Abstract

The passive Q-switching operation in a laser with a saturable absorber consists of an operation with high-intensity spikes in the output power. It is known that this particular regime corresponds mathematically to a large-amplitude time-periodic solution of the laser equations. In this paper we propose a new asymptotic method to construct this solution. We show how the time-periodic solution appears from an infinite-period solution (homo-clinic solution) and analyze the behavior of the maximum intensity and the period as a function of a control parameter. This asymptotic method is based on a recent evaluation of the parameters for CO2 lasers with different absorbers [ E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, Appl. Phys. B 30, 57 ( 1983)]. The relative magnitudes obtained by Arimondo et al. suggest a new approximation of the laser equations.

© 1988 Optical Society of America

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References

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  1. N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
    [CrossRef]
  2. P. Mandel, “Laser with a saturable absorber” (submitted to Progress in Optics, E. Wolf, ed.).
  3. E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
    [CrossRef]
  4. J. C. Antoranz, J. Gea, and M. G. Velarde, “Oscillatory phenomena and Q-switching in a model for a laser with a saturable absorber,” Phys. Rev. Lett. 47, 1895–1898 (1981).
    [CrossRef]
  5. P. Mandel and T. Erneux, “Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber. I.” Phys. Rev. A 30, 1893–1901 (1984).
    [CrossRef]
  6. E. Hofleich-Abate and F. Hofelich, “Time behavior of a laser with a saturable absorber as Q-switch,” J. Appl. Phys. 39, 4823–4827 (1968).
    [CrossRef]
  7. T. Erneux, “Analytic studies of a laser with a saturable absorber,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds.,Vol. 4 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1986), pp. 99–110.
  8. J. P. Keener, “Infinite period bifurcation and global bifurcation branches,” SIAM J. Appl. Math.41, 127–144 (1981).
    [CrossRef]
  9. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Pergamon, New York, 1966).
  10. N. Kopell and L. N. Howard, “Bifurcation and trajectories joining critical points,” Adv. Math. 18, 306–358 (1975).
    [CrossRef]
  11. T. Erneux and P. Mandel, “Bifurcation phenomena in a laser with saturable absorber II.” Z. Phys. B 44, 365–374 (1981).
    [CrossRef]
  12. J. F. Magnan and E. L. Reiss, “Double diffusive convection and λ-bifurcation,” Phys. Rev. A 31, 1841–1854 (1985).
    [CrossRef] [PubMed]
  13. W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
    [CrossRef]
  14. J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields(Springer-Verlag, New York, 1983).
  15. T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
    [CrossRef]
  16. G. Dangelmayr, D. Armbruster, and M. Neveling (University of Tübingen, Federal Republic of Germany), “A codimension three bifurcation for the laser with saturable absorber,” preprint.
  17. E. J. Doedel, “auto, a program for the automatic bifurcation analysis of autonomous systems,” Congr. Numer. 30, 265–284 (1981).

1987 (1)

T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
[CrossRef]

1986 (1)

W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
[CrossRef]

1985 (2)

J. F. Magnan and E. L. Reiss, “Double diffusive convection and λ-bifurcation,” Phys. Rev. A 31, 1841–1854 (1985).
[CrossRef] [PubMed]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
[CrossRef]

1984 (1)

P. Mandel and T. Erneux, “Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber. I.” Phys. Rev. A 30, 1893–1901 (1984).
[CrossRef]

1983 (1)

E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
[CrossRef]

1981 (3)

J. C. Antoranz, J. Gea, and M. G. Velarde, “Oscillatory phenomena and Q-switching in a model for a laser with a saturable absorber,” Phys. Rev. Lett. 47, 1895–1898 (1981).
[CrossRef]

T. Erneux and P. Mandel, “Bifurcation phenomena in a laser with saturable absorber II.” Z. Phys. B 44, 365–374 (1981).
[CrossRef]

E. J. Doedel, “auto, a program for the automatic bifurcation analysis of autonomous systems,” Congr. Numer. 30, 265–284 (1981).

1975 (1)

N. Kopell and L. N. Howard, “Bifurcation and trajectories joining critical points,” Adv. Math. 18, 306–358 (1975).
[CrossRef]

1968 (1)

E. Hofleich-Abate and F. Hofelich, “Time behavior of a laser with a saturable absorber as Q-switch,” J. Appl. Phys. 39, 4823–4827 (1968).
[CrossRef]

Abraham, N. B.

Andronov, A. A.

A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Pergamon, New York, 1966).

Antoranz, J. C.

J. C. Antoranz, J. Gea, and M. G. Velarde, “Oscillatory phenomena and Q-switching in a model for a laser with a saturable absorber,” Phys. Rev. Lett. 47, 1895–1898 (1981).
[CrossRef]

Arimondo, E.

E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
[CrossRef]

Armbruster, D.

G. Dangelmayr, D. Armbruster, and M. Neveling (University of Tübingen, Federal Republic of Germany), “A codimension three bifurcation for the laser with saturable absorber,” preprint.

Casagrande, F.

E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
[CrossRef]

Dangelmayr, G.

G. Dangelmayr, D. Armbruster, and M. Neveling (University of Tübingen, Federal Republic of Germany), “A codimension three bifurcation for the laser with saturable absorber,” preprint.

Davis, S. H.

W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
[CrossRef]

Doedel, E. J.

E. J. Doedel, “auto, a program for the automatic bifurcation analysis of autonomous systems,” Congr. Numer. 30, 265–284 (1981).

Erneux, T.

T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
[CrossRef]

P. Mandel and T. Erneux, “Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber. I.” Phys. Rev. A 30, 1893–1901 (1984).
[CrossRef]

T. Erneux and P. Mandel, “Bifurcation phenomena in a laser with saturable absorber II.” Z. Phys. B 44, 365–374 (1981).
[CrossRef]

T. Erneux, “Analytic studies of a laser with a saturable absorber,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds.,Vol. 4 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1986), pp. 99–110.

Gea, J.

J. C. Antoranz, J. Gea, and M. G. Velarde, “Oscillatory phenomena and Q-switching in a model for a laser with a saturable absorber,” Phys. Rev. Lett. 47, 1895–1898 (1981).
[CrossRef]

Glorieux, P.

E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
[CrossRef]

Guckenheimer, J.

J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields(Springer-Verlag, New York, 1983).

Hofelich, F.

E. Hofleich-Abate and F. Hofelich, “Time behavior of a laser with a saturable absorber as Q-switch,” J. Appl. Phys. 39, 4823–4827 (1968).
[CrossRef]

Hofleich-Abate, E.

E. Hofleich-Abate and F. Hofelich, “Time behavior of a laser with a saturable absorber as Q-switch,” J. Appl. Phys. 39, 4823–4827 (1968).
[CrossRef]

Holmes, P. J.

J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields(Springer-Verlag, New York, 1983).

Howard, L. N.

N. Kopell and L. N. Howard, “Bifurcation and trajectories joining critical points,” Adv. Math. 18, 306–358 (1975).
[CrossRef]

Jayakumar, P. K.

T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
[CrossRef]

Kath, W. L.

W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
[CrossRef]

Keener, J. P.

J. P. Keener, “Infinite period bifurcation and global bifurcation branches,” SIAM J. Appl. Math.41, 127–144 (1981).
[CrossRef]

Khaikin, S. E.

A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Pergamon, New York, 1966).

Kopell, N.

N. Kopell and L. N. Howard, “Bifurcation and trajectories joining critical points,” Adv. Math. 18, 306–358 (1975).
[CrossRef]

Lugiato, L. A.

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
[CrossRef]

E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
[CrossRef]

Magnan, J. F.

T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
[CrossRef]

J. F. Magnan and E. L. Reiss, “Double diffusive convection and λ-bifurcation,” Phys. Rev. A 31, 1841–1854 (1985).
[CrossRef] [PubMed]

Mandel, P.

P. Mandel and T. Erneux, “Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber. I.” Phys. Rev. A 30, 1893–1901 (1984).
[CrossRef]

T. Erneux and P. Mandel, “Bifurcation phenomena in a laser with saturable absorber II.” Z. Phys. B 44, 365–374 (1981).
[CrossRef]

P. Mandel, “Laser with a saturable absorber” (submitted to Progress in Optics, E. Wolf, ed.).

Narducci, L. M.

Neveling, M.

G. Dangelmayr, D. Armbruster, and M. Neveling (University of Tübingen, Federal Republic of Germany), “A codimension three bifurcation for the laser with saturable absorber,” preprint.

Olmstead, W. E.

W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
[CrossRef]

Reiss, E. L.

T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
[CrossRef]

J. F. Magnan and E. L. Reiss, “Double diffusive convection and λ-bifurcation,” Phys. Rev. A 31, 1841–1854 (1985).
[CrossRef] [PubMed]

Rosenblat, S.

W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
[CrossRef]

Velarde, M. G.

J. C. Antoranz, J. Gea, and M. G. Velarde, “Oscillatory phenomena and Q-switching in a model for a laser with a saturable absorber,” Phys. Rev. Lett. 47, 1895–1898 (1981).
[CrossRef]

Vitt, A. A.

A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Pergamon, New York, 1966).

Adv. Math. (1)

N. Kopell and L. N. Howard, “Bifurcation and trajectories joining critical points,” Adv. Math. 18, 306–358 (1975).
[CrossRef]

Appl. Phys. B (1)

E. Arimondo, F. Casagrande, L. A. Lugiato, and P. Glorieux, “Repetitive passive Q-switching and bistability in lasers with saturable absorbers,” Appl. Phys. B 30, 57–77 (1983).
[CrossRef]

Congr. Numer. (1)

E. J. Doedel, “auto, a program for the automatic bifurcation analysis of autonomous systems,” Congr. Numer. 30, 265–284 (1981).

J. Appl. Phys. (1)

E. Hofleich-Abate and F. Hofelich, “Time behavior of a laser with a saturable absorber as Q-switch,” J. Appl. Phys. 39, 4823–4827 (1968).
[CrossRef]

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

Phys. Rev. A (2)

P. Mandel and T. Erneux, “Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber. I.” Phys. Rev. A 30, 1893–1901 (1984).
[CrossRef]

J. F. Magnan and E. L. Reiss, “Double diffusive convection and λ-bifurcation,” Phys. Rev. A 31, 1841–1854 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. C. Antoranz, J. Gea, and M. G. Velarde, “Oscillatory phenomena and Q-switching in a model for a laser with a saturable absorber,” Phys. Rev. Lett. 47, 1895–1898 (1981).
[CrossRef]

SIAM J. Appl. Math. (2)

W. E. Olmstead, S. H. Davis, S. Rosenblat, and W. L. Kath, “Bifurcation with memory,” SIAM J. Appl. Math. 46, 171–188 (1986).
[CrossRef]

T. Erneux, E. L. Reiss, J. F. Magnan, and P. K. Jayakumar, “Nonlinear stability control and λ-bifurcation,” SIAM J. Appl. Math. 47, 1163–1176 (1987).
[CrossRef]

Z. Phys. B (1)

T. Erneux and P. Mandel, “Bifurcation phenomena in a laser with saturable absorber II.” Z. Phys. B 44, 365–374 (1981).
[CrossRef]

Other (6)

J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields(Springer-Verlag, New York, 1983).

G. Dangelmayr, D. Armbruster, and M. Neveling (University of Tübingen, Federal Republic of Germany), “A codimension three bifurcation for the laser with saturable absorber,” preprint.

P. Mandel, “Laser with a saturable absorber” (submitted to Progress in Optics, E. Wolf, ed.).

T. Erneux, “Analytic studies of a laser with a saturable absorber,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds.,Vol. 4 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1986), pp. 99–110.

J. P. Keener, “Infinite period bifurcation and global bifurcation branches,” SIAM J. Appl. Math.41, 127–144 (1981).
[CrossRef]

A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Pergamon, New York, 1966).

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Figures (6)

Fig. 1
Fig. 1

(a) Bifurcation diagram of the steady and time-periodic solutions of the LSA rate equations. Solid and dashed lines represent stable and unstable solutions, respectively. As corresponds to a bifurcation point from the zero intensity solutions to a subcritical branch of steady states. It also corresponds to the homoclinic solution from which a large-amplitude time-periodic solution emerges. AH and Ac denote the Hopf bifurcation point and a limit point of the periodic solutions, respectively. The values of the parameters are = − 3.4375, δ = δ ¯ = 0.04, and a = 5. (b) Bifurcation diagram of the steady-state solutions only. The branch of nonzero intensity states appears at A = As and has a limit point at a value below As and a Hopf bifurcation point at A = Ah.

Fig. 2
Fig. 2

(a) High-intensity pulses corresponding to PQS. I(t) has been obtained by solving numerically Eqs. (2.1) with A = As + 0.2, δ = δ = 0.04, = −3.4375, and a = 5. (b) Atomic inversion density of the active atoms D(t) as a function of t.

Fig. 3
Fig. 3

Phase-plane projection of the limit cycle. D(t) is represented as a function of I(t) for the same values of the parameters as in Fig. 2. Q and P correspond to the points on the D axis where I(t) suddenly increases and decreases, respectively. The arrows indicate that the time evolution is clockwise. The two dots correspond to two singular points: (I, D) = (0, 1) is a saddle point and (I, D) = (2.76, 0.27) is an unstable focus.

Fig. 4
Fig. 4

Period of the oscillations. Near the homoclinic solution located at A = As, the period of the oscillations is given in first approximation by Pe ∼ τ*/(αδ) = 0[(αδ)−1], where τ* satisfies Eq. (5.7) and α = AAs. The figure represents τ* as a function of − for different values of a = 3, 5, 7. All curves emerge from the − axis at − = F = (a − 1)−1 (a > 1). As − → ∞, τ* → 1.

Fig. 5
Fig. 5

Maximum intensity. Near the homoclinic solution located at A = As, the maximum intensity Im is in first approximation given by Imδ−1JM(α), where Jm is defined by Eq. (5.14). In the figure, we represent Jmj0 as a function of − for several values of a = 3, 5, 7). j0 is defined by Eq. (5.17). All curves emerge from the −axis at − = F = (a − 1) −1 (a > 1).

Fig. 6
Fig. 6

Exact and approximate limit cycles. The exac (denoted by N) has been obtained numerically by solvir for the same values of the parameters as in Figs. 2 an approximate limit cycle (denoted by A) has been obta asymptotic analysis of Eqs. (2.1) in the limit δ = δ ¯ 0 an Section 6.

Equations (75)

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I t = 2 I ( 1 + A D + D ¯ ) , D t = δ [ 1 D ( 1 + I ) ] , D ¯ t = δ ¯ [ 1 D ¯ ( 1 + a I ) ] .
δ = δ ¯ .
A s = 1 , A H = a + O ( δ ) .
a > 1 , < ( a 1 ) 1 ,
τ = δ t
I ( t , τ , δ ) = δ I 1 ( t , τ ) + δ 2 I 2 ( t , τ ) + ,
D ( t , τ , δ ) = D 0 ( t , τ ) + δ D 1 ( t , τ ) + ,
D ¯ ( t , τ , δ ) = D ¯ 0 ( t , τ ) + δ D ¯ 1 ( t , τ ) + ,
D 0 t = D ¯ 0 t = 0 ,
D 1 t = D 0 τ + 1 D 0 ,
D ¯ 1 t = D ¯ 0 τ + 1 D ¯ 0 ,
I 1 t = 2 I 1 ( 1 + A D 0 + D ¯ 0 ) .
D 0 = D 0 ( τ ) , D ¯ 0 = D ¯ 0 ( τ ) .
D 0 τ = 1 D 0 , D ¯ 0 τ = 1 D ¯ 0 .
D 0 = 1 + B e τ , D ¯ 0 = 1 + B ¯ e τ ,
I 1 ( t ) = C exp [ 2 α t 2 δ ( A B + B ¯ ) e δ t ]
I 1 ( τ ) = C exp { 2 δ [ α τ ( A B + B ¯ ) e τ } ,
α = A A s = 1 + A +
K = C exp [ 2 δ ( A B + B ¯ ) ] ,
I 1 ( τ ) = K exp { 2 δ [ α τ ( A B + B ¯ ) ( e τ 1 ) ] } .
I 1 ( S ) K exp [ 2 ( α + A B + B ¯ ) S ]
α + A B + B ¯ < 0 ,
α τ c ( A B + B ¯ ) [ exp ( τ c ) 1 ] = 0 .
α + ( A B + B ¯ ) exp ( τ c ) > 0 ,
Q : I = 0 , D = 1 + B exp ( τ c ) , D ¯ = 1 + B ¯ exp ( τ c ) ; P : I = 0 , D = 1 + B , D ¯ = 1 + B ¯ .
T = ( τ τ c ) δ 1 = t τ c δ 1 ,
I ( T , δ ) = δ 1 J 0 ( T ) + J 1 ( T ) + ,
D ( T , δ ) = G 0 ( T ) + δ G 1 ( T ) + ,
D ¯ ( T , δ ) = G ¯ 0 ( T ) + δ ¯ G ¯ 1 ( T ) + ,
J 0 T = 2 J 0 ( 1 + A G 0 + G ¯ 0 ) ,
G 0 T = G 0 J 0 ,
G ¯ 0 T = a G ¯ 0 J 0 .
J 0 0 , G 0 1 + B exp ( τ c ) , G 0 1 + B ¯ exp ( τ c ) as T
J 0 0 , G 0 1 + B , G ¯ 0 1 + B ¯ as T .
W ( T ) exp [ T J 0 ( s ) d s ] .
G 0 = c 1 W , G ¯ 0 = c 2 W a ,
J 0 T = 2 J 0 ( 1 + A c 1 W + c 2 W a ) .
W T = W J 0 .
d J 0 d W = 2 W ( 1 + A c 1 W + c 2 W a ) .
J 0 = 2 ln W 2 A c 1 W 2 a c 2 W a + c 3 ,
c 1 = 1 + B exp ( τ c ) , c 2 = 1 + B ¯ exp ( τ c ) , c 3 = 2 A c 1 + 2 a c 2 ,
c 1 W 1 + B , c 2 W 0 a = 1 + B ¯ , 2 ( ln W 0 A c 1 W 0 a c 2 W 0 a ) + c 3 = 0 ,
W 0 = W ( ) = exp [ + J 0 ( s ) d s ] .
W 0 [ 1 + B exp ( τ c ) ] = 1 + B ,
W 0 a [ 1 + B ¯ exp ( τ c ) ] = 1 + B ¯ ,
ln W 0 A [ 1 + B exp ( τ c ) ] ( W 0 1 ) a [ 1 + B ¯ exp ( τ c ) ] ( W 0 a 1 ) = 0 .
Pe ( δ ) τ c / δ .
1 + A c 1 W M + c 2 W M a = 0 , J M = 2 ln W M 2 A c 1 W M 2 a c 2 W M a + c 3 ,
α ln x + ( x 1 ) [ ( W 0 1 ) A ( 1 W 0 x ) + ( W 0 a 1 ) ( 1 W 0 a x ) ] = 0
ln W 0 + ( x 1 ) [ ( W 0 1 ) ( 1 W 0 x ) + ( W 0 a 1 ) ( 1 W 0 a x ) a ] = 0 ,
x = exp ( τ c ) .
ln W 0 + ( 1 W 0 ) A + ( 1 W 0 a ) a 0 .
ln x 1 α [ ( 1 W 0 ) A + ( 1 W 0 a ) ] .
W 0 = W * + O ( α , e 1 / α )
τ c = ln x = τ * α + O ( 1 , α 1 e 1 / α ) ,
ln W * + ( 1 W * ) ( 1 ) + ( 1 W * a ) a = 0
τ * = ( 1 W * ) ( 1 ) + ( 1 W * a ) .
> F = 1 / ( a 1 ) and a > 1 .
τ * 3 a ( a 1 ) 2 ( F ) 2 .
τ * 1 .
α > 0 or A > A s = 1 .
Pe 1 δ α .
I M = δ 1 J M + O ( 1 ) ,
J M = 2 [ ln W M + A ( 1 W M ) + a ( 1 W M a ) ] + O ( e 1 ) ,
1 + A W M + W M a + O ( e 1 / α ) = 0 .
W M = w 0 + O ( α , e 1 / α ) , J M = j 0 + O ( α , e 1 / α ) .
j 0 ( w 0 ) = 2 [ ln w 0 + ( 1 ) ( 1 w 0 ) + a ( 1 w 0 a ) ] ,
1 + ( 1 ) w 0 + w 0 a = 0 .
J M = J 0 4 ( a 1 ) 2 a 2 ( F ) 3 .
J M = J 0 2 [ ( 1 1 a ) + ln ( 1 ) ] .
I M δ 1 2 [ ( 1 1 a ) + ln ( 1 ) ] .
I ( t ) 0 , D ( t ) 1 + ( W 0 1 ) e δ t D ¯ ( t ) 1 + ( W 0 a 1 ) e δ t
ln ( W 0 ) A ( W 0 1 ) a ( W 0 a 1 ) = 0
τ c = α 1 [ A ( W 0 1 ) + ( W 0 1 ) ] .
I δ 1 2 [ ln ( W ) A ( W 1 ) a ( W a a ) ] , D W , D ¯ W a

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