Abstract

We study spatiotemporal instabilities in a bulk medium with Kerr-type nonlinearity and a volume Bragg grating along the direction of wave propagation. The continuous-wave beam propagation is unstable in such a periodic structure because of an interplay among grating-induced dispersion, diffraction, and nonlinear phase modulation. A linear stability analysis of the nonlinear coupled-mode equations predicts parameters for which novel self-pulsations that occur in both time and space can be observed experimentally.

© 2001 Optical Society of America

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References

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  1. A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, Berlin, 1975).
  2. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  3. C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
    [CrossRef]
  4. C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
    [CrossRef]
  5. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef] [PubMed]
  6. C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
    [CrossRef]
  7. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [CrossRef] [PubMed]
  8. L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
    [CrossRef] [PubMed]
  9. G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef] [PubMed]
  10. C. T. Law and A. E. Kaplan, “Dispersion-related multimode instabilities and self-sustained oscillations in nonlinear counterpropagating waves,” Opt. Lett. 14, 734–736 (1989).
    [CrossRef] [PubMed]
  11. B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
    [CrossRef]
  12. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  13. B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
    [CrossRef]
  14. P. Millar, R. M. De La Rue, T. F. Krauss, J. S. Aitchison, N. G. R. Broderick, and D. J. Richardson, “Nonlinear propagation effects in an AlGaAs Bragg grating filter,” Opt. Lett. 24, 685–687 (1999).
    [CrossRef]
  15. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
    [CrossRef]
  16. P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
    [CrossRef]
  17. N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
    [CrossRef]
  18. P. St. J. Russell and J.-L. Archambault, “Field microstructures and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media,” J. Phys. (Paris) III 4, 2471–2491 (1994).
  19. Yu. A. Logvin and V. M. Volkov, “Phase sensitivity of a nonlinear Bragg grating response under bidirectional illumination,” J. Opt. Soc. Am. B 16, 774–780 (1999).
    [CrossRef]
  20. M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
    [CrossRef]
  21. S. Spälter, G. Lenz, H. Y. Hwang, J. Zimmermann, S.-W. Cheong, T. Katsufuji, R. E. Slusher, “Nonlinear optics in chalcogenide waveguides,” presented at the 1999 Annual Meeting of the Optical Society of America.
  22. N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
    [CrossRef]
  23. G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of collinear waves,” J. Opt. Soc. Am. B 7, 1125–1141 (1990).
    [CrossRef]
  24. G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spälter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
    [CrossRef]
  25. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
    [CrossRef]

2000 (1)

1999 (3)

1998 (3)

1997 (3)

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

1996 (1)

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

1994 (2)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

P. St. J. Russell and J.-L. Archambault, “Field microstructures and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media,” J. Phys. (Paris) III 4, 2471–2491 (1994).

1992 (1)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

1991 (1)

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

1990 (3)

1989 (2)

1987 (1)

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

1986 (1)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Aceves, A. B.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Aggarwal, I. D.

Agrawal, G. P.

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
[CrossRef]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

Aitchison, J. S.

Archambault, J.-L.

P. St. J. Russell and J.-L. Archambault, “Field microstructures and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media,” J. Phys. (Paris) III 4, 2471–2491 (1994).

Asobe, M.

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[CrossRef]

Bingham, R.

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

Broderick, N. G. R.

Cao, X. D.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Cheong, S.-W.

De La Rue, R. M.

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Hwang, H. Y.

Kaplan, A. E.

Katsufuji, T.

Krauss, T. F.

Law, C. T.

Lenz, G.

Lines, M. E.

Liou, L. W.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Litchinitser, N. M.

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Logvin, Yu. A.

Luther, G. G.

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of collinear waves,” J. Opt. Soc. Am. B 7, 1125–1141 (1990).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

McKinstrie, C. J.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of collinear waves,” J. Opt. Soc. Am. B 7, 1125–1141 (1990).
[CrossRef]

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

Millar, P.

Patterson, D. B.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Richardson, D. J.

Russell, P. St. J.

P. St. J. Russell and J.-L. Archambault, “Field microstructures and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media,” J. Phys. (Paris) III 4, 2471–2491 (1994).

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

Sanghera, J. S.

Silberberg, Y.

Sipe, J. E.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Slusher, R. E.

Spälter, S.

Strasser, T. A.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Volkov, V. M.

Zimmermann, J.

Electron. Lett. (1)

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

J. Lightwave Technol. (2)

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
[CrossRef]

J. Mod. Opt. (1)

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. (Paris) III (1)

P. St. J. Russell and J.-L. Archambault, “Field microstructures and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media,” J. Phys. (Paris) III 4, 2471–2491 (1994).

Opt. Commun. (1)

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Opt. Fiber Technol. (1)

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[CrossRef]

Opt. Lett. (4)

Phys. Fluids B (1)

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

Phys. Rev. A (1)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulation instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Phys. Scr. (1)

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

Prog. Opt. (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Other (3)

A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, Berlin, 1975).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

S. Spälter, G. Lenz, H. Y. Hwang, J. Zimmermann, S.-W. Cheong, T. Katsufuji, R. E. Slusher, “Nonlinear optics in chalcogenide waveguides,” presented at the 1999 Annual Meeting of the Optical Society of America.

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Figures (10)

Fig. 1
Fig. 1

Schematic illustration of a bulk nonlinear periodic structure.

Fig. 2
Fig. 2

Dispersion curves for an infinite linear periodic structure.

Fig. 3
Fig. 3

(a) Instability domains in K space and (b) the corresponding gain for the top of the photonic bandgap (κ=10 cm-1, P=8 cm-1).

Fig. 4
Fig. 4

Instability gain under conditions identical to those of Fig. 3, except that κ=0 (no grating, no dispersion).

Fig. 5
Fig. 5

(a) Instability domains and (b) the corresponding gain for the bottom of the photonic bandgap (κ=10 cm-1, P=20 cm-1).

Fig. 6
Fig. 6

MI gain in the K space for values of f and P as shown. In all cases κ=10 cm-1.

Fig. 7
Fig. 7

MI gain in K space for values of f and P as shown.

Fig. 8
Fig. 8

FFWM-induced gain in K space: (a) P=8 cm-1, f=-1, κ=10 cm-1; (b) P=8 cm-1, f=-1, κ=0; (c) P=20 cm-1, f=1, κ=10 cm-1; (d) P=20 cm-1, f=1, κ=0.

Fig. 9
Fig. 9

BFWM-induced gain in K space: (a) P=8 cm-1, f=-1, κ=10 cm-1; (b) P=8 cm-1, f=-1, κ=0; (c) P=20 cm-1, f=1, κ=10 cm-1; (d) P=20 cm-1, f=1, κ=0.

Fig. 10
Fig. 10

Spatial period Λs (solid curves) and temporal period T (dashed curves) corresponding to maximum MI gain as functions of |Ω| for (a) anomalous dispersion and (b) normal dispersion with P=8 cm-1.

Equations (63)

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

E=[E+ exp(+ikBz)+E- exp(-ikBz)]exp(-iωBt)
i E+z+i 1V E+t+12kB 2E+x2+2E+y2+κE-
+Γ(|E+|2+2|E-|2)E+=0,
-i E-z+i 1V E-t+12kB 2E-x2+2E-y2+κE+
+Γ(|E-|2+2|E+|2)E-=0,
E+=af2+1 exp[i(Qz-ΩVt)]a+ exp[i(Qz-ΩVt)],
E-=aff2+1 exp[i(Qz-ΩVt)]a- exp[i(Qz-ΩVt)],
Ω=-κ2(f-1+f)-32Γa2,
Q=-κ2(f-1-f)-12Γa2 1-f21+f2.
E±=[a±+ε±(x, y, z, t)]exp[i(Qz-ΩVt)],
i ε+z+i 1V ε+t+12kB 2ε+x2+2ε+y2+κε--fκε+
+G[(ε++ε+*)+2 f(ε-+ε-*)]=0,
-i ε-z+i 1V ε-t+12kB 2ε-x2+2ε-y2+κε+
-f-1κε-+G[2 f(ε++ε+*)+f2(ε-+ε-*)]=0,
ε±(x, y, z, t)=b± exp[i(K·r-ω Vt)]+c± exp[-i(K·r-ω Vt)],
(ω-k-κf+G-kt2)b++Gc++(κ+2 fG)b-
+2 fGc-=0,
Gb++(-ω+k-κf+G-kt2)c++2 fGb-
+(κ+2 fG)c-=0,
(κ+2 fG)b++2 fGc++(ω+k-κf-1
+f2G-kt2)b-+f2Gc-=0,
2 fGb++(κ+2 fG)c++f2Gb-
+(-ω-k-κf-1+f2G-kt2)c-=0,
ω2=2κ2-2Gκ+k2-2κkt2-2Gkt2+kt4±2(κ4-2Gκ3+G2κ2+k2κ2+2Gk2κ-2kt2κ3+6Gkt2κ2-4G2kt2κ-2κk2kt2-2Gk2kt2+κ2kt4-4Gκkt4+4G2kt4+k2kt4)1/2.
k(1)=±(12Gκ-6Gkt2-2κkt2+kt4)1/2
k(2)=±(2Gkt2-2κkt2+kt4)1/2.
k(3)=±κ4-2Gκ3+G2κ2-2kt2κ3+6Gκ2kt2-4G2κkt2+κ2kt4-4Gκkt4+4G2kt4-κ2-2Gκ+2κkt2+2Gkt2-kt41/2,
ω2=k2-2Gkt2+kt4±2(4G2kt2-2G2k2kt2+k2kt4)1/2.
k(4)=±12Gκ,
ω2=2κ2+2Gκ+k2+2kt2κ-2Gkt2+kt4±2(κ4+2Gκ3+G2κ2+k2κ2-2Gκk2+2κ3kt2+6Gκ2kt2+4G2κkt2+2κk2kt2-2Gk2kt2+κ2kt4+4Gκkt4+4G2kt4+k2kt4)1/2.
k(5)=±(-12Gκ-6Gkt2+2κkt2+kt4)1/2
k(6)=±(2Gkt2+2κkt2+kt4).
k(7)=±κ4+2Gκ3+G2κ2+2kt2κ3+6Gκ2kt2+4G2κkt2+κ2kt4+4Gκkt4+4G2kt4-κ2+2Gκ-2κkt2+2Gkt2-kt41/2.
(ω-ω1)(ω-ω2)(ω-ω3)(ω-ω4)
-12 fG2kt2(κ+κf2+fkt2)-G[-12κfk2
-2k2kt2-2 f2k2kt2+4(f2-1)kkt2ω]
-2Gf-2[κ2kt2+5κ2f2kt2+5κ2f4kt2+κ2f6kt2
+2κfkt4+6κf3kt4+2κf5kt4+f2kt6+f4kt6
+2κf3ω2-f2kt2ω2-f4kt2ω2]=0,
ω1,2=κ2(f-1+f)±[k2-k(f-1-f)+¼(f-1+f)2]1/2+kt2,
ω3,4=-κ2(f-1+f)±[k2+k(f-1-f)+¼(f-1+f)2]1/2-kt2.
ω-k-κf+G-kt2GG-ω+k-κf+G-kt2
=0,
ω=k±κf+kt2κf-2G+kt2.
-κ|f|+kt2>0,-κ|f|-2G+kt2<0.
γ=-κ|f|+kt2κ|f|+2G-kt2.
kt=±G+κ|f|.
γmax=G.
γ=κf+kt22G-κf-kt2.
kt=0κf>G>κf/2,
kt=±G-κfG>κf,
γmax=κf2G-κf,
γmax=G,
ω-k-κf+G-kt22 fG2 fG-ω-k-κf-1+f2G-kt2
=0,
ω=0.5f-1{-fG+f3G-κ+κf2±[(fG-f3G+κ-κf2)2-4f(3f3G2+Gκ+κGf4-fκ2+fGk+f3Gk-κk-κf2k-fk2+fGkt2+f3Gkt2-κkt2-κf2kt2-2 fkkt2-fkt4)]1/2}.
ω2=-3G2+2G(κ-k-kt2)+(-κ+k+kt2)2
f=-1,
ω2=-3G2-2G(κ+k+kt2)+(κ+k+kt2)2
f=1.
γmax=2G,
kt2=G+κ-kf=-1,
kt2=G-κ-kf=1.

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