Abstract

Recently, an approximate boundary condition [Opt. Lett. 38, 3009 (2013)] was proposed for fast analysis of one-dimensional periodic arrays of graphene ribbons by using the Fourier modal method (FMM). Correct factorization rules are applicable to this approximate boundary condition where graphene is modeled as surface conductivity. We extend this approach to obtain the optical properties of two-dimensional periodic arrays of graphene. In this work, optical absorption of graphene squares in a checkerboard pattern and graphene nanodisks in a hexagonal lattice are calculated by the proposed formalism. The achieved results are compared with the conventional FMM, in which graphene is modeled as a finite thickness dielectric layer. We show that for the same truncation order, computation time can be reduced to one-ninth by the proposed formulation in comparison with the conventional FMM. Furthermore, the convergence rate is increased. Therefore, thanks to the improved convergence rate and reduced computational cost for a given truncation order, the computational time is saved more than 100 times for relative error of less than 1%. This is crucially important in analyzing two-dimensional periodic structures of graphene by the FMM.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
    [CrossRef]
  5. C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
    [CrossRef]
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    [CrossRef]
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2013 (2)

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

A. Khavasi, “Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons,” Opt. Lett. 38, 3009–3012 (2013).
[CrossRef]

2012 (5)

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

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

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

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

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef]

2011 (2)

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

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, 630–634 (2011).
[CrossRef]

2010 (1)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

2009 (1)

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

2006 (1)

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

2003 (1)

1997 (1)

1996 (2)

Ajayan, P. M.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

Atwater, H. A.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

Bao, Q.

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

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, 630–634 (2011).
[CrossRef]

Brar, V. W.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

Chang, D. E.

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

Christensen, J.

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

Fang, Z.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

García de Abajo, F. J.

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

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef]

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

Geng, B.

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, 630–634 (2011).
[CrossRef]

Giessen, H.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

Girit, C.

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, 630–634 (2011).
[CrossRef]

Granet, G.

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

Grigorenko, A. N.

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

Halas, N. J.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

Han, D.

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

Hao, Z.

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, 630–634 (2011).
[CrossRef]

Hentschel, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

Horng, J.

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, 630–634 (2011).
[CrossRef]

Ivaska, A.

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

Jang, M. S.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

Ju, L.

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, 630–634 (2011).
[CrossRef]

Khavasi, A.

Koppens, F. H. L.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef]

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

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

Li, L.

Liang, X.

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, 630–634 (2011).
[CrossRef]

Liu, N.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

Liu, Z.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

Loh, K. P.

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

Lopez, J. J.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

Manjavacas, A.

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

Martin, M.

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, 630–634 (2011).
[CrossRef]

Mesch, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

Niu, L.

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

Nordlander, P.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

Novoselov, K. S.

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

Polini, M.

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

Shan, C.

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

Shen, Y. R.

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, 630–634 (2011).
[CrossRef]

Sherrott, M.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

Song, J.

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Thongrattanasiri, S.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef]

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

Wang, F.

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, 630–634 (2011).
[CrossRef]

Wang, Y.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

Weiss, T.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

Yang, H.

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

Zettl, 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, 630–634 (2011).
[CrossRef]

ACS Nano (2)

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

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

Anal. Chem. (1)

C. Shan, H. Yang, J. Song, D. Han, A. Ivaska, and L. Niu, “Direct electrochemistry of glucose oxidase and biosensing for glucose based on graphene,” Anal. Chem. 81, 2378–2382 (2009).
[CrossRef]

J. Opt. A (1)

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A 8, 546–549 (2006).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nano Lett. (4)

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[CrossRef]

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

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[CrossRef]

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[CrossRef]

Nat. Nanotechnol. (1)

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, 630–634 (2011).
[CrossRef]

Nat. Photonics (1)

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

Opt. Lett. (1)

Phys. Rev. Lett. (1)

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef]

Other (1)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Checkerboard grating that is made of graphene squares of width w. (b) Hexagonal lattice of graphene nanodisks. The dielectric constants of the bottom and top layers are ϵ1 and ϵ2, respectively. The lattice constant of these structures is a. Also, c1 and c2 are two rectangles for applying boundary conditions to tangential magnetic components.

Fig. 2.
Fig. 2.

Absorption of a checkerboard composed of graphene squares. The parameters of this lattice are a=8μm and w=4μm, ϵ1=4ϵ0 and ϵ2=3ϵ0. The relaxation time and Fermi level are τ=0.25ps and EF=0.6eV, respectively. The new formulation of the FMM and conventional FMM results is depicted by solid and dotted lines, respectively. The truncation order is the same (M=N=20) in obtaining all the curves.

Fig. 3.
Fig. 3.

Ratio of the running time of the conventional FMM to the new formulation for different truncation orders in simulation of the checkerboard grating.

Fig. 4.
Fig. 4.

Relative error in the calculation of absorption of checkerboard grating. The convergence rate of the new formulation is more than the conventional FMM.

Fig. 5.
Fig. 5.

Absorption of a hexagonal lattice of graphene disks. The parameters of this lattice are a=12μm and D=6μm. Other parameters are the same as in Fig. 2

Equations (30)

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

x1=xytanψ,x2=ysecψ,x3=z.
a⃗1=r⃗x1=x^,a⃗2=r⃗x2=sinψx^+cosψy^,a⃗3=r⃗x3=z^,
a⃗1=x^tanψy^,a⃗2=secψy^,a⃗3=z^.
a⃗i·a⃗j=δij,
2Ep33Ep2=ik0secψμp(Hp1sinψHp2),3Ep11Ep3=ik0secψμp(Hp2sinψHp1),1Ep22Ep1=ik0cosψμpHp3,
2Hp33Hp2=ik0secψϵp(Ep1sinψEp2),3Hp11Hp3=ik0secψϵp(Ep2sinψEp1),1Hp22Hp1=ik0cosψϵpEp3.
Epj(r⃗)=m,n[ujmnpexp(ikx3mnpx3)+vjmnpexp(ikx3mnpx3)]exp[i(kx1mx1+kx2nx2)],Hpj(r⃗)=m,n[hjmnpexp(ikx3mnpx3)+gjmnpexp(ikx3mnpx3)]exp[i(kx1mx1+kx2nx2)].
kx3mnp=[k02ϵpμpsec2ψ(kx1m2+kx2n22kx1mkx2nsinψ)]1/2
h1mnp=+u1mnpFmnpu2mnpAmnp,g1mnp=v1mnpFmnp+v2mnpAmnp,
h2mnp=+u1mnpBmnpu2mnpFmnp,g2mnp=v1mnpBmnp+v2mnpFmnp,
Amnp=secψ[k0ϵpkx3mnpkx1m2kx3mnpk0μp],Bmnp=secψ[k0ϵpkx3mnpkx2n2kx3mnpk0μp],Fmnp=secψ[k0ϵpsinψkx3mnpkx1mkx2nkx3mnpk0μp],Dmnp=exp(ikx3mnpΔh2).
n^×(E⃗2E⃗1)=0.
E21=E11u1mn2+v1mn2=u1mn1+v1mn1,E22=E12u2mn2+v2mn2=u2mn1+v2mn1.
n^×(H⃗2H⃗1)=4πcσs(x1,x2)E⃗2,
C2H⃗·dl⃗=S2(ik0ϵ+4πcδ(x3)σs(x1,x2))E⃗·n^ds,
C2H⃗·dl⃗=Δl[H11(x1,x2)|x3=Δh/2H21(x1,x2)|x3=+Δh/2].
0ΔlΔh/2Δh/2(ik0ϵ+4πcδ(x3)σs(x1,x2))×E⃗2(x1,x2)·(a⃗1×a⃗3)|a⃗1×a⃗3|dx3dx1=Δlcosψσseff(x1,x2)E22(x1,x2)|x3=0,
σseff(x1,x2)=ik0ϵ1+ϵ22Δh+4πcσs(x1,x2).
H11(x1,x2)|x3=Δh/2H21(x1,x2)|x3=+Δh/2=cosψσseff(x1,x2)E22(x1,x2)|x3=0.
E22=E22sinψE21.
secψ[H11m|z=Δh/2H21m|z=+Δh/2]+j1σseffmj1E22j=sinψj1σseffmj1E2j1.
j,l[[1σseff]]mn,jl(secψ[H11jl|z=Δh/2H21jl|z=+Δh/2]+s,tσseffjl,stE22st)=sinψE2mn1.
cosψ[H11mn|z=Δh/2H21mn|z=+Δh/2]=sinψj,l[[1σseff]]mn,jl1E21jlj,l(cos2ψσseffjl,st+sin2ψ[[1σseff]]mn,jl1)E22jl.
[Fmn1u1mn1Amn1u2mn1]exp(ikx3mn1Δh2)[Fmn1v1mn1Amn1v2mn1]exp(ikx3mn1Δh2)=+[Fmn2exp(ikx3mn2Δh2)+tanψj,l[[1σseff]]mn,jl1]u1mn2[Amn2exp(ikx3mn2Δh2)+j,l(cosψσseffjl,st+tanψsinψ[[1σseff]]mn,jl1)]u2mn2[Fmn2exp(ikx3mn2Δh2)tanψj,l[[1σseff]]mn,jl1]v1mn2+[Amn2exp(ikx3mn2Δh2)j,l(cosψσseffjl,st+tanψsinψ[[1σseff]]mn,jl1)]v2mn2.
[Bmn1u1mn1Fmn1u2mn1]exp(ikx3mn1Δh2)[Bmn1v1mn1Fmn1v2mn1]exp(ikx3mn1Δh2)=+[Bmn2exp(ikx3mn2Δh2)+j,l(cosψσseffjl,st+tanψsinψ[[1σseff]]mn,jl1)]u1mn2[Fmn2exp(ikx3mn2Δh2)+tanψj,l[[1σseff]]mn,jl1]u2mn2[Bmn2exp(ikx3mn2Δh2)j,l(cosψσseffjl,st+tanψsinψ[[1σseff]]mn,jl1)]v1mn2+[Fmn2exp(ikx3mn2Δh2)tanψj,l[[1σseff]]mn,jl1]v2mn2.
[u2v1]=S1[u1v2]=[Tuu1Rud1Rdu1Tdd1][u1v2].
Rud1=(X1Y)1×(YX2),Tdd1=(X1Y)1×(X1X2),
Y=[B1D1F1D1F1D1A1D1],
X1=[B2D2+cosψσseff+tanψsinψ1σseff1F2D2tanψ1σseff1F2D2+tanψ1σseff1A2D2cosψσsefftanψsinψ1σseff1],X2=[B2(D2)1+cosψσseff+tanψsinψ1σseff1F2(D2)1tanψ1σseff1F2(D2)1+tanψ1σseff1A2(D2)1cosψσsefftanψsinψ1σseff1].
σs=e2EFπ2iω+iτ1+e24[H(ω2EF)+iπln|ω2EFω+2EF|],

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