Abstract

A graphene plasmonic structure consists of three graphene layers mingled with a silicon–air grating is proposed. We theoretically predict and numerically simulate the plasmon-induced transparency effect in this system at terahertz wavelengths, and a dual plasmon-induced transparency peaks can be successfully tuned by virtually shifting the desired Fermi energy on graphene layers. We investigate the surface plasmon dispersion relation by means of analytic calculations, and we can achieve the numerical solution of propagation constant got by the dispersion relation. A suitable theoretical model is established to study spectral features in the plasmonic graphene system, and the theoretical results agree well with the simulations. The proposed model and findings may provide guidance for fundamental research of highly tunable optoelectronic devices.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [PubMed]

2017 (5)

P. Zhang, N.-H. Shen, T. Koschny, and C. M. Soukoulis, “Surface-Plasmon-Mediated Gradient Force Enhancement and Mechanical State Transitions of Graphene Sheets,” ACS Photonics 4(1), 181–187 (2017).
[Crossref]

A. Y. Zhu, A. I. Kuznetsov, B. Luk’yanchuk, N. Engheta, and P. Genevet, “Traditional and emerging materials for optical metasurfaces,” Nanophotonics 6(2), 452–471 (2017).
[Crossref]

P. Qiu, W. Qiu, Z. Lin, H. Chen, J. Ren, J. X. Wang, Q. Kan, and J. Q. Pan, “Dynamically Tunable Plasmon-Induced Transparency in On-chip Graphene-Based Asymmetrical Nanocavity-Coupled Waveguide System,” Nanoscale Res. Lett. 12(1), 374 (2017).
[Crossref] [PubMed]

C. Sun, Z. Dong, J. Si, and X. Deng, “Independently tunable dual-band plasmonically induced transparency based on hybrid metal-graphene metamaterials at mid-infrared frequencies,” Opt. Express 25(2), 1242–1250 (2017).
[Crossref] [PubMed]

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref] [PubMed]

2016 (11)

H. Xu, H. Li, B. Li, Z. He, Z. Chen, and M. Zheng, “Influential and theoretical analysis of nano-defect in the stub resonator,” Sci. Rep. 6(1), 30877 (2016).
[Crossref] [PubMed]

S. Balci, O. Balci, N. Kakenov, F. B. Atar, and C. Kocabas, “Dynamic tuning of plasmon resonance in the visible using graphene,” Opt. Lett. 41(6), 1241–1244 (2016).
[Crossref] [PubMed]

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nat. Photonics 10(4), 227–238 (2016).
[Crossref]

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface Plasmon Polariton Graphene Photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

C. Caucheteur, T. Guo, F. Liu, B. O. Guan, and J. Albert, “Ultrasensitive plasmonic sensing in air using optical fibre spectral combs,” Nat. Commun. 7, 13371 (2016).
[Crossref] [PubMed]

S. X. Xia, X. Zhai, L. L. Wang, B. Sun, J. Q. Liu, and S. C. Wen, “Dynamically tunable plasmonically induced transparency in sinusoidally curved and planar graphene layers,” Opt. Express 24(16), 17886–17899 (2016).
[Crossref] [PubMed]

Z. Bai and G. Huang, “Plasmon dromions in a metamaterial via plasmon-induced transparency,” Phys. Rev. A 93(1), 013818 (2016).
[Crossref]

C. Sun, J. Si, Z. Dong, and X. Deng, “Tunable multispectral plasmon induced transparency based on graphene metamaterials,” Opt. Express 24(11), 11466–11474 (2016).
[Crossref] [PubMed]

Z. He, H. Li, B. Li, Z. Chen, H. Xu, and M. Zheng, “Theoretical analysis of ultrahigh figure of merit sensing in plasmonic waveguides with a multimode stub,” Opt. Lett. 41(22), 5206–5209 (2016).
[Crossref] [PubMed]

X. Zhao, L. Zhu, C. Yuan, and J. Yao, “Tunable plasmon-induced transparency in a grating-coupled double-layer graphene hybrid system at far-infrared frequencies,” Opt. Lett. 41(23), 5470–5473 (2016).
[Crossref] [PubMed]

H. Wang, J. Yang, J. Zhang, J. Huang, W. Wu, D. Chen, and G. Xiao, “Tunable band-stop plasmonic waveguide filter with symmetrical multiple-teeth-shaped structure,” Opt. Lett. 41(6), 1233–1236 (2016).
[Crossref] [PubMed]

2015 (1)

2012 (3)

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B 85(12), 125431 (2012).
[Crossref]

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

2011 (3)

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

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

L. A. Ponomarenko, A. K. Geim, A. A. Zhukov, R. Jalil, S. V. Morozov, K. S. Novoselov, I. V. Grigorieva, E. H. Hill, V. V. Cheianov, V. I. Fal’ko, K. Watanabe, T. Taniguchi, and R. V. Gorbachev, “Tunable metal–insulator transition in double-layer graphene heterostructures,” Nat. Phys. 7(12), 958–961 (2011).
[Crossref]

2009 (1)

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

2008 (1)

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

2007 (1)

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

1997 (1)

S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

1991 (1)

H. A. Haus and W. Huang, “Coupled-Mode Theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

1969 (1)

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
[Crossref]

Albert, J.

C. Caucheteur, T. Guo, F. Liu, B. O. Guan, and J. Albert, “Ultrasensitive plasmonic sensing in air using optical fibre spectral combs,” Nat. Commun. 7, 13371 (2016).
[Crossref] [PubMed]

Altug, H.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref] [PubMed]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Atar, F. B.

Bai, Z.

Z. Bai and G. Huang, “Plasmon dromions in a metamaterial via plasmon-induced transparency,” Phys. Rev. A 93(1), 013818 (2016).
[Crossref]

Balci, O.

Balci, S.

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Basov, D. N.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

Boyd, R. W.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Buljan, H.

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

Castro Neto, A. H.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Caucheteur, C.

C. Caucheteur, T. Guo, F. Liu, B. O. Guan, and J. Albert, “Ultrasensitive plasmonic sensing in air using optical fibre spectral combs,” Nat. Commun. 7, 13371 (2016).
[Crossref] [PubMed]

Cheianov, V. V.

L. A. Ponomarenko, A. K. Geim, A. A. Zhukov, R. Jalil, S. V. Morozov, K. S. Novoselov, I. V. Grigorieva, E. H. Hill, V. V. Cheianov, V. I. Fal’ko, K. Watanabe, T. Taniguchi, and R. V. Gorbachev, “Tunable metal–insulator transition in double-layer graphene heterostructures,” Nat. Phys. 7(12), 958–961 (2011).
[Crossref]

Chen, D.

Chen, H.

P. Qiu, W. Qiu, Z. Lin, H. Chen, J. Ren, J. X. Wang, Q. Kan, and J. Q. Pan, “Dynamically Tunable Plasmon-Induced Transparency in On-chip Graphene-Based Asymmetrical Nanocavity-Coupled Waveguide System,” Nanoscale Res. Lett. 12(1), 374 (2017).
[Crossref] [PubMed]

Chen, Z.

Z. He, H. Li, B. Li, Z. Chen, H. Xu, and M. Zheng, “Theoretical analysis of ultrahigh figure of merit sensing in plasmonic waveguides with a multimode stub,” Opt. Lett. 41(22), 5206–5209 (2016).
[Crossref] [PubMed]

H. Xu, H. Li, B. Li, Z. He, Z. Chen, and M. Zheng, “Influential and theoretical analysis of nano-defect in the stub resonator,” Sci. Rep. 6(1), 30877 (2016).
[Crossref] [PubMed]

Chu, H. S.

C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B 85(12), 125431 (2012).
[Crossref]

Dastmalchi, P.

Deng, X.

Dominguez, G.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Dong, Z.

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Echtermeyer, T. J.

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface Plasmon Polariton Graphene Photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Economou, E. N.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

Eiden, A.

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface Plasmon Polariton Graphene Photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Engheta, N.

A. Y. Zhu, A. I. Kuznetsov, B. Luk’yanchuk, N. Engheta, and P. Genevet, “Traditional and emerging materials for optical metasurfaces,” Nanophotonics 6(2), 452–471 (2017).
[Crossref]

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

Fal’ko, V. I.

L. A. Ponomarenko, A. K. Geim, A. A. Zhukov, R. Jalil, S. V. Morozov, K. S. Novoselov, I. V. Grigorieva, E. H. Hill, V. V. Cheianov, V. I. Fal’ko, K. Watanabe, T. Taniguchi, and R. V. Gorbachev, “Tunable metal–insulator transition in double-layer graphene heterostructures,” Nat. Phys. 7(12), 958–961 (2011).
[Crossref]

Fei, Z.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Ferrari, A. C.

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface Plasmon Polariton Graphene Photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Fogler, M. M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Gan, C. H.

C. H. Gan, H. S. Chu, and E. P. Li, “Synthesis of highly confined surface plasmon modes with doped graphene sheets in the midinfrared and terahertz frequencies,” Phys. Rev. B 85(12), 125431 (2012).
[Crossref]

Geim, A. K.

L. A. Ponomarenko, A. K. Geim, A. A. Zhukov, R. Jalil, S. V. Morozov, K. S. Novoselov, I. V. Grigorieva, E. H. Hill, V. V. Cheianov, V. I. Fal’ko, K. Watanabe, T. Taniguchi, and R. V. Gorbachev, “Tunable metal–insulator transition in double-layer graphene heterostructures,” Nat. Phys. 7(12), 958–961 (2011).
[Crossref]

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

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Genevet, P.

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic illustration of the three layers graphene structure. The red dielectric is silicon, the blue is a silica substrate, the yellows are the electrode and the other dielectrics are air. (b) A front view of Fig. 1(a) which is a period surrounded by the black frame (The period L = 400nm, d1 = 250nm, d2 = 250nm, l1 = 250nm, and l2 = 150nm). (c) An equivalent theoretical coupled model for this graphene-based plasmonic resonators.
Fig. 2
Fig. 2 (a) The dispersion relation of this TM SPP surface wave with different Fermi energy. (b) The real parts of effective refractive index of the plasmonic mode at the three layers graphene structure with different Fermi energy. (c) Transmission spectra of the hybrid system with three layers graphene (bule), only top-layer graphene (red), only middle-layer graphene (green), and only bottom-layer graphene (purple) as Fermi energy EF = 1.0eV.
Fig. 3
Fig. 3 The simulated transmittance (blue solid lines) and theoretical fitting (red cycle lines) as EF = 1.05eV, 1.00eV, 0.95eV, 0.90eV in the three layers graphene structure, respectively. For theoretical transmission spectra and structural properties of graphene, the decay rates are γw1 = 0.38 × 1012 rad/s, γw2 = 3.15 × 1012 rad/s, γw3 = 1.52 × 1012 rad/s, γi1 = 2.55 × 1011 rad/s, γi2 = 1.21 × 1011 rad/s, and γi3 = 0.23 × 1011 rad/s, respectively. The coupling coefficients are μ12 = 6.38 × 1011 rad/s, μ21 = 6.38 × 1011 rad/s, μ13 = 2.27 × 1011 rad/s, μ31 = 2.27 × 1011 rad/s, μ23 = 6.38 × 1011 rad/s, and μ32 = 6.38 × 1011 rad/s, respectively.
Fig. 4
Fig. 4 (a) The wavelength values of dip as a function of Fermi energy EF. (b) The wavelength values of peak as a function of Fermi energy EF. (c)-(e) Simulated electric field intensities profile with the graphene Fermi energy EF = 0.9eV at λ = 16369.9nm, 16858.6nm, 17569.4nm, respectively. (f)-(h) Simulated electric field intensities profile with the graphene Fermi energy EF = 0.95eV at λ = 15934.1nm, 16410nm, 17102.7nm, respectively.
Fig. 5
Fig. 5 (a) The Fermi energy EF as a function of the applied bias voltage Vg (b) Evolution of the transmission spectra versus Fermi energy EF and wavelength λ.

Equations (26)

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σ = i e 2 E F π 2 ( ω + i τ 1 )
R e g i o n 1 ( z > d 1 ) : H y 1 = A e i β x e k 1 z
E x 1 = i A k 1 ω ε 0 ε 1 e i β x e k 1 z
E z 1 = A β ω ε 0 ε 1 e i β x e k 1 z
R e g i o n 2 ( 0 < z < d 1 ) : H y 2 = B e i β x e k 2 z + C e i β x e k 2 z
E x 2 = i B k 2 ω ε 0 ε 2 e i β x e k 2 z + i C k 2 ω ε 0 ε 2 e i β x e k 2 z
E z 2 = B β ω ε 0 ε 2 e i β x e k 2 z C β ω ε 0 ε 2 e i β x e k 2 z
H y 3 = D e i β x e k 3 z + E e i β x e k 3 z
E x 3 = i D k 3 ω ε 0 ε 3 e i β x e k 3 z + i E k 3 ω ε 0 ε 3 e i β x e k 3 z
E z 3 = D β ω ε 0 ε 3 e i β x e k 3 z E β ω ε 0 ε 3 e i β x e k 3 z
R e g i o n 4 ( z < d 2 ) : H y 4 = F e i β x e k 4 z
E x 4 = i F k 4 ω ε 0 ε 4 e i β x e k 4 z
E z 4 = F β ω ε 0 ε 4 e i β x e k 4 z
( ε 2 k 2 + ε 4 k 1 + i σ ω ε 0 ) e 2 k 2 d 1 ε 2 k 2 + ε 4 k 1 + i σ ω ε 0 = ( ε 2 k 2 + ε 2 k 2 i σ ω ε 0 ) ( ε 2 k 2 + ε 1 k 1 + i σ ω ε 0 ) + i σ ω ε 0 ( ε 2 k 2 + ε 1 k 1 + i σ ω ε 0 ) e 2 k 2 d 1 i σ ω ε 0 ( ε 2 k 2 + ε 1 k 1 + i σ ω ε 0 ) + ( ε 2 k 2 + ε 2 k 2 + i σ ω ε 0 ) ( ε 2 k 2 + ε 1 k 1 + i σ ω ε 0 ) e 2 k 2 d 1
( γ 1 i μ 12 i μ 13 i μ 21 γ 2 i μ 23 i μ 31 i μ 32 γ 3 ) ( a 1 a 2 a 3 ) = ( 1 τ w 1 0 0 0 1 τ w 2 0 0 0 1 τ w 3 ) ( A 1 + i n + A 1 i n A 2 + i n + A 2 i n A 3 + i n + A 3 i n )
A n + i n = A ( n 1 ) + o u t e i φ n 1 , A ( n 1 ) i n = A n o u t e i φ n 1 ( n = 2 , 3 )
A n + o u t = A n + i n 1 τ w n a n , A n o u t = A n i n 1 τ w n a n ( n = 1 , 2 , 3 )
t = A 3 + o u t A 1 + i n = t 0 b 1 t 1 b 2 t 2 b 3 γ 1
t 0 = e 2 i φ + 1 τ ω 1 γ 1 e 2 i φ
t 1 = 1 τ ω 3 γ 1 ( γ 1 γ 2 - χ 12 χ 21 ) + 1 τ ω 1 e 2 i φ χ 12 ( χ 13 χ 21 + γ 1 χ 23 ) + 1 τ ω 1 e 2 i φ ( γ 1 γ 2 χ 12 χ 21 ) χ 13 + 1 τ ω 2 e i φ γ 1 ( χ 13 χ 21 + γ 1 χ 23 )
t 2 = 1 τ ω 3 γ 1 ( χ 31 γ 2 + χ 21 χ 32 ) + 1 τ ω 1 e 2 i φ χ 12 ( χ χ 31 23 + χ 21 γ 3 ) + 1 τ ω 1 e 2 i φ ( χ 31 γ 2 + χ 21 χ 32 ) χ 13 + 1 τ ω 2 e i φ γ 1 ( χ 31 χ 23 + χ 21 γ 3 )
b 1 = χ 31 e i φ 1 τ ω 2 χ 21 e 2 i φ 1 τ ω 3
b 2 = γ 1 e i φ 1 τ ω 2 + χ 21 1 τ ω 1
b 3 = ( γ 1 γ 2 χ 12 χ 21 ) ( χ 31 χ 23 + χ 21 γ 3 ) ( χ 31 γ 2 + χ 21 χ 32 ) ( χ 13 χ 21 + γ 1 χ 23 )
χ m n = 1 τ w m τ w n e i φ + i μ m n ( m = 1 , 2 , 3 ; n = 1 , 2 , 3 ; m n )
E F = v F π ε 0 ε d V g d s u b e

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