Abstract

We theoretically explore the electromagnetic modes specific to a topological insulator superlattice in which topological and conventional insulator thin films are stacked periodically. In particular, we obtain analytic formulas for low energy mode that corresponds to a helicon wave, as well as those for photonic bands. We illustrate that the system can be modeled as a stack of quantum Hall layers whose conductivity tensors alternately change signs, and then we analyze the photonic band structures. This subject is a natural extension of a previous study by Tselis et al., which took into consideration a stack of identical quantum Hall layers but their discussion was limited into a low energy mode. Thus we provide analytic formulas for photonic bands and compare their features between the two systems. Our central findings in the topological insulator superlattice are that a low energy mode corresponding to a helicon wave has linear dispersion instead of the conventional quadratic form, and that a robust gapless photonic band appears although the system considered has spacial periodicity. In addition, we demonstrate that the photonic bands agree with the numerically calculated transmission spectra.

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  1. E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33, 1193–1322 (1970).
    [CrossRef]
  2. P. M. Platzman and P. A. Wolff, Waves and Interactions in Solids State Plasma, Solid State Phys. 13 (Academic Press, 1972).
  3. J. J. Quinn and K-s. Yi, Solid State Physics: Principles and Modern Applications (Springer, 2009).
  4. A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B29, 2021–2027 (1984).
    [CrossRef]
  5. M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys.82, 3045–3067 (2010).
    [CrossRef]
  6. X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys.83, 1057–1110 (2011).
    [CrossRef]
  7. M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys.2, 55–78 (2011).
    [CrossRef]
  8. A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805-1–146805-4 (2009).
    [CrossRef]
  9. C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).
  10. T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn.81, 094401–094408 (2012).
    [CrossRef]
  11. K. W. Chiu and J. J. Quinn, Phys. Rev. B9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974).
    [CrossRef]
  12. X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008).
    [CrossRef]
  13. X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009).
    [CrossRef] [PubMed]
  14. J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
    [CrossRef]
  15. W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett.105, 057401-1–057401-4 (2010).
    [CrossRef]
  16. M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304-1–113304-4 (2009).
    [CrossRef]
  17. F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett.58, 1799–1802 (1987).
    [CrossRef] [PubMed]
  18. E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).
  19. J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express21, 8564–8569 (2013).
    [CrossRef] [PubMed]
  20. J. Inoue and A. Tanaka, “Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry,” Phys. Rev. B85, 125425-1–125425-7 (2012).
    [CrossRef]
  21. W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).
  22. M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys.38, R123–R152 (2005).
    [CrossRef]

2013 (1)

2012 (2)

T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn.81, 094401–094408 (2012).
[CrossRef]

J. Inoue and A. Tanaka, “Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry,” Phys. Rev. B85, 125425-1–125425-7 (2012).
[CrossRef]

2011 (2)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys.83, 1057–1110 (2011).
[CrossRef]

M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys.2, 55–78 (2011).
[CrossRef]

2010 (5)

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys.82, 3045–3067 (2010).
[CrossRef]

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
[CrossRef]

W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett.105, 057401-1–057401-4 (2010).
[CrossRef]

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

2009 (3)

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304-1–113304-4 (2009).
[CrossRef]

X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009).
[CrossRef] [PubMed]

A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805-1–146805-4 (2009).
[CrossRef]

2008 (1)

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008).
[CrossRef]

2005 (1)

M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys.38, R123–R152 (2005).
[CrossRef]

1987 (1)

F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett.58, 1799–1802 (1987).
[CrossRef] [PubMed]

1984 (1)

A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B29, 2021–2027 (1984).
[CrossRef]

1974 (1)

K. W. Chiu and J. J. Quinn, Phys. Rev. B9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974).
[CrossRef]

1970 (1)

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33, 1193–1322 (1970).
[CrossRef]

Chang, M. C.

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304-1–113304-4 (2009).
[CrossRef]

Chikamatsu, A.

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

Chiu, K. W.

K. W. Chiu and J. J. Quinn, Phys. Rev. B9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974).
[CrossRef]

Dittrich, W.

W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).

Drew, H. D.

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
[CrossRef]

Essin, A. M.

A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805-1–146805-4 (2009).
[CrossRef]

Fiebig, M.

M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys.38, R123–R152 (2005).
[CrossRef]

Furdyna, J. K.

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33, 1193–1322 (1970).
[CrossRef]

Hasan, M. Z.

M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys.2, 55–78 (2011).
[CrossRef]

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys.82, 3045–3067 (2010).
[CrossRef]

Hasegawa, T.

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

Hirose, Y.

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

Hughes, T. L.

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008).
[CrossRef]

Inoue, J.

J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express21, 8564–8569 (2013).
[CrossRef] [PubMed]

J. Inoue and A. Tanaka, “Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry,” Phys. Rev. B85, 125425-1–125425-7 (2012).
[CrossRef]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys.82, 3045–3067 (2010).
[CrossRef]

Kriegler, C. E.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).

Linden, S.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).

MacDonald, A. H.

W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett.105, 057401-1–057401-4 (2010).
[CrossRef]

Maciejko, J.

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
[CrossRef]

Moore, J. E.

M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys.2, 55–78 (2011).
[CrossRef]

A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805-1–146805-4 (2009).
[CrossRef]

Ochiai, T.

T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn.81, 094401–094408 (2012).
[CrossRef]

Palik, E. D.

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33, 1193–1322 (1970).
[CrossRef]

Platzman, P. M.

P. M. Platzman and P. A. Wolff, Waves and Interactions in Solids State Plasma, Solid State Phys. 13 (Academic Press, 1972).

Qi, X. L.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys.83, 1057–1110 (2011).
[CrossRef]

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
[CrossRef]

X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009).
[CrossRef] [PubMed]

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008).
[CrossRef]

Quinn, J. J.

A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B29, 2021–2027 (1984).
[CrossRef]

K. W. Chiu and J. J. Quinn, Phys. Rev. B9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974).
[CrossRef]

J. J. Quinn and K-s. Yi, Solid State Physics: Principles and Modern Applications (Springer, 2009).

Reuter, M.

W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).

Rill, M. S.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).

Sakai, E.

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

Shimada, T.

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

Tanaka, A.

J. Inoue and A. Tanaka, “Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry,” Phys. Rev. B85, 125425-1–125425-7 (2012).
[CrossRef]

Tse, W. K.

W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett.105, 057401-1–057401-4 (2010).
[CrossRef]

Tselis, A. C.

A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B29, 2021–2027 (1984).
[CrossRef]

Vanderbilt, D.

A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805-1–146805-4 (2009).
[CrossRef]

Wegener, M.

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).

Wilczek, F.

F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett.58, 1799–1802 (1987).
[CrossRef] [PubMed]

Wolff, P. A.

P. M. Platzman and P. A. Wolff, Waves and Interactions in Solids State Plasma, Solid State Phys. 13 (Academic Press, 1972).

Yang, M. F.

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304-1–113304-4 (2009).
[CrossRef]

Yi, K-s.

J. J. Quinn and K-s. Yi, Solid State Physics: Principles and Modern Applications (Springer, 2009).

Zang, J.

X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009).
[CrossRef] [PubMed]

Zhang, S. C.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys.83, 1057–1110 (2011).
[CrossRef]

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
[CrossRef]

X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009).
[CrossRef] [PubMed]

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008).
[CrossRef]

Ann. Rev. Condens. Matter Phys. (1)

M. Z. Hasan and J. E. Moore, “Three-dimensional topological insulators,” Ann. Rev. Condens. Matter Phys.2, 55–78 (2011).
[CrossRef]

Appl. Phys. Express (1)

E. Sakai, A. Chikamatsu, Y. Hirose, T. Shimada, and T. Hasegawa, “Magnetic and transport properties of anatase TiO2 codoped with Fe and Nb,” Appl. Phys. Express3, 043001-1–043001-3 (2010).

IEEE J. Sel. Top. Quant. Electron. (1)

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,”IEEE J. Sel. Top. Quant. Electron.16, 367–375 (2010).

J. Phys. D: Appl. Phys. (1)

M. Fiebig, “Revival of the magnetoelectric effect,” J. Phys. D: Appl. Phys.38, R123–R152 (2005).
[CrossRef]

J. Phys. Soc. Jpn. (1)

T. Ochiai, “Theory of Light Scattering in Axion Electrodynamics,” J. Phys. Soc. Jpn.81, 094401–094408 (2012).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (5)

K. W. Chiu and J. J. Quinn, Phys. Rev. B9, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” 4724–4732 (1974).
[CrossRef]

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B78, 195424-1–195424-43 (2008).
[CrossRef]

A. C. Tselis and J. J. Quinn, “Retardation effects and transverse collective excitations in semiconductor superlattices,” Phys. Rev. B29, 2021–2027 (1984).
[CrossRef]

J. Inoue and A. Tanaka, “Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry,” Phys. Rev. B85, 125425-1–125425-7 (2012).
[CrossRef]

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B80, 113304-1–113304-4 (2009).
[CrossRef]

Phys. Rev. Lett. (4)

F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett.58, 1799–1802 (1987).
[CrossRef] [PubMed]

A. M. Essin, J. E. Moore, and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805-1–146805-4 (2009).
[CrossRef]

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett.105, 166803-1–166803-4 (2010).
[CrossRef]

W. K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett.105, 057401-1–057401-4 (2010).
[CrossRef]

Rep. Prog. Phys. (1)

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33, 1193–1322 (1970).
[CrossRef]

Rev. Mod. Phys. (2)

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys.82, 3045–3067 (2010).
[CrossRef]

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys.83, 1057–1110 (2011).
[CrossRef]

Science (1)

X. L. Qi, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science323, 1184–1187 (2009).
[CrossRef] [PubMed]

Other (3)

P. M. Platzman and P. A. Wolff, Waves and Interactions in Solids State Plasma, Solid State Phys. 13 (Academic Press, 1972).

J. J. Quinn and K-s. Yi, Solid State Physics: Principles and Modern Applications (Springer, 2009).

W. Dittrich and M. Reuter, Selected Topics in Gauge Theories (Springer, 1986).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of a stack of quantum Hall layers, QHLS, which was considered in [4]. This system shows distinct EM properties between the two circular polarizations E± and in the long wavelength limit, it has a gapless quadratic dispersion relation that is active only for E+. (b) Schematic of our system, AQHLS, which is a stack of quantum Hall layers whose conductivities change signs layer by layer. The EM properties for the two polarizations are identical. (c) Topological insulator superlattice, which can be model by the system in (b). Gray (white) regions indicate topological (conventional) insulator thin films. (d) Asymmetric AQHLS (0 < α < 2).

Fig. 2
Fig. 2

Two branches of Ω1, from Eq. (20), in the vicinity of K = 0 and Ω1 = 2π.

Fig. 3
Fig. 3

Photonic band structures (a) Ω+ and (b) Ω for E+ and E, respectively, in a QHLS as functions of K. The right panels in (a) and (b) are corresponding transmission spectra T as functions of Ω.

Fig. 4
Fig. 4

(a) Photonic band structure Ω and transmission spectrum in AQHLS. (b) “Vacuum” like photonic band Ω found in AQHLS.

Equations (29)

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( 2 z 2 β 2 ) E ± ( z ) = 4 π i ω c 2 J ± ( z ) ,
E ± ( z ) = 1 ξ d z exp ( β | z z | ) J ± ( z ) ,
σ ( z ) = ( σ ^ x x ( z ) σ ^ x y ( z ) σ ^ x y ( z ) σ ^ x x ( z ) ) .
E ± ( z ) = 1 ξ d z exp ( β | z z | ) σ ± ( z ) E ± ( z ) .
σ ± ( z ) = σ ± m δ ( z 2 d m ) .
ξ = σ ± S ,
S i sin Ω cos Ω cos K ,
Ω + ~ K 2 .
D = ε ( z ) E a top Θ ( z ) B ,
H = 1 μ 0 B + b top Θ ( z ) E ,
1 μ 0 z ( B y B x ) = i ε 2 ω ( E x E y ) + b top δ ( z ) ( E y E x ) ,
1 μ 0 z ( B y B x ) = i ε 2 ω ( E x E y ) + σ ^ x y δ ( z ) ( E y E x ) .
σ ± ( z ) = σ ± [ m δ ( z 2 d m ) m δ ( z 2 d m d ) ] ,
E ± ( z ) = σ ± ξ [ z : + exp ( β | z z | ) E ± ( + ) ( z ) z : exp ( β | z z | ) E ± ( ) ( z ) ] ,
ξ E ± 0 ( + ) = σ ± S E ± 0 ( + ) σ ± S E ± 0 ( ) ,
S = 2 i sin Ω 2 cos K 2 cos Ω cos K ,
ξ E ± 0 ( ) = σ ± S E ± 0 ( + ) σ ± S E ± 0 ( ) .
ξ 2 σ ± 2 ( S 2 S 2 ) = 0 .
Ω 1 ± K ( 1 + A ) 1 / 2 ,
Ω 1 = cos 1 [ cos K + ( A 2 + A sin 2 K ) 1 / 2 1 + A ] ,
Ω 2 = cos 1 [ cos K ( A 2 + A sin 2 K ) 1 / 2 1 + A ] .
Ω ± ( n ) = cos 1 [ cos K ( 1 ) n ( A 2 + A sin 2 K ) 1 / 2 1 + A ] .
A ( S 2 S 2 ) = 1 .
Ω 1 ± = cos 1 [ A + cos K 1 + A ] ,
Ω 2 ± = K .
2 A ( cos Ω cos K ) sin ( α 2 Ω ) sin ( 2 α 2 Ω ) = ( cos Ω cos K ) 2 .
2 A sin ( α 2 Ω 1 ± ) sin ( 2 α 2 Ω 1 ± ) = cos Ω 1 ± cos K ,
cos Ω 2 ± cos K = 0 .
Ω 1 ± K { 1 + A α ( 2 α ) } 1 / 2 ,

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