Abstract

Modulation instability (MI) in a coupled resonator optical waveguide (CROW) and photonic-crystal waveguide (PCW) with nonlinear Kerr media was studied by using the tight-binding theory. By considering the coupling between the defects, we obtained a discrete nonlinear evolution equation and termed it the extended discrete nonlinear Schrödinger (EDNLS) equation. By solving this equation for CROWs and PCWs, we obtained the MI region and the MI gains, G(p,q), for different wavevectors of the incident plane wave (p) and perturbation (q) analytically. In CROWs, the MI region, in which solitons can be formed, can only occur for pa being located either before or after π/2, where a is the separation of the cavities. The location of the MI region is determined by the number of the separation rods between defects and the sign of the Kerr coefficient. However, in the PCWs, pa in the MI region can exceed the π/2. For those wavevectors close to π/2, the MI profile, G(q), can possess two gain maxima at fixed pa. It is quite different from the results of the nonlinear CROWs and optical fibers. By numerically solving the EDNLS equation using the 4th order Runge-Kutta method to observe exponential growth of small perturbation in the MI region, we found it is consistent with our analytic solutions.

© 2009 Optical Society of America

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  1. E. Yablonovitch, "Photonic band-gap crystals," J. Phys. Condens. Matter 5, 2443-2460 (1993).
    [CrossRef]
  2. E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993).
    [CrossRef]
  3. D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
    [CrossRef]
  4. N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12127-12133 (1998).
    [CrossRef]
  5. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  6. A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
    [CrossRef]
  7. W. J. Kim, W. Kuang, and J. D. O'Brien, "Dispersion characteristics of photonic crystal coupled resonator optical waveguides," Opt. Express 11, 3431-3437 (2003).
    [CrossRef] [PubMed]
  8. S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
    [CrossRef]
  9. S. F. Mingaleev, and Y. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86, 5474-5477 (2001).
    [CrossRef] [PubMed]
  10. D. N. Christodoulides and N. K. Efremidis, "Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals," Opt. Lett. 27, 568-570 (2002).
    [CrossRef]
  11. S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
    [CrossRef]
  12. A. G. Shagalov, "Modulational instability of nonlinear waves in the range of zero dispersion," Physics Lett. A 239, 41-45 (1998).
    [CrossRef]
  13. L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003).
    [CrossRef]
  14. F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
    [CrossRef]
  15. F. M. Mitschke and L. F. Mollenauer, "Discovery of the soliton self-frequency shift," Opt. Lett. 11, 659-661 (1986).
    [CrossRef] [PubMed]
  16. T. Kamalakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron. 41, 1419-1425 (2005).
    [CrossRef]
  17. S. Mookherjea, "Dispersion characteristics of coupled-resonator optical waveguides," Opt. Lett. 30, 2406-2408 (2005).
    [CrossRef] [PubMed]
  18. F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
    [CrossRef]
  19. K. Hosomi and T. Katsuyama, "A dispersion compensator using coupled defects in a photonic crystal," IEEE J. Quantum Electron. 38, 825-829 (2002).
    [CrossRef]

2007

F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
[CrossRef]

F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
[CrossRef]

2006

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
[CrossRef]

2005

T. Kamalakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron. 41, 1419-1425 (2005).
[CrossRef]

S. Mookherjea, "Dispersion characteristics of coupled-resonator optical waveguides," Opt. Lett. 30, 2406-2408 (2005).
[CrossRef] [PubMed]

2003

W. J. Kim, W. Kuang, and J. D. O'Brien, "Dispersion characteristics of photonic crystal coupled resonator optical waveguides," Opt. Express 11, 3431-3437 (2003).
[CrossRef] [PubMed]

L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003).
[CrossRef]

2002

K. Hosomi and T. Katsuyama, "A dispersion compensator using coupled defects in a photonic crystal," IEEE J. Quantum Electron. 38, 825-829 (2002).
[CrossRef]

D. N. Christodoulides and N. K. Efremidis, "Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals," Opt. Lett. 27, 568-570 (2002).
[CrossRef]

2001

S. F. Mingaleev, and Y. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86, 5474-5477 (2001).
[CrossRef] [PubMed]

2000

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

1999

A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

1998

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

A. G. Shagalov, "Modulational instability of nonlinear waves in the range of zero dispersion," Physics Lett. A 239, 41-45 (1998).
[CrossRef]

1993

E. Yablonovitch, "Photonic band-gap crystals," J. Phys. Condens. Matter 5, 2443-2460 (1993).
[CrossRef]

E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993).
[CrossRef]

1986

Abdullaev, F. K.

F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
[CrossRef]

Bouketir, A.

F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
[CrossRef]

Busch, K.

S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
[CrossRef]

Chen, C. H.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

Cheng, S. C.

F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
[CrossRef]

Chien, F. S. S.

F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
[CrossRef]

Christodoulides, D. N.

Efremidis, N. K.

Hadzievski, L.

L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003).
[CrossRef]

Hosomi, K.

K. Hosomi and T. Katsuyama, "A dispersion compensator using coupled defects in a photonic crystal," IEEE J. Quantum Electron. 38, 825-829 (2002).
[CrossRef]

Hsieh, W. F.

F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
[CrossRef]

Imhof, A.

A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
[CrossRef]

Kamalakis, T.

T. Kamalakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron. 41, 1419-1425 (2005).
[CrossRef]

Katsuyama, T.

K. Hosomi and T. Katsuyama, "A dispersion compensator using coupled defects in a photonic crystal," IEEE J. Quantum Electron. 38, 825-829 (2002).
[CrossRef]

Kim, W. J.

Kivshar, Y. S.

S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
[CrossRef]

S. F. Mingaleev, and Y. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86, 5474-5477 (2001).
[CrossRef] [PubMed]

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

Kuang, W.

Lagendijk, A.

A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
[CrossRef]

Lee, R. K.

Messikh, A.

F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
[CrossRef]

Miao, B. L.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

Mingaleev, S. F.

S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
[CrossRef]

S. F. Mingaleev, and Y. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86, 5474-5477 (2001).
[CrossRef] [PubMed]

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

Miroshnichenko, A. E.

S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
[CrossRef]

Mitschke, F. M.

Modinos, A.

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

Mollenauer, L. F.

Mookherjea, S.

Murakowski, J.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

O'Brien, J. D.

Prather, D. W.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

Sammut, R. A.

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

Scherer, A.

Schneider, G. J.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

Shagalov, A. G.

A. G. Shagalov, "Modulational instability of nonlinear waves in the range of zero dispersion," Physics Lett. A 239, 41-45 (1998).
[CrossRef]

Sharkawy, A.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

Shi, S. Y.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

Skoric, M. M.

L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003).
[CrossRef]

Sphicopoulos, T.

T. Kamalakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron. 41, 1419-1425 (2005).
[CrossRef]

Sprik, R.

A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
[CrossRef]

Stefanou, N.

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

Stepic, M.

L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003).
[CrossRef]

Tu, J. B.

F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
[CrossRef]

Umarov, B. A.

F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
[CrossRef]

Vos, W. L.

A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
[CrossRef]

Xu, Y.

Yablonovitch, E.

E. Yablonovitch, "Photonic band-gap crystals," J. Phys. Condens. Matter 5, 2443-2460 (1993).
[CrossRef]

E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993).
[CrossRef]

Yariv, A.

IEEE J. Quantum Electron.

K. Hosomi and T. Katsuyama, "A dispersion compensator using coupled defects in a photonic crystal," IEEE J. Quantum Electron. 38, 825-829 (2002).
[CrossRef]

T. Kamalakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron. 41, 1419-1425 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

E. Yablonovitch, "Photonic band-gap crystals," J. Phys. Condens. Matter 5, 2443-2460 (1993).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007).
[CrossRef]

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003).
[CrossRef]

Phys. Rev. E

S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006).
[CrossRef]

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

Phys. Rev. Lett.

S. F. Mingaleev, and Y. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86, 5474-5477 (2001).
[CrossRef] [PubMed]

A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999).
[CrossRef]

Physica D

F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007).
[CrossRef]

Physics Lett. A

A. G. Shagalov, "Modulational instability of nonlinear waves in the range of zero dispersion," Physics Lett. A 239, 41-45 (1998).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

The structures of (a) a PCW, (b) a CROW with one separation rod and (c) a CROW with two separation rods, where a is the length of successive defect points and aL is the lattice constant of a PC.

Fig. 2.
Fig. 2.

(a) The electric field distribution (Ez) of point defect mode simulated by the plane wave expansion method in the square lattice with the dielectric constant, radii of dielectric rods and the radius (rd) of the defect rods being 12, 0.2aL and 0.05aL for frequency f = 0.364 c/aL . (b) The field distribution of the red dash line in (a).

Fig. 3.
Fig. 3.

The dispersion relations of (a) a CROW with one separation rod and (b) a PCW in square lattices, which are simulated by the plane wave expansion method. The dash red lines are the edges of the band gaps.

Fig. 4.
Fig. 4.

(a) The values of A and (b) the gains and regions of the MI of the CROW with γ|ϕ0|2=0.01 (2πc/aL).

Fig. 5.
Fig. 5.

(a) (b)The values of A in the PCW. The region and gains of MI with (c) positive Kerr media (γ|ϕ0|2=0.01*2πc/a) and (d) negative Kerr media (γ|ϕ0|2=-0.01*2πc/a).

Fig. 6.
Fig. 6.

The MI gain profiles gotten by analytic solution and the simulation by 4th order Runge-Kutta method in different qa with γ|ϕ0 | 2 = 0.01 (2πc/a).

Fig. 7.
Fig. 7.

The evolution of the perturbation in the PCW with (a) pa=0.4π and qa=0.1π (b) pa=0.6π and qa=0.1π.

Tables (1)

Tables Icon

Table 1 MI regions of CROWs

Equations (10)

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

i d b n dt + ( ω 0 + c 0 ) b n + c 1 ( b n + 1 + b n 1 ) + c 2 ( b n + 2 + b n 2 ) + γ b n 2 b n = 0 .
c m = ω 0 ∫∫∫ Δ ε E 0 n E 0 n + m ∫∫∫ ( μ 0 H 0 n 2 + ε E 0 n 2 ) ,
γ = 2 n 0 n 2 ε 0 ω 0 ∫∫∫ E 0 n 4 ∫∫∫ ( μ 0 H 0 n 2 + ε E 0 n 2 ) ,
ω ( pa ) = ω 0 c 0 2 c 1 cos ( pa ) 2 c 2 cos ( 2 pa ) γ ϕ 0 2 .
b n = ( ϕ 0 + ν n ( t ) ) e i ( pna ωt ) ,
i d ν n dt + c 1 ( v n + 1 e ipa + v n 1 e ipa 2 cos ( pa ) v n ) +
c 2 ( v n + 2 e ipa + v n 2 e ipa 2 cos ( 2 pa ) v n ) + γ ϕ 0 2 ( v n + v n * ) = 0 .
v n ( t ) = ( V 1 e iqna + V 2 * e iqna ) e i Ω t ,
Ω p q = B ± A ( A γ ϕ 0 2 ) .
G p q = 2 * Im ( Ω p q ) = Re ( 2 · A ( γ ϕ 0 2 A ) 1 2 ) = 2 · Re ( A 0.5 γ ϕ 0 2 ) 2 + 0.25 γ 2 ϕ 0 4 .

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