

Kockaert, P.

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics 4(8), 535–544 (2010).
[Crossref]

Koppens, F. H. L.

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

Koster, A.

Kriezis, E. E.

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Kurdyukov, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Lamont, M. R.

Lee, C.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics 4(8), 535–544 (2010).
[Crossref]

Li, S.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Lim, C. H. Y. X.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Lipson, M.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
[Crossref]

Liu, C. H.

C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
[Crossref]

Liu, M.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Liu, X. H.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

Loh, K. P.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

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

H. Zhang, S. Virally, Q. Bao, K. P. Loh, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Lu, Y.

Luo, Z.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Luther-Davies, B.

Madden, S.

Magnusson, R.

Marburger, J. H.

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[Crossref]

Massar, S.

Mazurenko, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Meric, I.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

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]

Min, C.

Ming, H.

Mingaleev, S. F.

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]

Moharam, M. G.

Morandotti, R.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
[Crossref]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Mortensen, N. A.

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

Moss, D. J.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
[Crossref]

V. G. Taeed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[Crossref]

Neviere, M.

Ngo, Q. M.

Nguyen, H. C.

Ni, Z.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Nihei, H.

H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
[Crossref]

Nikolaenko, A. E.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Ning, T.

Norris, T. B.

C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
[Crossref]

Novoselov, K. S.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Okamoto, A.

H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
[Crossref]

Papasimakis, N.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Paraire, N.

Peres, N. M. R.

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Pevtsov, A. B.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Pollard, R.

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

Pommet, D. A.

Qiu, C.

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]

Reinisch, R.

Santos, J. E.

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[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]

Selkin, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

Shen, J.

Shen, Z. X.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Shepard, K. L.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Shu, J.

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]

Sikes, K.

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

Simon, H. J.

Soljacic, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

Song, S. H.

Sorgenfrei, S.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Stormer, H.

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

Sun, Z.

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

Taeed, V. G.

Tang, D. Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Tang, S.

Taniguchi, T.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Tsilipakos, O.

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Van, V.

VanStryland, E. W.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. VanStryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67(15), 2120–2122 (1995).
[Crossref]

Vasilevskiy, M. I.

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[Crossref]

Venkatesan, T.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Vincent, P.

Virally, S.

Wang, B.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Wang, L.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Wang, P.

Wang, Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Wang, Z.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. VanStryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67(15), 2120–2122 (1995).
[Crossref]

Watanabe, K.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Wubs, M.

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

Wurtz, G. A.

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

Wysin, G. M.

Xiang, Y.

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmon,” Sci. Rep. 5, 12271 (2015).
[Crossref]

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Xiao, S.

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]

Xu, Q. H.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Yan, W.

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

Yang, G.

Yin, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Young, A. F.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Zayats, A. V.

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Zhan, T. R.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

Zhang, H.

H. Zhang, S. Virally, Q. Bao, K. P. Loh, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Zhang, K.

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Zhang, X.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Zhao, F. Y.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

Zheludev, N. I.

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

Zhong, Z.

C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
[Crossref]

Zhou, L.

Zhou, Y.

Zhu, B.

Zi, J.

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

ACS nano (2)

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

P. Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Adv. Opt. Mater. (1)

Q. Bao, J. Chen, Y. Xiang, K. Zhang, S. Li, X. Jiang, Q. H. Xu, K. P. Loh, and T. Venkatesan, “Graphene nanobubbles: a new optical nonlinear material,” Adv. Opt. Mater. 3, (6)744–749 (2015).
[Crossref]

Appl. Phys. Lett. (3)

A. E. Nikolaenko, N. Papasimakis, E. Atmatzakis, Z. Luo, Z. X. Shen, F. D. Angelis, S. A. Boden, E. D. Fabrizio, and N. I. Zheludev, “Nonlinear graphene metamaterial,” Appl. Phys. Lett. 100(18), 181109 (2012).
[Crossref]

G. Assanto, Z. Wang, D. J. Hagan, and E. W. VanStryland, “All-optical modulation via nonlinear cascading in type II second-harmonic generation,” Appl. Phys. Lett. 67(15), 2120–2122 (1995).
[Crossref]

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[Crossref]

J. Appl. Phys. (1)

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Nano Lett. (1)

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

Nat. Nanotech. (1)

C. H. Liu, Y. C. Chang, T. B. Norris, and Z. Zhong, “Graphene photodetectors with ultra-broadband and high responsivity at room temperature,” Nat. Nanotech. 9(4), 273–278 (2014).
[Crossref]

Nat. Nanotechnol. (1)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010).
[Crossref] [PubMed]

Nat. Photonics (3)

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

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

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Nature (1)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Nature Photonics (2)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics 4(8), 535–544 (2010).
[Crossref]

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7(8), 597–607 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. B (3)

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86(16), 165416 (2012).
[Crossref]

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

N. M. R. Peres, Yu. V. Bludov, J. E. Santos, A. P. Jauho, and M. I. Vasilevskiy, “Optical bistability of graphene in the terahertz range,” Phys. Rev. B 90(12), 125425 (2014)
[Crossref]

Phys. Rev. E (2)

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinearphotonic crystals,” Phys. Rev. E 66(5), 055601 (2002).
[Crossref]

Phys. Rev. Lett. (3)

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Selkin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).
[Crossref] [PubMed]

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[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]

Proc. SPIE (1)

H. Nihei and A. Okamoto, “Photonic crystal systems for high-speed optical memory device on an atomic scale,” Proc. SPIE 4416, 470–473 (2001).
[Crossref]

Rev. Mod. Phys. (1)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Sci. Rep. (1)

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmon,” Sci. Rep. 5, 12271 (2015).
[Crossref]

Science (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric feld effect in atomically thin carbon flms,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Solid State Commun. (1)

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

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 (a) Schematic of the proposed structure. (b) Distribution of normalized electric filed intensity. (c) Linear transmittance in wavelength spectrum of graphene SPs with different EF, calculated by FEM (solid lines) and RCWA (blue circles).
Fig. 2
Fig. 2 (a) Nonlinear transmittance in wavelength spectrum with different Iin, EF = 0.535eV and τ = 400fs. (b) Dependence of transmittance on Iin for different EF, τ = 400fs. (c) Dependence of transmittance on Iin for different τ, EF = 0.535eV.
Fig. 3
Fig. 3 Temporal response of bistability using (a) EF = 0.535eV, τ = 400fs (b) EF = 0.535eV, τ = 600fs.
Fig. 4
Fig. 4 (a) Angular dependence of linear transmittance with different EF, and (b) Optical bistability with different incident angle.
Fig. 5
Fig. 5 (a) Schematic of the proposed structure. (b) Distribution of normalized electric filed intensity, with log10 scale, in one period. (c) Nonlinear transmittance in wavelength spectrum with different Iin.
Fig. 6
Fig. 6 (a) Optical bistability with different τ. (b) Temporal response of bistability.

Equations (3)

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

σ intra = i e 2 k B T π 2 ( ω + i Γ ) [ E F k B T + 2 ln ( e E F k B T + 1 ) ] ,
σ inter = i e 2 4 π ln [ 2 E F ( ω + i Γ ) 2 E F + ( ω + i Γ ) ] ,
σ 3 = i 9 8 e 4 v f 2 π E F 2 ω 3 ,

Metrics