Abstract

In this paper, dual optical bistability in a sandwich structure including graphene monolayers, dielectric layers, and a composite of golden plasmonic nanoparticles layers is proposed and studied. The theoretical and numerical results show that the asymmetric sandwich structure offers suitable dual optical bistability in THz range. Moreover, wide dual optical bistability behavior can be achieved via inserting plasmonic nanoparticle composite as two sided layers in the multilayer graphene structure, in which the said composite counterbalances the reduced graphene nonlinearities at higher frequencies. Therefore, we can have appropriate optical bistability in the favorable THz frequency range. The results provide using multilayer graphene structures for future all-optical bistability-based, low-power, and tunable optical devices in all-optical communication technology.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (4)

J. L. Cheng, J. E. Sipe, S. W. Wu, and Ch. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, J. E. Sipe, S. W. Wu, and C. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

X. Jiang, J. Gao, and X. Sun, “Control of dispersion properties in a nonlinear dielectric-graphene plasmonic waveguide,” Phys. E (Amsterdam, Neth.) 106, 176–179 (2019).
[Crossref]

L. Jiang, J. Tang, J. Xu, Z. Zheng, J. Dong, J. Guo, S. Qian, X. Dai, and Y. Xiang, “Graphene Tamm plasmon-induced low-threshold optical bistability at terahertz frequencies,” Opt. Mater. Express 9(1), 139–150 (2019).
[Crossref]

2018 (11)

L. Guo, Y. He, Y. Chen, and C. Yin, “Controllable transition between optical bistability and multistability in graphene/dielectric/graphene structure,” Eur. Phys. J. B 91(5), 79 (2018).
[Crossref]

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

T. Naseri and M. Balaei, “Enhanced nonlinear optical response of core?shell graphene-wrapped spherical nanoparticles,” J. Opt. Soc. Am. B 35(9), 2278–2285 (2018).
[Crossref]

N. Daneshfar, T. Naseri, and M. Jalilian, “Effect of gain medium and graphene on the resonance energy transfer between two molecules positioned near a plasmonic multilayer nanoparticle,” Phys. Plasmas 25(9), 093301 (2018).
[Crossref]

T. Naseri, N. Daneshfar, M. Moradi-Dangi, and F. Eynipour-Malaee, “Terahertz optical bistability of graphene-coated cylindrical core-shell nanoparticles,” Theor. Appl. Phys. 12(4), 257–263 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

X. Feng, J. Zou, W. Xu, Z. H. Zhu, X. Yuan, J. Zhang, and S. H. Qin, “Coherent perfect absorption and asymmetric interferometric light-light control in graphene with resonant dielectric nanostructures,” Opt. Express 26(22), 29183–29191 (2018).
[Crossref]

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

S. Baher and Z. Lorestaniweiss, “Propagation of surface plasmon polaritons in monolayer graphene surrounded by nonlinear dielectric media,” J. Appl. Phys. 124(7), 073103 (2018).
[Crossref]

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

2017 (5)

S. A. Mikhailov, “Nonperturbative quasiclassical theory of the nonlinear electrodynamic response of graphene,” Phys. Rev. B 95(8), 085432 (2017).
[Crossref]

X. Jiang, J. Bao, B. Zhang, and X. Sun, “Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide,” Nanoscale Res. Lett. 12(1), 395 (2017).
[Crossref]

E. Yarmoghaddam and S. Rakheja, “Dispersion characteristics of THz surface plasmons in nonlinear graphene-based parallel-plate waveguide with Kerr-type core dielectric,” J. Appl. Phys. 122(8), 083101 (2017).
[Crossref]

J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

J. Guo, L. Jiang, Y. Jia, X. Dai, Y. Xiang, and D. Fan, “Low threshold optical bistability in one-dimensional gratings based on graphene plasmonics,” Opt. Express 25(6), 5972–5981 (2017).
[Crossref]

2016 (4)

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
[Crossref]

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
[Crossref]

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

2015 (3)

2014 (5)

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

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

J. Zhang, C. Guo, K. Liu, Z. Zhu, W. Ye, X. Yuan, and S. Qin, “Coherent perfect absorption and transparency in a nanostructured graphene film,” Opt. Express 22(10), 12524–12532 (2014).
[Crossref]

2011 (1)

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

2010 (3)

M. J. Allen, V. C. Tung, and R. B. Kaner, “Honeycomb carbon: a review of graphene,” Chem. Rev. 110(1), 132–145 (2010).
[Crossref]

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]

2008 (2)

X. Wang, L. Zhi, and K. Mullen, “Transparent, conductive graphene electrodes for dye-sensitized solar cells,” Nano Lett. 8(1), 323–327 (2008).
[Crossref]

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]

2007 (1)

D. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

1983 (1)

Alexander, R. W.

Allen, M. J.

M. J. Allen, V. C. Tung, and R. B. Kaner, “Honeycomb carbon: a review of graphene,” Chem. Rev. 110(1), 132–145 (2010).
[Crossref]

Andreeva, V.

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

Ang, L. K.

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

Baher, S.

S. Baher and Z. Lorestaniweiss, “Propagation of surface plasmon polaritons in monolayer graphene surrounded by nonlinear dielectric media,” J. Appl. Phys. 124(7), 073103 (2018).
[Crossref]

Balaei, M.

Bao, J.

X. Jiang, J. Bao, B. Zhang, and X. Sun, “Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide,” Nanoscale Res. Lett. 12(1), 395 (2017).
[Crossref]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Bludov, Y. V.

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

Chen, Y.

L. Guo, Y. He, Y. Chen, and C. Yin, “Controllable transition between optical bistability and multistability in graphene/dielectric/graphene structure,” Eur. Phys. J. B 91(5), 79 (2018).
[Crossref]

Cheng, J. L.

J. L. Cheng, J. E. Sipe, S. W. Wu, and Ch. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, J. E. Sipe, S. W. Wu, and C. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

Dai, X.

Daneshfar, N.

N. Daneshfar, T. Naseri, and M. Jalilian, “Effect of gain medium and graphene on the resonance energy transfer between two molecules positioned near a plasmonic multilayer nanoparticle,” Phys. Plasmas 25(9), 093301 (2018).
[Crossref]

T. Naseri, N. Daneshfar, M. Moradi-Dangi, and F. Eynipour-Malaee, “Terahertz optical bistability of graphene-coated cylindrical core-shell nanoparticles,” Theor. Appl. Phys. 12(4), 257–263 (2018).
[Crossref]

Das, S.

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

Dong, J.

Eynipour-Malaee, F.

T. Naseri, N. Daneshfar, M. Moradi-Dangi, and F. Eynipour-Malaee, “Terahertz optical bistability of graphene-coated cylindrical core-shell nanoparticles,” Theor. Appl. Phys. 12(4), 257–263 (2018).
[Crossref]

Fan, D.

Feng, X.

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]

Gao, J.

X. Jiang, J. Gao, and X. Sun, “Control of dispersion properties in a nonlinear dielectric-graphene plasmonic waveguide,” Phys. E (Amsterdam, Neth.) 106, 176–179 (2019).
[Crossref]

Gao, L.

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

Geim, D. K.

D. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

Gibbs, H.

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).

Guo, C.

J. L. Cheng, J. E. Sipe, S. W. Wu, and C. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. Zhang, C. Guo, K. Liu, Z. Zhu, W. Ye, X. Yuan, and S. Qin, “Coherent perfect absorption and transparency in a nanostructured graphene film,” Opt. Express 22(10), 12524–12532 (2014).
[Crossref]

Guo, Ch.

J. L. Cheng, J. E. Sipe, S. W. Wu, and Ch. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

Guo, J.

L. Jiang, J. Tang, J. Xu, Z. Zheng, J. Dong, J. Guo, S. Qian, X. Dai, and Y. Xiang, “Graphene Tamm plasmon-induced low-threshold optical bistability at terahertz frequencies,” Opt. Mater. Express 9(1), 139–150 (2019).
[Crossref]

J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

J. Guo, L. Jiang, Y. Jia, X. Dai, Y. Xiang, and D. Fan, “Low threshold optical bistability in one-dimensional gratings based on graphene plasmonics,” Opt. Express 25(6), 5972–5981 (2017).
[Crossref]

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
[Crossref]

L. Jiang, J. Guo, L. Wu, X. Dai, and Y. Xiang, “Manipulating the optical bistability at terahertz frequency in the Fabry-Perot cavity with graphene,” Opt. Express 23(24), 31181–31191 (2015).
[Crossref]

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014).
[Crossref]

Guo, L.

L. Guo, Y. He, Y. Chen, and C. Yin, “Controllable transition between optical bistability and multistability in graphene/dielectric/graphene structure,” Eur. Phys. J. B 91(5), 79 (2018).
[Crossref]

Guo, S.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[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]

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]

He, D.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

He, Y.

L. Guo, Y. He, Y. Chen, and C. Yin, “Controllable transition between optical bistability and multistability in graphene/dielectric/graphene structure,” Eur. Phys. J. B 91(5), 79 (2018).
[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]

Huang, W.

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

Huang, Y.

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

Jalilian, M.

N. Daneshfar, T. Naseri, and M. Jalilian, “Effect of gain medium and graphene on the resonance energy transfer between two molecules positioned near a plasmonic multilayer nanoparticle,” Phys. Plasmas 25(9), 093301 (2018).
[Crossref]

Jauho, A. P.

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

Jia, Y.

Jiang, L.

Jiang, X.

X. Jiang, J. Gao, and X. Sun, “Control of dispersion properties in a nonlinear dielectric-graphene plasmonic waveguide,” Phys. E (Amsterdam, Neth.) 106, 176–179 (2019).
[Crossref]

X. Jiang, J. Bao, B. Zhang, and X. Sun, “Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide,” Nanoscale Res. Lett. 12(1), 395 (2017).
[Crossref]

Jiang, Y.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Joung, D.

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

Kaner, R. B.

M. J. Allen, V. C. Tung, and R. B. Kaner, “Honeycomb carbon: a review of graphene,” Chem. Rev. 110(1), 132–145 (2010).
[Crossref]

Khondaker, S. I.

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

Kyoseva, E.

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

Li, C.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Liang, S.-J.

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

Liu, K.

Liu, L.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Liu, Z.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Long, L. L.

Lorestaniweiss, Z.

S. Baher and Z. Lorestaniweiss, “Propagation of surface plasmon polaritons in monolayer graphene surrounded by nonlinear dielectric media,” J. Appl. Phys. 124(7), 073103 (2018).
[Crossref]

Luskin, M.

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

Margetis, D.

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

Markel, V. A.

Mikhailov, S. A.

S. A. Mikhailov, “Nonperturbative quasiclassical theory of the nonlinear electrodynamic response of graphene,” Phys. Rev. B 95(8), 085432 (2017).
[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]

Miroshnichenko, A. E.

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

Y. Huang, A. E. Miroshnichenko, and L. Gao, “Low-threshold optical bistability of graphene-wrapped dielectric composite,” Sci. Rep. 6(1), 23354 (2016).
[Crossref]

Moger, J.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent Nonlinear Optical Response of Graphene,” Phys. Rev. Lett. 105(9), 097401 (2010).
[Crossref]

Moradi-Dangi, M.

T. Naseri, N. Daneshfar, M. Moradi-Dangi, and F. Eynipour-Malaee, “Terahertz optical bistability of graphene-coated cylindrical core-shell nanoparticles,” Theor. Appl. Phys. 12(4), 257–263 (2018).
[Crossref]

Mullen, K.

X. Wang, L. Zhi, and K. Mullen, “Transparent, conductive graphene electrodes for dye-sensitized solar cells,” Nano Lett. 8(1), 323–327 (2008).
[Crossref]

Naseri, T.

T. Naseri, N. Daneshfar, M. Moradi-Dangi, and F. Eynipour-Malaee, “Terahertz optical bistability of graphene-coated cylindrical core-shell nanoparticles,” Theor. Appl. Phys. 12(4), 257–263 (2018).
[Crossref]

N. Daneshfar, T. Naseri, and M. Jalilian, “Effect of gain medium and graphene on the resonance energy transfer between two molecules positioned near a plasmonic multilayer nanoparticle,” Phys. Plasmas 25(9), 093301 (2018).
[Crossref]

T. Naseri and M. Balaei, “Enhanced nonlinear optical response of core?shell graphene-wrapped spherical nanoparticles,” J. Opt. Soc. Am. B 35(9), 2278–2285 (2018).
[Crossref]

Novoselov, K. S.

D. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

Ordal, M. A.

Peng, H.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Peres, N. M. R.

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

Qian, S.

Qin, S.

Qin, S. H.

Rakheja, S.

E. Yarmoghaddam and S. Rakheja, “Dispersion characteristics of THz surface plasmons in nonlinear graphene-based parallel-plate waveguide with Kerr-type core dielectric,” J. Appl. Phys. 122(8), 083101 (2017).
[Crossref]

Ruan, B.

J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

Santos, J. E.

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

Seal, S.

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

Singh, V.

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

Sipe, J. E.

J. L. Cheng, J. E. Sipe, S. W. Wu, and Ch. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, J. E. Sipe, S. W. Wu, and C. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

Sun, X.

X. Jiang, J. Gao, and X. Sun, “Control of dispersion properties in a nonlinear dielectric-graphene plasmonic waveguide,” Phys. E (Amsterdam, Neth.) 106, 176–179 (2019).
[Crossref]

X. Jiang, J. Bao, B. Zhang, and X. Sun, “Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide,” Nanoscale Res. Lett. 12(1), 395 (2017).
[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]

Tang, D.

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014).
[Crossref]

Tang, J.

Tung, V. C.

M. J. Allen, V. C. Tung, and R. B. Kaner, “Honeycomb carbon: a review of graphene,” Chem. Rev. 110(1), 132–145 (2010).
[Crossref]

Vasilevskiy, M. I.

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

Vermeulen, N.

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

Wang, H.

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
[Crossref]

Wang, J.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Wang, X.

X. Wang, L. Zhi, and K. Mullen, “Transparent, conductive graphene electrodes for dye-sensitized solar cells,” Nano Lett. 8(1), 323–327 (2008).
[Crossref]

Ward, C. A.

Wen, S.

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
[Crossref]

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014).
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Wolf, E. L.

E. L. Wolf, Applications of graphene: an overview (Springer Science, Business Media, 2014).

Wu, J.

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
[Crossref]

Wu, L.

Wu, S. W.

J. L. Cheng, J. E. Sipe, S. W. Wu, and Ch. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, J. E. Sipe, S. W. Wu, and C. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

Xiang, Y.

L. Jiang, J. Tang, J. Xu, Z. Zheng, J. Dong, J. Guo, S. Qian, X. Dai, and Y. Xiang, “Graphene Tamm plasmon-induced low-threshold optical bistability at terahertz frequencies,” Opt. Mater. Express 9(1), 139–150 (2019).
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J. Guo, L. Jiang, Y. Jia, X. Dai, Y. Xiang, and D. Fan, “Low threshold optical bistability in one-dimensional gratings based on graphene plasmonics,” Opt. Express 25(6), 5972–5981 (2017).
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J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
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L. Jiang, J. Guo, L. Wu, X. Dai, and Y. Xiang, “Manipulating the optical bistability at terahertz frequency in the Fabry-Perot cavity with graphene,” Opt. Express 23(24), 31181–31191 (2015).
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X. Dai, L. Jiang, and Y. Xiang, “Tunable optical bistability of dielectric/nonlinear graphene/dielectric heterostructures,” Opt. Express 23(5), 6497–6508 (2015).
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X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5(1), 12271 (2015).
[Crossref]

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014).
[Crossref]

Xu, J.

L. Jiang, J. Tang, J. Xu, Z. Zheng, J. Dong, J. Guo, S. Qian, X. Dai, and Y. Xiang, “Graphene Tamm plasmon-induced low-threshold optical bistability at terahertz frequencies,” Opt. Mater. Express 9(1), 139–150 (2019).
[Crossref]

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Xu, W.

Yarmoghaddam, E.

E. Yarmoghaddam and S. Rakheja, “Dispersion characteristics of THz surface plasmons in nonlinear graphene-based parallel-plate waveguide with Kerr-type core dielectric,” J. Appl. Phys. 122(8), 083101 (2017).
[Crossref]

Ye, W.

Yin, C.

L. Guo, Y. He, Y. Chen, and C. Yin, “Controllable transition between optical bistability and multistability in graphene/dielectric/graphene structure,” Eur. Phys. J. B 91(5), 79 (2018).
[Crossref]

Ying, X.

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

Yuan, X.

Zhai, L.

V. Singh, D. Joung, L. Zhai, S. Das, S. I. Khondaker, and S. Seal, “Graphene based materials: past, present and future,” Prog. Mater. Sci. 56(8), 1178–1271 (2011).
[Crossref]

Zhang, B.

X. Jiang, J. Bao, B. Zhang, and X. Sun, “Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide,” Nanoscale Res. Lett. 12(1), 395 (2017).
[Crossref]

Zhang, H.

J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

Zhang, J.

Zheng, Z.

Zhi, L.

X. Wang, L. Zhi, and K. Mullen, “Transparent, conductive graphene electrodes for dye-sensitized solar cells,” Nano Lett. 8(1), 323–327 (2008).
[Crossref]

Zhu, J.

J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

Zhu, Z.

Zhu, Z. H.

Zou, J.

AIP Adv. (1)

J. Wang, X. Ying, D. He, C. Li, S. Guo, H. Peng, L. Liu, Y. Jiang, J. Xu, and Z. Liu, “Enhanced absorption of graphene with variable bandwidth in quarter-wavelength cavities,” AIP Adv. 8(12), 125301 (2018).
[Crossref]

APL Photonics (2)

J. L. Cheng, J. E. Sipe, S. W. Wu, and Ch. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

J. L. Cheng, J. E. Sipe, S. W. Wu, and C. Guo, “Intraband divergences in third order optical response of 2D systems,” APL Photonics 4(3), 034201 (2019).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014).
[Crossref]

Carbon (2)

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang, “Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration,” Carbon 127, 187–192 (2018).
[Crossref]

Chem. Rev. (1)

M. J. Allen, V. C. Tung, and R. B. Kaner, “Honeycomb carbon: a review of graphene,” Chem. Rev. 110(1), 132–145 (2010).
[Crossref]

Eur. Phys. J. B (1)

L. Guo, Y. He, Y. Chen, and C. Yin, “Controllable transition between optical bistability and multistability in graphene/dielectric/graphene structure,” Eur. Phys. J. B 91(5), 79 (2018).
[Crossref]

J. Appl. Phys. (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]

S. Baher and Z. Lorestaniweiss, “Propagation of surface plasmon polaritons in monolayer graphene surrounded by nonlinear dielectric media,” J. Appl. Phys. 124(7), 073103 (2018).
[Crossref]

E. Yarmoghaddam and S. Rakheja, “Dispersion characteristics of THz surface plasmons in nonlinear graphene-based parallel-plate waveguide with Kerr-type core dielectric,” J. Appl. Phys. 122(8), 083101 (2017).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys. (2)

J. Guo, B. Ruan, J. Zhu, X. Dai, Y. Xiang, and H. Zhang, “Low-threshold optical bistability in a metasurface with graphene,” J. Phys. D: Appl. Phys. 50(43), 434003 (2017).
[Crossref]

H. Wang, J. Wu, J. Guo, L. Jiang, Y. Xiang, and S. Wen, “Low-threshold optical bistability with multilayer graphene-covering Otto configuration,” J. Phys. D: Appl. Phys. 49(25), 255306 (2016).
[Crossref]

Nano Lett. (1)

X. Wang, L. Zhi, and K. Mullen, “Transparent, conductive graphene electrodes for dye-sensitized solar cells,” Nano Lett. 8(1), 323–327 (2008).
[Crossref]

Nanoscale Res. Lett. (1)

X. Jiang, J. Bao, B. Zhang, and X. Sun, “Dual nonlinearity Controlling of Mode and Dispersion Properties in Graphene-Dielectric Plasmonic Waveguide,” Nanoscale Res. Lett. 12(1), 395 (2017).
[Crossref]

Nat. Mater. (1)

D. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

Nat. Photonics (1)

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

New J. Phys. (2)

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16(5), 053014 (2014).
[Crossref]

Opt. Express (5)

Opt. Mater. Express (1)

Phys. E (Amsterdam, Neth.) (1)

X. Jiang, J. Gao, and X. Sun, “Control of dispersion properties in a nonlinear dielectric-graphene plasmonic waveguide,” Phys. E (Amsterdam, Neth.) 106, 176–179 (2019).
[Crossref]

Phys. Plasmas (1)

N. Daneshfar, T. Naseri, and M. Jalilian, “Effect of gain medium and graphene on the resonance energy transfer between two molecules positioned near a plasmonic multilayer nanoparticle,” Phys. Plasmas 25(9), 093301 (2018).
[Crossref]

Phys. Rev. B (4)

V. Andreeva, M. Luskin, and D. Margetis, “Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials,” Phys. Rev. B 98(19), 195407 (2018).
[Crossref]

S. A. Mikhailov, “Nonperturbative quasiclassical theory of the nonlinear electrodynamic response of graphene,” Phys. Rev. B 95(8), 085432 (2017).
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Figures (6)

Fig. 1.
Fig. 1. Dependence of bistability of the local shell electric field intensity $|E_{s}|$ versus the external incident field $|E_{0}|$ on graphene. Parameters are $\textit {b}=15nm$, $\eta = 0.7$, $E_{F} = 0.4 ev$, $\epsilon _{s}= 5.59$, $\epsilon _{h}=2.25$, $\tau = 0.1ps$, and $\lambda = 22\mu m$.
Fig. 2.
Fig. 2. (a) The output field versus the incident field $E_{0}$, (b) Transmission versus the incident field $E_{0}$. Parameters are $E_{F} = 0.6 eV$, $\epsilon _{1}=\epsilon _{2}= 2.55$, $\epsilon _{h}= 2.55$, $d_{1}=400 nm$, $d_{2} = 2200nm$ , $f = 0.01$ and $\lambda = 390\mu m$.
Fig. 3.
Fig. 3. (a) The output field versus the incident field $E_{0}$, (b) Transmission versus the incident field $E_{0}$. Parameters are $E_{F} = 0.6 eV$, $\epsilon _{1}=\epsilon _{2}= 2.55$, $\epsilon _{h}= 2.55$, $f = 0.01$ and $\lambda = 390\mu m$, and $d_{1}=d_{2} =400 nm$, or $d_{1}=400 nm$ and $d_{2} = 2200nm$.
Fig. 4.
Fig. 4. The output field versus the incident field $E_{0}$. Parameters are $E_{F} = 0.6 eV$, $\epsilon _{1}=\epsilon _{2}= 2.55$, $\epsilon _{h}= 2.55$, $d_{1}=400 nm$, $d_{2} = 2200nm$ , $f = 0.01$ and $\lambda = 390\mu m$.
Fig. 5.
Fig. 5. The output field versus the incident field $E_{0}$. Parameters are $E_{F} = 0.6 eV$, $\epsilon _{1}=\epsilon _{2}= 2.55$, $\epsilon _{h}= 2.55$, $d_{1}=400 nm$, $d_{2} = 2200nm$, and $f = 0.01$.
Fig. 6.
Fig. 6. The output field versus the incident field $E_{0}$. Parameters are $\epsilon _{1}=\epsilon _{2}= 2.55$, $\epsilon _{h}= 2.55$, $d_{1}=400 nm$, $d_{2} = 2200nm$, $f = 0.01$, and $\lambda = 390\mu m$.

Equations (13)

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σ i = σ 0 i + σ 3 i | E | g i 2 ; i = 1 , 2 , 3
σ 0 i = σ i n t r a + σ i n t e r , σ i n t r a = i e 2 k B T π 2 ( ω + i / τ ) [ E F k B T + 2 L n ( e E F k B T + 1 ] , σ i n t e r = i e 2 4 π L n [ 2 E F ( ω + i / τ ) 2 E F + ( ω + i / τ ) ] ,
σ 3 = i 9 8 e 2 π 2 ( e v F ) 2 E F ω 3 ,
ϵ 3 = ϵ 4 = ϵ h [ ( ϵ c + 2 ϵ h ) + 2 f ( ϵ c ϵ h ) ] ( ϵ c + 2 ϵ h ) f ( ϵ c ϵ h )
E x ( z ) = { E 3 ( z , x ) = E 0 e i k 3 z ( z + d 1 ) e i k x x + r e i k 3 z ( z + d 1 ) e i k x x ; z < d 1 E 1 ( z , x ) = K 1 e i k 1 z z e i k x x + G 1 e i k 1 z z e i k x ; d 1 < z < 0 E 2 ( z , x ) = K 2 e i k 2 z ( z d 2 ) e i k x x + G 2 e i k 2 z ( z d 2 ) e i k x x ; 0 < z < d 2 E 4 ( z , x ) = t e i k 4 z ( z d 2 ) e i k x x ; z > d 2
E 0 = A t + B t 3 + C t 5 + D t 7 + H t 9 + F t 11
A = i = 1 7 A i A 1 = 1 2 ( a 1 + b 1 ) ( a 2 + b 2 ) + k 2 2 k 1 ( a 1 b 1 ) ( a 2 b 2 ) σ 01 A 2 = k 2 2 cos ( k 2 d 2 ) ( a 1 b 1 ) 2 k 1 ϵ 0 ϵ 2 ω σ 02 i k 2 sin ( k 2 d 2 ) ( a 1 + b 1 ) 2 ϵ 0 ϵ 2 ω σ 02 A 3 = k 3 cos ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 ϵ 3 ω σ 03 + i k 2 k 3 sin ( k 1 d 1 ) ( b 1 a 2 ) 2 k 1 ϵ 0 ϵ 3 ω σ 03 A 4 = k 1 k 2 sin ( k 2 d 2 ) ( b 1 a 1 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ω 2 σ 01 σ 02 A 5 = i k 2 k 3 cos ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 2 ϵ 3 ω 2 σ 02 σ 03 i k 2 2 k 3 cos ( k 2 d 2 ) sin ( k 1 d 1 ) 2 k 1 ϵ 0 2 ϵ 2 ϵ 3 ω 2 σ 02 σ 03 A 6 = i k 1 k 3 sin ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 2 ϵ 1 ϵ 3 ω 2 σ 01 σ 03 A 7 = k 1 k 2 k 3 sin ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ϵ 3 ω 3 σ 01 σ 02 σ 03
B = i = 1 7 B i B 1 = k 1 ( a 1 b 1 ) ( a 2 + b 2 ) 2 ω ϵ 0 ϵ 1 σ 31 | η | 2 B 2 = k 2 2 cos ( k 2 d 2 ) ( a 1 b 1 ) 2 k 1 ϵ 0 ϵ 2 ω σ 32 i k 2 sin ( k 2 d 2 ) ( a 1 + b 1 ) 2 ϵ 0 ϵ 2 ω σ 32 B 3 = [ k 3 cos ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 ϵ 3 ω + i k 2 k 3 sin ( k 1 d 1 ) ( b 1 a 2 ) 2 k 1 ϵ 0 ϵ 3 ω ] σ 33 | γ | 2 B 4 = k 1 k 2 sin ( k 2 d 2 ) ( b 1 a 1 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ω 2 ( σ 01 σ 32 + σ 31 σ 02 | η | 2 ) B 5 = [ i k 2 k 3 cos ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 2 ϵ 3 ω 2 i k 2 2 k 3 cos ( k 2 d 2 ) sin ( k 1 d 1 ) 2 k 1 ϵ 0 2 ϵ 2 ϵ 3 ω 2 ] × ( σ 02 σ 33 | γ | 2 + σ 32 σ 03 ) ( σ 01 σ 33 | γ | 2 + σ 31 σ 03 | η | 2 ) B 7 = k 1 k 2 k 3 sin ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ϵ 3 ω 3 ( σ 01 σ 02 σ 33 | γ | 2 + σ 03 ( σ 01 σ 32 + σ 02 σ 31 | η | 2 ) )
C = i = 1 5 C i C 1 = k 3 cos ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 ϵ 3 ω + i k 2 k 3 sin ( k 1 d 1 ) ( b 1 a 2 ) 2 k 1 ϵ 0 ϵ 3 ω 2 σ 33 ( γ ξ ) C 2 = k 1 k 2 sin ( k 2 d 2 ) ( b 1 a 1 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ω 2 σ 31 σ 32 | η | 2 C 3 = [ i k 2 k 3 cos ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 2 ϵ 3 ω 2 i k 2 2 k 3 cos ( k 2 d 2 ) sin ( k 1 d 1 ) 2 k 1 ϵ 0 2 ϵ 2 ϵ 3 ω 2 ] × ( 2 σ 02 σ 33 ( γ ξ + σ 32 σ 33 | γ | 2 ) C 4 = i k 1 k 3 sin ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 2 ϵ 1 ϵ 3 ω 2 ( 2 σ 01 σ 33 ( γ ξ + σ 33 | η | 2 | γ | 2 ) ) C 5 = k 1 k 2 k 3 sin ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ϵ 3 ω 3 [ σ 01 σ 32 + σ 02 σ 31 | η | 2 σ 33 | γ | 2 + σ 31 σ 32 σ 03 | η | 2 + 2 σ 02 σ 33 ( γ ξ ) ]
D = i = 1 4 D i D 1 = [ k 3 cos ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 ϵ 3 ω + i k 2 k 3 sin ( k 1 d 1 ) ( b 1 a 2 ) 2 k 1 ϵ 0 ϵ 3 ω ] σ 33 | ξ | 2 D 2 = [ i k 2 k 3 cos ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 2 ϵ 3 ω 2 i k 2 2 k 3 cos ( k 2 d 2 ) sin ( k 1 d 1 ) 2 k 1 ϵ 0 2 ϵ 2 ϵ 3 ω 2 ] × ( σ 02 σ 33 | ξ | 2 + 2 σ 32 σ 33 ( γ ξ ) ) D 3 = i k 1 k 3 sin ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 2 ϵ 1 ϵ 3 ω 2 × ( σ 01 σ 33 | ξ | 2 + 2 σ 31 σ 33 ( γ ξ ) | η | 2 ) D 4 = k 1 k 2 k 3 sin ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ϵ 3 ω 3 × ( σ 01 σ 02 σ 33 | ξ | 2 + ( σ 01 σ 32 + σ 02 σ 31 | η | 2 ) × σ 33 2 ( γ ξ ) + σ 33 σ 32 | γ | 2 | η | 2 )
H = i = 1 3 H i H 1 = [ i k 2 k 3 cos ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 2 ϵ 3 ω 2 i k 2 2 k 3 cos ( k 2 d 2 ) sin ( k 1 d 1 ) 2 k 1 ϵ 0 2 ϵ 2 ϵ 3 ω 2 ] × σ 32 σ 33 | ξ | 2 H 2 = i k 1 k 3 sin ( k 1 d 1 ) ( a 2 + b 2 ) 2 ϵ 0 2 ϵ 1 ϵ 3 ω 2 σ 31 σ 33 | η | 2 | ξ | 2 H 3 = k 1 k 2 k 3 sin ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ϵ 3 ω 3 ] × ( σ 01 σ 32 σ 02 σ 31 σ 33 | η | 2 | ξ | 2 + 2 σ 31 σ 32 σ 33 ( γ ξ ) | η | 2 )
F = k 1 k 2 k 3 sin ( k 1 d 1 ) sin ( k 2 d 2 ) 2 ϵ 0 2 ϵ 1 ϵ 2 ϵ 3 ω 3 σ 31 σ 32 σ 33 | η | 2 | ξ | 2
a 1 = 1 2 ( 1 + k 1 k 3 ) e i k 1 d 1 b 1 = 1 2 ( 1 k 1 k 3 ) e i k 1 d 1 a 2 = 1 2 ( 1 + k 4 k 2 ) e i k 2 d 2 b 2 = 1 2 ( 1 k 4 k 2 ) e i k 2 d 2 η = a 2 + b 2 i k 2 s i n ( k 2 d 2 ) σ 02 ϵ 0 ϵ 2 ω γ = ( a 2 + b 2 ) c o s ( k 1 d 1 ) i k 2 σ 02 c o s ( k 1 d 1 ) s i n ( k 2 d 2 ) ϵ 0 ϵ 2 ω + i k 2 s i n ( k 1 d 1 ) ( b 2 a 2 ) k 1 k 2 k 1 σ 01 σ 02 s i n ( k 1 d 1 ) s i n ( k 2 d 2 ) ϵ 1 ϵ 2 ϵ 0 2 ω 2 i k 1 s i n ( k 1 d 1 ) ( a 2 + b 2 ) σ 01 ϵ 0 ϵ 1 ω i k 2 2 s i n ( k 1 d 1 ) c o s ( k 2 d 2 ) σ 02 k 1 ϵ 0 ϵ 2 ω ξ = i c o s ( k 1 d 1 ) s i n ( k 2 d 2 ) σ 32 ϵ 0 ϵ 2 ω ( σ 01 σ 32 + σ 02 σ 31 | η | 2 ) k 1 k 2 s i n ( k 1 d 1 ) s i n ( k 2 d 2 ) ϵ 0 2 ϵ 1 ϵ 2 ω 2 i k 1 s i n ( k 1 d 1 ) σ 31 | η | 2 ( a 2 + b 2 ) ϵ 0 ϵ 1 ω i k 2 2 σ 32 s i n ( k 1 d 1 ) s i n ( k 2 d 2 ) k 1 ϵ 0 ϵ 2 ω