Abstract

The dispersion equations for polariton waves in dielectric ionic crystal with the absorption are obtained. The self-consistent solutions of the system of Maxwell electromagnetic field equations and the equations of motion of ions have been used. The elastic and absorption properties of the crystal are taken into account in the ion equations of motion. It is shown that the separated equations of motion for positive and negative ions allow obtaining all branches of phonon and polariton spectrum by the example of the ionic crystal of cubic symmetry at the terahertz range. It has been shown that the variation of absorption in the crystal leads to changing of the character of spectrum branch and the polariton velocities.

© 2014 Optical Society of America

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References

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  1. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, 2009).
  2. H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
    [CrossRef]
  3. R. H. Poolman, E. A. Muljarov, and A. L. Ivanov, “Terahertz response of acoustically driven optical phonons,” Phys. Rev. B 81, 245208 (2010).
    [CrossRef]
  4. M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
    [CrossRef]
  5. M. F. Pereira and I. A. Faragai, “Coupling of THz radiation with intervalence band transitions in microcavities,” Opt. Express 22, 3439–3446 (2014).
    [CrossRef]
  6. C. Kittel, Quantum Theory of Solids (Wiley, 1963).
  7. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).
  8. A. S. Davydov, Solid State Physics (Nauka, 1976) (in Russian).
  9. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, 1984).
  10. E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).
  11. I. V. Dzedolik, Polaritons in Optical Fibers and Dielectric Resonators (DIP, 2007) (in Russian).
  12. W.-C. Wang and D.-M. Hwang, “Raman scattering by polariton in potassium bromate crystals,” Chin. J. Phys. 15, 147–155 (1977).
  13. V. S. Podolsky, L. I. Deych, and A. A. Lisyansky, “Local polariton states in impure ionic crystals,” Phys. Rev. B 57, 5168–5176 (1998).
    [CrossRef]
  14. S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
    [CrossRef]
  15. Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
    [CrossRef]
  16. I. Carusotto, T. Voltz, and A. Imamoglu, “Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states,” Europhys. Lett. 90, 37001 (2010).
    [CrossRef]
  17. W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
    [CrossRef]
  18. I. V. Dzedolik, Electromagnetic Field in Active and Passive Media (DIP, 2012) (in Russian).
  19. I. V. Dzedolik and O. Karakchieva, “Polariton spectrum in nonlinear dielectric medium,” Appl. Opt. 52, 3073–3078 (2013).
    [CrossRef]
  20. I. V. Dzedolik and O. Karakchieva, “Control of polariton spectrum in bigyrotropic medium,” Appl. Opt. 52, 6112–6118 (2013).
    [CrossRef]
  21. C. Kittel, Introduction to Solid State Physics (Wiley, 1962).
  22. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. V. 2. The Classical Theory of Fields (Nauka, 1988) (in Russian).
  23. G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).
  24. J. A. Reissland, The Physics of Phonons (Wiley, 1973).

2014 (1)

2013 (2)

2012 (1)

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

2011 (1)

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

2010 (2)

I. Carusotto, T. Voltz, and A. Imamoglu, “Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states,” Europhys. Lett. 90, 37001 (2010).
[CrossRef]

R. H. Poolman, E. A. Muljarov, and A. L. Ivanov, “Terahertz response of acoustically driven optical phonons,” Phys. Rev. B 81, 245208 (2010).
[CrossRef]

2009 (2)

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

2003 (1)

S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
[CrossRef]

1998 (1)

V. S. Podolsky, L. I. Deych, and A. A. Lisyansky, “Local polariton states in impure ionic crystals,” Phys. Rev. B 57, 5168–5176 (1998).
[CrossRef]

1977 (1)

W.-C. Wang and D.-M. Hwang, “Raman scattering by polariton in potassium bromate crystals,” Chin. J. Phys. 15, 147–155 (1977).

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, 1984).

Albuquerque, E. L.

E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).

Bai, W.-C.

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

Beck, M.

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

Carusotto, I.

I. Carusotto, T. Voltz, and A. Imamoglu, “Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states,” Europhys. Lett. 90, 37001 (2010).
[CrossRef]

Castellano, F.

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

Cottam, M. G.

E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).

Davydov, A. S.

A. S. Davydov, Solid State Physics (Nauka, 1976) (in Russian).

Deych, L. I.

V. S. Podolsky, L. I. Deych, and A. A. Lisyansky, “Local polariton states in impure ionic crystals,” Phys. Rev. B 57, 5168–5176 (1998).
[CrossRef]

Dzedolik, I. V.

I. V. Dzedolik and O. Karakchieva, “Control of polariton spectrum in bigyrotropic medium,” Appl. Opt. 52, 6112–6118 (2013).
[CrossRef]

I. V. Dzedolik and O. Karakchieva, “Polariton spectrum in nonlinear dielectric medium,” Appl. Opt. 52, 3073–3078 (2013).
[CrossRef]

I. V. Dzedolik, Electromagnetic Field in Active and Passive Media (DIP, 2012) (in Russian).

I. V. Dzedolik, Polaritons in Optical Fibers and Dielectric Resonators (DIP, 2007) (in Russian).

Faist, J.

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

Faragai, I. A.

Geiser, M.

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, 1984).

Huang, C.-P.

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

Hwang, D.-M.

W.-C. Wang and D.-M. Hwang, “Raman scattering by polariton in potassium bromate crystals,” Chin. J. Phys. 15, 147–155 (1977).

Imamoglu, A.

I. Carusotto, T. Voltz, and A. Imamoglu, “Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states,” Europhys. Lett. 90, 37001 (2010).
[CrossRef]

Inoue, H.

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Ivanov, A. L.

R. H. Poolman, E. A. Muljarov, and A. L. Ivanov, “Terahertz response of acoustically driven optical phonons,” Phys. Rev. B 81, 245208 (2010).
[CrossRef]

Jiang, L.

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

Karakchieva, O.

Katayma, K.

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Kittel, C.

C. Kittel, Quantum Theory of Solids (Wiley, 1963).

C. Kittel, Introduction to Solid State Physics (Wiley, 1962).

Kojima, S.

S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. V. 2. The Classical Theory of Fields (Nauka, 1988) (in Russian).

Lee, Y.-S.

Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, 2009).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. V. 2. The Classical Theory of Fields (Nauka, 1988) (in Russian).

Lisyansky, A. A.

V. S. Podolsky, L. I. Deych, and A. A. Lisyansky, “Local polariton states in impure ionic crystals,” Phys. Rev. B 57, 5168–5176 (1998).
[CrossRef]

Muljarov, E. A.

R. H. Poolman, E. A. Muljarov, and A. L. Ivanov, “Terahertz response of acoustically driven optical phonons,” Phys. Rev. B 81, 245208 (2010).
[CrossRef]

Nelson, K.

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Nishizawa, S.

S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
[CrossRef]

Pantell, R. H.

R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

Pereira, M. F.

Podolsky, V. S.

V. S. Podolsky, L. I. Deych, and A. A. Lisyansky, “Local polariton states in impure ionic crystals,” Phys. Rev. B 57, 5168–5176 (1998).
[CrossRef]

Poolman, R. H.

R. H. Poolman, E. A. Muljarov, and A. L. Ivanov, “Terahertz response of acoustically driven optical phonons,” Phys. Rev. B 81, 245208 (2010).
[CrossRef]

Puthoff, H. E.

R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

Qi, Z.

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

Reissland, J. A.

J. A. Reissland, The Physics of Phonons (Wiley, 1973).

Scalari, G.

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

Shen, Q.

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Shen, Z.-Q.

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

Takeda, M. W.

S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
[CrossRef]

Toyoda, T.

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Tsumura, N.

S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
[CrossRef]

Voltz, T.

I. Carusotto, T. Voltz, and A. Imamoglu, “Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states,” Europhys. Lett. 90, 37001 (2010).
[CrossRef]

Wang, W.-C.

W.-C. Wang and D.-M. Hwang, “Raman scattering by polariton in potassium bromate crystals,” Chin. J. Phys. 15, 147–155 (1977).

Zhang, H.

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

Zhang, H.-Z.

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

Zhang, L.-Q.

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

Zhu, S.-N.

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

Zhu, Y.-Y.

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M. Geiser, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Room temperature terahertz polariton emitter,” Appl. Phys. Lett. 101, 141118 (2012).
[CrossRef]

Chin. J. Phys. (1)

W.-C. Wang and D.-M. Hwang, “Raman scattering by polariton in potassium bromate crystals,” Chin. J. Phys. 15, 147–155 (1977).

Europhys. Lett. (1)

I. Carusotto, T. Voltz, and A. Imamoglu, “Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states,” Europhys. Lett. 90, 37001 (2010).
[CrossRef]

J. Appl. Phys. (2)

Z. Qi, Z.-Q. Shen, C.-P. Huang, S.-N. Zhu, and Y.-Y. Zhu, “Phonon polaritons in a nonaxial aligned piezoelectric superlattice,” J. Appl. Phys. 105, 074102 (2009).
[CrossRef]

H. Inoue, K. Katayma, Q. Shen, T. Toyoda, and K. Nelson, “Terahertz reflection response measurement using a phonon polariton wave,” J. Appl. Phys. 105, 054902 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (3)

R. H. Poolman, E. A. Muljarov, and A. L. Ivanov, “Terahertz response of acoustically driven optical phonons,” Phys. Rev. B 81, 245208 (2010).
[CrossRef]

V. S. Podolsky, L. I. Deych, and A. A. Lisyansky, “Local polariton states in impure ionic crystals,” Phys. Rev. B 57, 5168–5176 (1998).
[CrossRef]

S. Kojima, N. Tsumura, M. W. Takeda, and S. Nishizawa, “Far-infrared phonon-polariton dispersion probe by terahertz time-domain spectroscopy,” Phys. Rev. B 67, 035102 (2003).
[CrossRef]

Solid State Commun. (1)

W.-C. Bai, H. Zhang, L. Jiang, H.-Z. Zhang, and L.-Q. Zhang, “Theoretical investigation of phonon–polariton modes in undoped and ion-doped PPLN crystals,” Solid State Commun. 151, 1261–1265 (2011).
[CrossRef]

Other (12)

I. V. Dzedolik, Electromagnetic Field in Active and Passive Media (DIP, 2012) (in Russian).

Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, 2009).

C. Kittel, Quantum Theory of Solids (Wiley, 1963).

R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

A. S. Davydov, Solid State Physics (Nauka, 1976) (in Russian).

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, 1984).

E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).

I. V. Dzedolik, Polaritons in Optical Fibers and Dielectric Resonators (DIP, 2007) (in Russian).

C. Kittel, Introduction to Solid State Physics (Wiley, 1962).

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. V. 2. The Classical Theory of Fields (Nauka, 1988) (in Russian).

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

J. A. Reissland, The Physics of Phonons (Wiley, 1973).

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Figures (4)

Fig. 1.
Fig. 1.

Phonon and photon spectra without interaction of photons and phonons in the crystal of cubic symmetry.

Fig. 2.
Fig. 2.

Phonon and polariton spectra with interaction of photons and phonons in the crystal of cubic symmetry.

Fig. 3.
Fig. 3.

Phonon and polariton spectra with interaction of photons and phonons in the crystal of cubic symmetry with absorption.

Fig. 4.
Fig. 4.

Normalized velocity v¯=c1εdω/dk of longitudinal polaritons: curve 1 at Γ/Ω=0.002; curve 2 at Γ/Ω=0.02; curve 3 at Γ/Ω=0.2.

Equations (33)

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(×B)j=εcEjt+4πcjj,(×E)j=1cBjt,
2Rj(1n)t2=Ωj124(2Rj(1n)Rj(2n)Rj(2n1))Γj1Rj(1n)t+eeffM1{Ej+[1c(Rt)×B]j},2Rj(2n)t2=Ωj224(2Rj(2n)Rj(1n)Rj(1n+1))Γj2Rj(2n)teeffM2{Ej+[1c(Rt)×B]j},
Fx=Mdvx/dt=eExe(vz/c)By=e(1kvz/ω)Ex,Fz=e(vx/c)By=e(kvx/ω)Ex,
D(ω,kx,ky,kz)=0.
kEy=ωcBx,kEx=ωcBy,Bz=0,0=εEz+2πeeffen=1N(Rz(1n)Rz(2n)),kBy=εωcEx+2πeeffωcn=1N(Rx(1n)Rx(2n)),kBx=εωcEy+2πeeffωcn=1N(Ry(1n)Ry(2n)).
2Rx(1n)t2=14Ωx12(2Rx(1n)Rx(2n)Rx(2n1))Γx1Rx(1n)t+eeffM1Ex(n),2Rx(2n)t2=14Ωx22(2Rx(2n)Rx(1n)Rx(1n+1))Γx2Rx(2n)teeffM2Ex(n),2Ry(1n)t2=14Ωy12(2Ry(1n)Ry(2n)Ry(2n1))Γy1Ry(1n)t,2Ry(2n)t2=14Ωy22(2Ry(2n)Ry(1n)Ry(1n+1))Γy2Ry(2n)t,2Rz(1n)t2=14Ωz12(2Rz(1n)Rz(2n)Rz(2n1))Γz1Rz(1n)t+eeffM1Ez(n),2Rz(2n)t2=14Ωz22(2Rz(2n)Rz(1n)Rz(1n+1))Γz2Rz(2n)teeffM2Ez(n).
(12Ω12ω2+12ωP12ε1ω2ω2c2ε1k2iΓ1ω)Rx1(14Ω12(1+eika)+12ωP12ε1ω2ω2c2ε1k2)Rx2=0,(14Ω22(1+eika)+12ωP22ε1ω2ω2c2ε1k2)Rx1(12Ω22ω2+12ωP22ε1ω2ω2c2ε1k2iΓ2ω)Rx2=0,
(12Ω12ω2iΓ1ω)Ry114Ω12(1+eika)Ry2=0,14Ω22(1+eika)Ry1(12Ω22ω2iΓ2ω)Ry2=0,
(12Ω12+12ωP12ε1ω2iΓ1ω)Rz1(14Ω12(1+eika)+12ωP12ε1)Rz2=0,(14Ω22(1+eika)+12ωP22ε1)Rz1(12Ω22+12ωP22ε1ω2iΓ2ω)Rz2=0,
b1,2=ωP1,22ε1Ω1,22.
[12Ω12(1+b1ω2ω2c2ε1k2)ω2iΓ1ω]×[12Ω22(1+b2ω2ω2c2ε1k2)ω2iΓ2ω]116Ω12Ω22(1+eika+2b1ω2ω2c2ε1k2)×(1+eika+2b2ω2ω2c2ε1k2)=0,
(12Ω12ω2iΓ1ω)(12Ω22ω2iΓ2ω)116Ω12Ω22(1+eika)(1+eika)=0.
[12Ω12(1+b1)ω2iΓ1ω][12Ω22(1+b2)ω2iΓ2ω]116Ω12Ω22(1+eika+2b1)(1+eika+2b2)=0.
(ω2c2ε1k2){ω4+i2Γω3(Γ2+Ω2)ω2iΓΩ2ω+18Ω4[1cos(ka)]}=0,
ω4+i2Γω3(Γ2+Ω2)ω2iΓΩ2ω+18Ω4[1cos(ka)]=0,
ω4+i2Γω3(Γ2+Ω2)ω2iΓΩ2ω+18Ω4[1cos(ka)]=0;
ω6+i2Γω5(Ω2+Γ2+ωP2ε1+c2ε1k2)ω4iΓ(Ω2+ωP2ε1+2c2ε1k2)ω3+{(Ω4/2+Ω2ωP2ε1)[1cos(ka)]/4+(Ω2+Γ2)c2ε1k2}ω2+iΓΩ2c2ε1k2ωΩ4[1cos(ka)]c2ε1k2/8=0,
ω4+i2Γω3(Ω2+Γ2)ω2iΓΩ2ω+Ω4[1cos(ka)]/8=0,
ω4+i2Γω3(Ω2+Γ2+ωP2ε1)ω2iΓ(Ω2+ωP2ε1)ω+(Ω4/2+Ω2ωP2ε1)[1cos(ka)]/4=0.
ω¯2k¯2=0,
ω¯4+i2a1x,y,zω¯3a20x,y,zω¯2ia1x,y,zω¯+[1cos(k¯a¯)]/8=0,
ω¯6+i2a1xω¯5(k¯2+a2x)ω¯4ia1x(2k¯2+a3)ω¯3+{a4[1cos(k¯a¯)]/4+a5xk¯2}ω¯2+ia1xk¯2ω¯[1cos(k¯a¯)]k¯2/8=0,
ω¯4+i2a1zω¯3a2zω¯2ia1za3ω¯+a4[1cos(k¯a¯)]/4=0,
ω¯±2=12±12{112[1cos(k¯a¯)]}1/2.
ω¯12=A+Bα/3,ω¯2,32=(A+B)/2±i(AB)3/2α/3,
ω¯6,72=12a3±12{a32a4[1cos(k¯a¯)]}1/2.
ω¯4(k¯2+a3)ω¯2+k¯2=0,
ω¯±2=12(k¯2+a3)±[14(k¯2+a3)2k¯2]1/2.
ω¯2a3=0,
ω¯2=a3,
ω12=Ω2(A0+B0+a3/3),ω2,32=Ω2[(A0+B0)/2±i(A0B0)3/2+a3/3],ω4,52=Ω2/2±Ω2/2,ω6,72=(Ω2+ωP2ε1)/2±(Ω2+ωP2ε1)/2,
ε˜j=ωP+2(Ω˜j122+Ω˜j212)+ωP2(Ω˜j212+Ω˜j122+)Ω˜j122+Ω˜j212+Ω˜j122Ω˜j212+,
|Ω˜1200Ω1()200α11000000Ω˜1200Ω1()200α11000000Ω˜1200Ω1()200α11000Ω(+)200Ω˜2200α12000000Ω(+)200Ω˜2200α12000000Ω(+)200Ω˜2200α12000α21ω00α22ω00εω000ckzcky0α21ω00α22ω00εω0ckz0ckx00α21ω00α22ω00εωckyckx00000000ckzckyω00000000ckz0ckx0ω0000000ckyckx000ω|=0,

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