Abstract

We consider the electromagnetic field due to the reflection and transmission of a plane wave by a bounded periodic structure containing chiral layers. The problem is solved using the 2×2-block-representation transfer-matrix formulation. The frequency and angular dependences of the reflection and transmission coefficients are given. The boundaries of the passbands and stopbands are determined from the basic-element transfer-matrix eigenvalue analysis.

© 2008 Optical Society of America

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  1. M. Born and E. Wolf, Principle of Optics (Pergamon, 1968).
  2. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423-438 (1977).
    [CrossRef]
  3. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438-448 (1977).
    [CrossRef]
  4. G. Kim and E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132-137 (1992).
    [CrossRef]
  5. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265-4274 (1996).
    [CrossRef]
  6. F. G. Bass, A. A. Bulgakov, and A. P. Tetervov, High-Frequency Properties of Semiconductors with Super-Lattice (Nauka, 1989) (in Russian).
  7. H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995).
  8. K. Iga, “Surface-emitting laser--its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6, 1201-1215 (2000).
    [CrossRef]
  9. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer-Verlag, 1989).
  10. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).
  11. D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
    [CrossRef]
  12. M. Norgen and S. He, “General scheme for electromagnetic reflection and transmission for composite structures of complex materials,” IEE Proc., Part H: Microwaves, Antennas Propag. 142, 52-56 (1995).
    [CrossRef]
  13. K. M. Flood and D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339-345 (1994).
    [CrossRef]
  14. W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagn. Res. PIER 20, pp. 153-178 (1998). doi:10.2528/PIER98020400. Available on http://ceta.mit.edu/PIER/pier.php?volume=20. (October 2008).
    [CrossRef]
  15. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
    [CrossRef]
  16. V. B. Kazanskiy, V. R. Tuz, and V. V. Khardikov, “Quasiperiodic metal-dielectric structure as a multifunctional control system,” Radioelectron. Commun. Syst. 45, 38-46 (2002).
  17. V. B. Kazanskiy and V. R. Tuz, “Influence of natural fields of multi-section waveguide filters on dissipation and conversion of the TEmn- and TMmn-waves,” in Kharkov University Bulletin, Radiophysics and Electronics, 544, 83-86 (in Russian) (2002).
  18. A. A. Bulgakov and V. K. Kononenko, “Effect of translation symmetry on electrodynamic properties on the semiconductor-dielectric structure placed in a magnetic field,” Telecommun. Radio Eng. (Engl. Transl.) 55, 369-378 (2001).

2002 (1)

V. B. Kazanskiy, V. R. Tuz, and V. V. Khardikov, “Quasiperiodic metal-dielectric structure as a multifunctional control system,” Radioelectron. Commun. Syst. 45, 38-46 (2002).

2001 (1)

A. A. Bulgakov and V. K. Kononenko, “Effect of translation symmetry on electrodynamic properties on the semiconductor-dielectric structure placed in a magnetic field,” Telecommun. Radio Eng. (Engl. Transl.) 55, 369-378 (2001).

2000 (1)

K. Iga, “Surface-emitting laser--its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6, 1201-1215 (2000).
[CrossRef]

1998 (1)

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagn. Res. PIER 20, pp. 153-178 (1998). doi:10.2528/PIER98020400. Available on http://ceta.mit.edu/PIER/pier.php?volume=20. (October 2008).
[CrossRef]

1996 (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265-4274 (1996).
[CrossRef]

1995 (1)

M. Norgen and S. He, “General scheme for electromagnetic reflection and transmission for composite structures of complex materials,” IEE Proc., Part H: Microwaves, Antennas Propag. 142, 52-56 (1995).
[CrossRef]

1994 (1)

K. M. Flood and D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339-345 (1994).
[CrossRef]

1992 (1)

1989 (1)

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

1977 (2)

1972 (1)

Bass, F. G.

F. G. Bass, A. A. Bulgakov, and A. P. Tetervov, High-Frequency Properties of Semiconductors with Super-Lattice (Nauka, 1989) (in Russian).

Berreman, D. W.

Born, M.

M. Born and E. Wolf, Principle of Optics (Pergamon, 1968).

Bulgakov, A. A.

A. A. Bulgakov and V. K. Kononenko, “Effect of translation symmetry on electrodynamic properties on the semiconductor-dielectric structure placed in a magnetic field,” Telecommun. Radio Eng. (Engl. Transl.) 55, 369-378 (2001).

F. G. Bass, A. A. Bulgakov, and A. P. Tetervov, High-Frequency Properties of Semiconductors with Super-Lattice (Nauka, 1989) (in Russian).

Engheta, N.

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Flood, K. M.

K. M. Flood and D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339-345 (1994).
[CrossRef]

Garmire, E.

He, S.

M. Norgen and S. He, “General scheme for electromagnetic reflection and transmission for composite structures of complex materials,” IEE Proc., Part H: Microwaves, Antennas Propag. 142, 52-56 (1995).
[CrossRef]

Hong, C.-S.

Iga, K.

K. Iga, “Surface-emitting laser--its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6, 1201-1215 (2000).
[CrossRef]

Jaggard, D.

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Jaggard, D. L.

K. M. Flood and D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339-345 (1994).
[CrossRef]

Kazanskiy, V. B.

V. B. Kazanskiy, V. R. Tuz, and V. V. Khardikov, “Quasiperiodic metal-dielectric structure as a multifunctional control system,” Radioelectron. Commun. Syst. 45, 38-46 (2002).

V. B. Kazanskiy and V. R. Tuz, “Influence of natural fields of multi-section waveguide filters on dissipation and conversion of the TEmn- and TMmn-waves,” in Kharkov University Bulletin, Radiophysics and Electronics, 544, 83-86 (in Russian) (2002).

Khardikov, V. V.

V. B. Kazanskiy, V. R. Tuz, and V. V. Khardikov, “Quasiperiodic metal-dielectric structure as a multifunctional control system,” Radioelectron. Commun. Syst. 45, 38-46 (2002).

Kim, G.

Kim, Y.

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Kononenko, V. K.

A. A. Bulgakov and V. K. Kononenko, “Effect of translation symmetry on electrodynamic properties on the semiconductor-dielectric structure placed in a magnetic field,” Telecommun. Radio Eng. (Engl. Transl.) 55, 369-378 (2001).

Kowarz, M. W.

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer-Verlag, 1989).

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Liu, J. C.

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Nan, G. H.

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagn. Res. PIER 20, pp. 153-178 (1998). doi:10.2528/PIER98020400. Available on http://ceta.mit.edu/PIER/pier.php?volume=20. (October 2008).
[CrossRef]

Norgen, M.

M. Norgen and S. He, “General scheme for electromagnetic reflection and transmission for composite structures of complex materials,” IEE Proc., Part H: Microwaves, Antennas Propag. 142, 52-56 (1995).
[CrossRef]

Pelet, P.

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Schubert, M.

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265-4274 (1996).
[CrossRef]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Tetervov, A. P.

F. G. Bass, A. A. Bulgakov, and A. P. Tetervov, High-Frequency Properties of Semiconductors with Super-Lattice (Nauka, 1989) (in Russian).

Tretyakov, S. A.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Tuz, V. R.

V. B. Kazanskiy, V. R. Tuz, and V. V. Khardikov, “Quasiperiodic metal-dielectric structure as a multifunctional control system,” Radioelectron. Commun. Syst. 45, 38-46 (2002).

V. B. Kazanskiy and V. R. Tuz, “Influence of natural fields of multi-section waveguide filters on dissipation and conversion of the TEmn- and TMmn-waves,” in Kharkov University Bulletin, Radiophysics and Electronics, 544, 83-86 (in Russian) (2002).

Ujihara, K.

H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995).

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer-Verlag, 1989).

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer-Verlag, 1989).

Viitanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Wolf, E.

M. Born and E. Wolf, Principle of Optics (Pergamon, 1968).

Wolff, I.

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagn. Res. PIER 20, pp. 153-178 (1998). doi:10.2528/PIER98020400. Available on http://ceta.mit.edu/PIER/pier.php?volume=20. (October 2008).
[CrossRef]

Yariv, A.

Yeh, P.

Yin, W. Y.

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagn. Res. PIER 20, pp. 153-178 (1998). doi:10.2528/PIER98020400. Available on http://ceta.mit.edu/PIER/pier.php?volume=20. (October 2008).
[CrossRef]

Yokoyama, H.

H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995).

IEE Proc., Part H: Microwaves, Antennas Propag. (1)

M. Norgen and S. He, “General scheme for electromagnetic reflection and transmission for composite structures of complex materials,” IEE Proc., Part H: Microwaves, Antennas Propag. 142, 52-56 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. M. Flood and D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339-345 (1994).
[CrossRef]

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

K. Iga, “Surface-emitting laser--its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6, 1201-1215 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Phys. Rev. B (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265-4274 (1996).
[CrossRef]

Prog. Electromagn. Res. (1)

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagn. Res. PIER 20, pp. 153-178 (1998). doi:10.2528/PIER98020400. Available on http://ceta.mit.edu/PIER/pier.php?volume=20. (October 2008).
[CrossRef]

Radioelectron. Commun. Syst. (1)

V. B. Kazanskiy, V. R. Tuz, and V. V. Khardikov, “Quasiperiodic metal-dielectric structure as a multifunctional control system,” Radioelectron. Commun. Syst. 45, 38-46 (2002).

Telecommun. Radio Eng. (Engl. Transl.) (1)

A. A. Bulgakov and V. K. Kononenko, “Effect of translation symmetry on electrodynamic properties on the semiconductor-dielectric structure placed in a magnetic field,” Telecommun. Radio Eng. (Engl. Transl.) 55, 369-378 (2001).

Other (6)

V. B. Kazanskiy and V. R. Tuz, “Influence of natural fields of multi-section waveguide filters on dissipation and conversion of the TEmn- and TMmn-waves,” in Kharkov University Bulletin, Radiophysics and Electronics, 544, 83-86 (in Russian) (2002).

M. Born and E. Wolf, Principle of Optics (Pergamon, 1968).

F. G. Bass, A. A. Bulgakov, and A. P. Tetervov, High-Frequency Properties of Semiconductors with Super-Lattice (Nauka, 1989) (in Russian).

H. Yokoyama and K. Ujihara, Spontaneous Emission and Laser Oscillation in Microcavities (CRC Press, 1995).

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics (Springer-Verlag, 1989).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

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Figures (4)

Fig. 1
Fig. 1

Bounded periodic sequence of material and chiral layers.

Fig. 2
Fig. 2

(a) Frequency and (b) angular dependences of the reflection and transmission coefficient magnitudes for a sequence of isotropic layers (without chirality): ϵ j = μ j = 1 , j 2 , ϵ 2 = 2 , μ 2 = 1 , ρ = 0 , N = 5 , d 1 L = d 2 L = 0.5 . (a) φ 0 = 25 ° , (b) k 0 L = 6 .

Fig. 3
Fig. 3

(a) Frequency and (b) angular dependences of the reflection and transmission coefficient magnitudes for a sequence of chiral layers: ϵ j = μ j = 1 , j 2 , ϵ 2 = 2 , μ 2 = 1 , ρ = 0.1 , N = 5 , d 1 L = d 2 L = 0.5 . (a) φ 0 = 25 ° , (b) k 0 L = 6 .

Fig. 4
Fig. 4

(a) Frequency and (b) angular dependences of the reflection coefficient magnitudes for a sequence of chiral layers when the structure is placed on a perfectly reflecting surface: ϵ j = μ j = 1 , j 2 , 3 , ϵ 2 = 2 , μ 2 = 1 , N = 5 , d 1 L = d 2 L = 0.5 . (a) φ 0 = 25 ° , (b) k 0 L = 6 , ρ = 0.1 .

Equations (27)

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Δ E x + k 0 2 ( n 2 2 + ρ 2 ) E x 2 i k 0 2 ρ μ 2 H x = 0 ,
Δ H x + k 0 2 ( n 2 2 + ρ 2 ) H x + 2 i k 0 2 ρ ϵ 2 E x = 0 ,
E x e = Q e + + Q e , H x e = i η 2 1 ( Q e + Q e ) ,
E x h = i η 2 ( Q h + Q h ) , H x h = Q h + + Q h ,
Δ Q s + + ( γ + ) 2 Q s + = 0 , Δ Q s + ( γ ) 2 Q s = 0 .
Q e ± = 1 2 Y e ± { A e ± exp [ i ( γ y ± y + γ z ± z ) ] + B e ± exp [ i ( γ y ± y γ z ± z ) ] } ,
Q h ± = Y h ± 2 { A h ± exp [ i ( γ y ± y + γ z ± z ) ] + B h ± exp [ i ( γ y ± y γ z ± z ) ] } ,
{ E x 1 e E y 1 h } = ± { 1 Y 1 e i Y 1 h } ( { A m e A m h } e i k z 1 ( z z m ) ± { B m e B m h } e i k z 1 ( z z m ) ) ,
{ H y 1 e H x 1 h } = { Y 1 e i Y 1 h } ( { A m e A m h } e i k z 1 ( z z m ) { B m e B m h } e i k z 1 ( z z m ) ) ,
{ E x 2 E y 2 } = ± { 1 2 Y 2 e + i 2 Y 2 h + } [ A m + e i γ z + ( z z m 1 ) ± B m + e i γ z + ( z z m 1 ) ] + { 1 2 Y 2 e i 2 Y 2 h } [ A m e i γ z ( z z m 1 ) ± B m e i γ z ( z z m 1 ) ] ,
{ H y 2 H x 2 } = { Y 2 e + 2 i Y 2 h + 2 } [ A m + e i γ z + ( z z m 1 ) B m + e i γ z + ( z z m 1 ) ] ± { Y 2 e 2 i Y 2 h 2 } [ A m e i γ z ( z z m 1 ) B m e i γ z ( z z m 1 ) ] ,
{ E x 3 e E y 3 h } = ± { ( 1 Y 3 e ) A N + 1 e ( i Y 3 h ) A N + 1 h } e i k z 3 ( z N L ) , { H y 3 e H x 3 h } = { Y 3 e A N + 1 e i Y 3 h A N + 1 h } e i k z 3 ( z N L ) .
( A 0 s B 0 s 0 B 0 s ) = T 01 T N 1 T ( A N + 1 s 0 A N + 1 s 0 ) , ( A m s B m s A m s B m s ) = T 1 ( A m + B m + A m B m ) ,
( A m + B m + A m B m ) = T 2 ( A m + 1 s B m + 1 s A m + 1 s B m + 1 s ) , ( A m s B m s A m s B m s ) = T 1 T 2 ( A m + 1 s B m + 1 s A m + 1 s B m + 1 s ) ,
T p ν = ( ( T p ν s ) 0 0 ( T p ν s ) ) , T 1 = ( ( T 1 + s s ) ( T 1 s s ) ( T 1 + s s ) ( T 1 s s ) ) ,
T 2 = ( ( T 2 + s s ) ( T 2 + s s ) ( T 2 s s ) ( T 2 s s ) ) ,
T p ν s = 1 2 Y p s Y ν s ( Y p s + Y ν s ± ( Y p s Y ν s ) ± ( Y p s Y ν s ) Y p s + Y ν s ) ,
T 1 ± e e = 1 4 Y 1 e Y 2 e ± ( Y 1 e + Y 2 e ± Y 1 e Y 2 e ± Y 1 e Y 2 e ± Y 1 e + Y 2 e ± ) E 1 , T 1 ± e h = ± 1 4 Y 1 h Y 2 h ± ( Y 2 h ± + Y 1 h Y 2 h ± Y 1 h Y 2 h ± Y 1 h Y 2 h ± + Y 1 h ) E 1 ,
T 1 ± h h = 1 4 Y 1 h Y 2 h ± ( Y 2 h ± + Y 1 h Y 2 h ± Y 1 h Y 2 h ± Y 1 h Y 2 h ± + Y 1 h ) E 1 , T 1 ± h e = 1 4 Y 1 e Y 2 e ± ( Y 1 e + Y 2 e ± Y 1 e Y 2 e ± Y 1 e Y 2 e ± Y 1 e + Y 2 e ± ) E 1 ,
T 2 ± e e = 1 4 Y 2 e Y 1 e Y 2 e ± × ( ( Y 2 e + Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e + Y 1 e ) ( Y 2 e Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e + Y 1 e ) ( Y 2 e Y 2 e ± ) ( Y 2 e + Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e Y 2 e ± ) ) E 2 ± ,
T 2 ± e h = 1 4 Y 2 h Y 1 h Y 2 h ± × ( ( Y 2 h + Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h Y 2 h ± ) ( Y 2 h + Y 1 h ) ( Y 2 h Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h + Y 1 h ) ( Y 2 h Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h + Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h Y 2 h ± ) ) E 2 ± ,
T 2 ± h h = 1 4 Y 2 h Y 1 h Y 2 h ± × ( ( Y 2 h + Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h Y 2 h ± ) ( Y 2 h + Y 1 h ) ( Y 2 h Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h + Y 1 h ) ( Y 2 h Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h + Y 1 h ) ( Y 2 h + Y 2 h ± ) ( Y 2 h Y 1 h ) ( Y 2 h Y 2 h ± ) ) E 2 ± ,
T 2 ± h e = 1 4 Y 2 e Y 1 e Y 2 e ± × ( ( Y 2 e + Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e + Y 1 e ) ( Y 2 e Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e + Y 1 e ) ( Y 2 e Y 2 e ± ) ( Y 2 e + Y 1 e ) ( Y 2 e + Y 2 e ± ) ( Y 2 e Y 1 e ) ( Y 2 e Y 2 e ± ) ) E 2 ± ,
Det ( T λ I ) λ 4 a 1 λ 3 + a 2 λ 2 a 3 λ + 1 = 0 ,
[ λ 2 ( λ 1 + λ 1 1 ) λ + 1 ] [ λ 2 ( λ 3 + λ 3 1 ) λ + 1 ] = 0 .
( A m s B m s A m s B m s ) = T ( A m + 1 s B m + 1 s A m + 1 s B m + 1 s ) = exp ( i Γ L ) ( A m + 1 s B m + 1 s A m + 1 s B m + 1 s ) ,
λ j ± 1 = cos Γ j L ± i sin Γ j L .

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