Abstract

The relaxation oscillations that are present in the output of a high-power free-running diode-pumped Tm-doped silica fiber laser are characterized and modeled. The laser operates on the  3H43H6 quasi-three-level transition and operates efficiently from ∼1.9 to ∼2.0 µm. From the measurements of the  3F43H4 fluorescence after 1.98-µm pumping of a separate, heavily Tm-doped silica fiber, it is established that the  3H4,  3H43H6,  3F4 upconversion process is significant in heavily Tm-doped silica fibers. This process causes the saturable absorption that is indicated by the presence of the relaxation oscillations. A detailed theoretical model that describes the ion-pair dynamics relevant to the Tm-doped silica system is presented. Equations for the steady-state intracavity photon density and for the steady-state population densities of the isolated and paired ions are derived. A linear stability analysis and numerical analysis are also carried out. It is established that, for large emission-to-absorption cross-section ratios that are relevant to Tm-doped silica and for pump rates for which stable output is predicted, the oscillations are weakly damped before the steady state is reached. Best agreement between the experimental and model results is achieved when all the Tm3+ ions within the present fiber are considered paired.

© 1999 Optical Society of America

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References

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  1. D. Marcuse, “Pulsing behavior of a three-level laser with saturable absorber,” IEEE J. Quantum Electron. 29, 2390–2396 (1993).
    [CrossRef]
  2. F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209–213 (1997).
    [CrossRef]
  3. F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
    [CrossRef] [PubMed]
  4. P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
    [CrossRef]
  5. P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
    [CrossRef]
  6. S. Colin, E. Contesse, P. Le Boudec, G. Stéphan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
    [CrossRef] [PubMed]
  7. F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
    [CrossRef]
  8. P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
    [CrossRef]
  9. F. Sanchez and G. Stéphan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
    [CrossRef]
  10. W. H. Loh, “Suppression of self-pulsing behavior in erbium-doped fiber lasers with resonant pumping,” Opt. Lett. 21, 734–736 (1996).
    [CrossRef] [PubMed]
  11. W. H. Loh and J. P. de Sandro, “Suppression of self-pulsing behavior in erbium-doped fiber lasers with resonant pumping: experimental results,” Opt. Lett. 21, 1475–1477 (1996).
    [CrossRef] [PubMed]
  12. L. Luo and P. L. Chu, “Suppression of self-pulsing in an erbium-doped fiber laser,” Opt. Lett. 22, 1174–1176 (1997).
    [CrossRef] [PubMed]
  13. S. D. Jackson and T. A. King, “High-power diode-cladding-pumped Tm-doped silica fiber laser,” Opt. Lett. 23, 1462–1464 (1998).
    [CrossRef]
  14. S. D. Jackson and T. A. King, “High power diode-pumped Tm-doped silica fibre laser in a double-clad pumping arrangement,” in Conference on Lasers and Electro-Optics (CLEO/Europe) (Optical Society of America, Washington, D.C., 1998), paper CTuF1.
  15. R. V. Johnson and J. H. Marburger, “Relaxation oscillations in stimulated Raman and Brillouin scattering,” Phys. Rev. B 4, 1175–1182 (1971).
    [CrossRef]
  16. I. Bar-Joseph, A. A. Freisem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of stimulated Brillouin scattering in single-mode optical fibres,” J. Opt. Soc. Am. B 2, 1606–1611 (1985).
    [CrossRef]
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  18. M. Ding and P. K. Cheo, “Dependence of ion-pair induced self-pulsing in Er-doped fiber lasers on emission to absorption ratio,” IEEE Photon. Technol. Lett. 8, 1627–1629 (1996).
    [CrossRef]
  19. H. L. An, E. Y. B. Pun, H. D. Lui, and X. Z. Lin, “Effects of ion clusters on the performance of a heavily doped erbium-doped fiber laser,” Opt. Lett. 23, 1197–1199 (1998).
    [CrossRef]
  20. A. Wood, Acoustics (Blackie, London, 1960), Chap. 16, p. 411.

1998 (2)

1997 (2)

1996 (5)

1995 (1)

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

1993 (5)

P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

D. Marcuse, “Pulsing behavior of a three-level laser with saturable absorber,” IEEE J. Quantum Electron. 29, 2390–2396 (1993).
[CrossRef]

1985 (1)

1971 (1)

R. V. Johnson and J. H. Marburger, “Relaxation oscillations in stimulated Raman and Brillouin scattering,” Phys. Rev. B 4, 1175–1182 (1971).
[CrossRef]

An, H. L.

Bar-Joseph, I.

Bayon, J.-F.

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
[CrossRef]

Besnard, P.

Cheo, P. K.

M. Ding and P. K. Cheo, “Dependence of ion-pair induced self-pulsing in Er-doped fiber lasers on emission to absorption ratio,” IEEE Photon. Technol. Lett. 8, 1627–1629 (1996).
[CrossRef]

Chu, P. L.

Colin, S.

Contesse, E.

de Sandro, J. P.

Delevaque, E.

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

Ding, M.

M. Ding and P. K. Cheo, “Dependence of ion-pair induced self-pulsing in Er-doped fiber lasers on emission to absorption ratio,” IEEE Photon. Technol. Lett. 8, 1627–1629 (1996).
[CrossRef]

François, P. L.

P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
[CrossRef]

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

François, P.-L.

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

Freisem, A. A.

Jackson, S. D.

Jaouen, C.

Johnson, R. V.

R. V. Johnson and J. H. Marburger, “Relaxation oscillations in stimulated Raman and Brillouin scattering,” Phys. Rev. B 4, 1175–1182 (1971).
[CrossRef]

Kellou, A.

King, T. A.

Le Boudec, P.

S. Colin, E. Contesse, P. Le Boudec, G. Stéphan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
[CrossRef] [PubMed]

P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
[CrossRef]

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

Le Flohic, M.

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

LeBoudec, P.

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

LeFlohic, M.

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

Lichtman, E.

Lin, X. Z.

Loh, W. H.

Lui, H. D.

Luo, L.

Marburger, J. H.

R. V. Johnson and J. H. Marburger, “Relaxation oscillations in stimulated Raman and Brillouin scattering,” Phys. Rev. B 4, 1175–1182 (1971).
[CrossRef]

Marcuse, D.

D. Marcuse, “Pulsing behavior of a three-level laser with saturable absorber,” IEEE J. Quantum Electron. 29, 2390–2396 (1993).
[CrossRef]

Pun, E. Y. B.

Sanchez, F.

F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209–213 (1997).
[CrossRef]

S. Colin, E. Contesse, P. Le Boudec, G. Stéphan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
[CrossRef] [PubMed]

F. Sanchez and G. Stéphan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

Stéphan, G.

S. Colin, E. Contesse, P. Le Boudec, G. Stéphan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
[CrossRef] [PubMed]

F. Sanchez and G. Stéphan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

P. Le Boudec, C. Jaouen, P. L. François, J.-F. Bayon, F. Sanchez, P. Besnard, and G. Stéphan, “Antiphase dynamics and chaos in self-pulsing erbium-doped fiber laser,” Opt. Lett. 18, 1890–1892 (1993).
[CrossRef]

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Stéphan, G. M.

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

Waarts, R. G.

IEEE J. Quantum Electron. (2)

D. Marcuse, “Pulsing behavior of a three-level laser with saturable absorber,” IEEE J. Quantum Electron. 29, 2390–2396 (1993).
[CrossRef]

F. Sanchez, M. LeFlohic, G. M. Stéphan, P. LeBoudec, and P.-L. François, “Quasi-periodic route to chaos in erbium-doped fiber laser,” IEEE J. Quantum Electron. 31, 481–488 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. Ding and P. K. Cheo, “Dependence of ion-pair induced self-pulsing in Er-doped fiber lasers on emission to absorption ratio,” IEEE Photon. Technol. Lett. 8, 1627–1629 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (7)

Opt. Quantum Electron. (2)

P. Le Boudec, M. Le Flohic, P. L. François, F. Sanchez, and G. Stéphan, “Self-pulsing in Er3+-doped fibre laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

P. Le Boudec, P. L. François, E. Delevaque, J.-F. Bayon, F. Sanchez, and G. Stéphan, “Influence of ion pairs on the dynamical behaviour of Er3+-doped fibre lasers,” Opt. Quantum Electron. 25, 501–507 (1993).
[CrossRef]

Phys. Rev. A (1)

F. Sanchez, P. Le Boudec, P. L. François, and G. Stéphan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. V. Johnson and J. H. Marburger, “Relaxation oscillations in stimulated Raman and Brillouin scattering,” Phys. Rev. B 4, 1175–1182 (1971).
[CrossRef]

Phys. Rev. E (1)

F. Sanchez and G. Stéphan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

Other (3)

A. Wood, Acoustics (Blackie, London, 1960), Chap. 16, p. 411.

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

S. D. Jackson and T. A. King, “High power diode-pumped Tm-doped silica fibre laser in a double-clad pumping arrangement,” in Conference on Lasers and Electro-Optics (CLEO/Europe) (Optical Society of America, Washington, D.C., 1998), paper CTuF1.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the free-running Tm-doped silica fiber laser. The inset shows a simplified energy-level diagram for 790 nm-pumping of Tm3+.

Fig. 2
Fig. 2

Measured rf spectra for a 0.675-m-long fiber laser cavity with variation in the normalized pump rate r that had (a) Fresnel reflection for feedback only (laser wavelength, 1885 nm; cavity decay rate, 9.9×108 s-1) and (b) a dichroic mirror at the input end to the fiber cavity (laser wavelength, 1913 nm; cavity decay rate, 4.9×108 s-1).

Fig. 3
Fig. 3

Measured absorption and emission cross sections as a function of the double-clad Tm-doped silica fiber laser wavelength. Inset, the  3F43H4 fluorescence spectrum measured when a single-clad Tm-doped silica fiber is pumped at 1.98 µm.

Fig. 4
Fig. 4

Simplified energy-level diagrams for isolated ions and paired ions, showing the major processes and nomenclature used in the ion-pair model.

Fig. 5
Fig. 5

Numerical calculations of il for variation of normalized pump rate for the Er-doped silica fiber laser (where N0=1025 m-3, L=4 m, and R1R2=0.04) for (a) γ=1, x=0.1, y=0.2, and σa=5.6×10-25 m2 (inset, the long-term temporal behavior of il when r=3), (b) γ=1.5, x=0.1, y=0.2, and σa=3.0×10-25 m2 (inset, the amplitude envelope for il when r=3), and (c) γ=1, x=0.1, y=0.2, σa=5.6×10-25 m2, and τ2 reduced to 340 µs.

Fig. 6
Fig. 6

Calculated values from the linear stability analysis for the Er-doped silica fiber laser system of (a) Re(λ) when γ and τ2 are varied and (b) Im(λ) when γ and τ2 are varied. Here y=0.2, N0=1025 m-3, L=4 m, and R1R2=0.04.

Fig. 7
Fig. 7

Calculated values of (a) Re(λ) and (b) Im(λ) for L=0.675 m, R1R2=0.04, r=3.0, x=0.1, and σa=7.67×10-26 m2.

Fig. 8
Fig. 8

Fourier transform of the calculated temporal behavior of il for the Tm-doped silica fiber laser system with variation of normalized pump rate r. In this case, L=0.675 m, R1R2=0.04, γ=6.52, σa=7.67×10-26 m2, y=0.08, and x=0.5.

Fig. 9
Fig. 9

Measured angular frequency of the relaxation oscillations for various fiber laser configurations. λ=1885 nm for a 0.675-m-long fiber used with a mirror reflectivity product R1R2=0.0016 and γ=4.35. λ=1913 mm for a 0.675-m-long fiber used with a mirror reflectivity product R1R2=0.04 and γ=6.52. λ=1967 nm for a 2.84-m long fiber used with a mirror reflectivity product R1R2=0.0016 and γ=19.6. λ=1996nm for a 5.11-m long fiber used with a mirror reflectivity product R1R2=0.04.

Fig. 10
Fig. 10

Oscilloscope trace of fiber laser output when the proximal end of the fiber was gradually misaligned (from A to C) with the focus of the pump source. The fiber length was 2.6 m, R1R2=0.0016, and 6.4 W of pump power was launched into the fiber. The free end length of the fiber was 38.5 mm.

Tables (1)

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Table 1 Values of Parameters Used for the Numerical Calculations

Equations (39)

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dnidt=2Λ-a2(ni+1)-il[ni(γ+1)+(γ-1)],
dn+dt=a12(1-n+)-a222(n++n-)+yil2[(2-3n+)(γ+1)-n-(γ-1)],
dn-dt=2Λ-a12(1-n+)-a222(n++n-)-yil2[(2-n+)(γ-1)+n-(γ+1)],
dildt=-il+(1-2x)2Ail[ni(γ+1)+(γ-1)]+xBil2[(2-n+)(γ-1)+n-(γ+1)].
Λth=a2a22a12[1+(1-2x)A+xB]γa22a2xB+(γ+1)a12[a22(1-2x)A+a2xB].
dnidt=2Λ-(ni+1)(a2+ilγ).
v=0.56vkl2,
A1il¯3+B1il¯2+C1il¯+D=0.
A1=y2(γ+1)[(γ-1)2+3(γ+1)2],
B1=y2(γ+1)23a21-(γ-1)xB2+3(γ+1)(1-2x)A2(a2-2Λ)+y2(γ-1)2 (1-2x)A2[(γ+1)(a2-2Λ)+a2(γ-1)]+y(γ+1)2{xB[(γ+1)(a12+a22-6Λ)-2a22(γ-1)]+2a12+4a22}+y(γ-1)2xB(γ+1)(a22-a12-2Λ)-2y(γ+1)(γ-1)a12,
C1=y(γ+1)2[(1-2x)A(2a22a2+a12a2-4a22Λ-2a12Λ)+xBa2(a22+a12-6Λ)]+y(γ-1)2[xBa2(a22-2Λ-a12)-(1-2x)Aa2a12]+2y(γ+1)(γ-1)×[(1-2x)A(-a22a2-a12Λ)-a22a2xB]+2(γ+1)2xB(a22a12-Λa22-2a12Λ)-2(γ+1)(γ-1)xBa22(a12+Λ)+2y(γ+1)a2(a12+2a22)+2y(γ-1)a2a12-4(γ+1)a12a22,
D1=2(γ+1)[xBa2(a22a12-a22Λ-4a12Λ)+a12a22(1-2x)A(a2-2Λ)]-2a22a2(γ-1)[xB(Λ+a12)+a12(1-2x)A]+4a12a22a2.
ni¯=2Λ-a2-il¯(γ-1)a2+il¯(γ+1).
n+¯=A2il¯2+B2il¯+C2xB[a2+il¯(γ+1)][a12(γ+1)-a22γ].
A2=y(γ+1)2,
B2=(γ+1)2xB(a12-2Λ)+(1-2x)×Aya22-Λ+(γ+1)(a22+ya2)-(γ+1)×(γ-1)ya2(1-2x)A2+a22xB,
C2=(γ+1)xBa2(a12-2Λ)+(1-2x)×Aa22a22-Λ-(γ-1)a2a22×(1-2x)A2+xB+a2a22.
n-¯=A3il¯2+B3il¯+C3xB[a2+il¯(γ+1)][a12(γ+1)-a22γ].
A3=y(γ+1)(γ-1),
B3=(γ+1)(γ-1)xB(a22+2Λ-a12)+(1-2x)Aya22-Λ+(γ+1)(2a12-a22)+a2(γ-1)y-a2(γ-1)2y (1-2x)A2,
C3=(γ+1)a22a2xB+(1-2x)Aa12a2-a22a22-2Λa12+a22Λ+(γ-1)xBa2(-2Λ-a2)+(1-2x)Aa2a222-a12+2a2a12.
det(L-λI)=0,
L=z1100z140z22z23z240z32z33z34z41z42z430.
z11=-a2-il¯(γ+1),
z14=-ni¯(γ+1)-(γ-1),
z22=-a12-a222-3yil¯2(γ+1),
z23=-a222-yil¯2(γ-1),
z24=y2[(2-3n+¯)(γ+1)-n-¯(γ-1)],
z32=a12-a222+yil¯2(γ-1),
z33=-a222-yil¯2(γ+1),
z34=-y2[(2-n+¯)(γ-1)+n-¯(γ+1)],
z41=(1-2x)2Ail¯(γ+1),
z42=xBil¯2(γ-1),
z43=xBil¯2(γ+1).
λ4+A4λ3+B4λ2+C4λ+D4=0.
A4=-z11-z22-z33,
B4=z11z33+z11z22-z41z14+z22z33-z34z43-z32z23-z42z24,
C4=-z11z22z33+z11z34z43+z11z32z23+z22z34z43-z42z23z34+z41z14z22+z11z42z24-z32z24z43,
D4=z11z42z23z34-z41z14z22z33+z41z32z14z23+z11z32z24z43-z11z42z24z33-z11z22z43z34.

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