Abstract

We study the process of second-harmonic generation when the nonlinear crystal is in an active laser cavity. This system is characterized by isolated bifurcations (steady and Hopf) but also by interacting bifurcations. They are studied both analytically and numerically.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Haken, Synergetics, 2nd ed. (Springer-Verlag, Heidelberg, 1978).
    [CrossRef]
  2. L. A. Lugiato, “Theory of optical Instability,” in Progress in Otpics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–216.
    [CrossRef]
  3. E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
    [CrossRef]
  4. C. M. Bowden, M. Ciftan, and H. R. Robl, eds., Optical Bistability (Plenum, New York, 1981).
    [CrossRef]
  5. C. M. Bowden, H. M. Gibbs, and S. L. McCall, eds., Optical Bistability 2 (Plenum, New York, 1984).
    [CrossRef]
  6. N. B. Abraham, L. A. Lugiato, and L. M. Narducci, eds., feature issue on instabilities in optically active media, J. Opt. Soc. Am. B 2, 1–272 (1985).
  7. K. J. McNeil, P. D. Drummond, and D. F. Walls, “Selfpulsing in second harmonic generation,” Opt. Commun. 27, 292 (1978).
    [CrossRef]
  8. P. D. Drummond, K. J. McNeil, and D. F. Walls, “Bistability and photon antibunching in sub/second harmonic generation,” Opt. Commun. 28, 255 (1979).
    [CrossRef]
  9. P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” I, Opt. Acta 27, 321–335 (1980); II, Opt. Acta 28, 211–225 (1981).
    [CrossRef]
  10. D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden, M. Ciftan, and H. R. Robl, eds. (Plenum, New York, 1984).
  11. P. Mandel and T. Erneux, “Amplitude self-modulation of intra-cavity second harmonic generation,” Opt. Acta 29, 7–21 (1982).
    [CrossRef]
  12. C. M. Savage and D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557–561 (1983).
    [CrossRef]
  13. P. Mandel and T. Erneux, Second Harmonic Generation in a Resonant Cavity, Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1983), pp. 132–141.
  14. X.-G. Wu and P. Mandel, “Intracavity second-harmonic generation with a periodic modulation,” J. Opt. Soc. Am. B 2, 1678–1681 (1986).
    [CrossRef]
  15. S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
    [CrossRef]
  16. N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–34 (1985).
    [CrossRef]
  17. P. Mandel, “On the transition from absorptive to dispersive behavior of a laser due to nonlinear losses,” Phys. Lett. A83207–210 (1981).
  18. P. Mandel and T. Erneux, “Laser-Lorenz equations with a time-dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).
    [CrossRef]

1986 (1)

1985 (2)

N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–34 (1985).
[CrossRef]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, eds., feature issue on instabilities in optically active media, J. Opt. Soc. Am. B 2, 1–272 (1985).

1984 (1)

P. Mandel and T. Erneux, “Laser-Lorenz equations with a time-dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).
[CrossRef]

1983 (1)

C. M. Savage and D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557–561 (1983).
[CrossRef]

1982 (2)

P. Mandel and T. Erneux, “Amplitude self-modulation of intra-cavity second harmonic generation,” Opt. Acta 29, 7–21 (1982).
[CrossRef]

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

1981 (1)

P. Mandel, “On the transition from absorptive to dispersive behavior of a laser due to nonlinear losses,” Phys. Lett. A83207–210 (1981).

1980 (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” I, Opt. Acta 27, 321–335 (1980); II, Opt. Acta 28, 211–225 (1981).
[CrossRef]

1979 (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Bistability and photon antibunching in sub/second harmonic generation,” Opt. Commun. 28, 255 (1979).
[CrossRef]

1978 (2)

K. J. McNeil, P. D. Drummond, and D. F. Walls, “Selfpulsing in second harmonic generation,” Opt. Commun. 27, 292 (1978).
[CrossRef]

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

Abraham, E.

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

Abraham, N. B.

Dangoisse, D.

Dembinski, S. T.

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

Drummond, P. D.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” I, Opt. Acta 27, 321–335 (1980); II, Opt. Acta 28, 211–225 (1981).
[CrossRef]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Bistability and photon antibunching in sub/second harmonic generation,” Opt. Commun. 28, 255 (1979).
[CrossRef]

K. J. McNeil, P. D. Drummond, and D. F. Walls, “Selfpulsing in second harmonic generation,” Opt. Commun. 27, 292 (1978).
[CrossRef]

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden, M. Ciftan, and H. R. Robl, eds. (Plenum, New York, 1984).

Erneux, T.

P. Mandel and T. Erneux, “Laser-Lorenz equations with a time-dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).
[CrossRef]

P. Mandel and T. Erneux, “Amplitude self-modulation of intra-cavity second harmonic generation,” Opt. Acta 29, 7–21 (1982).
[CrossRef]

P. Mandel and T. Erneux, Second Harmonic Generation in a Resonant Cavity, Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1983), pp. 132–141.

Glorieux, P.

Haken, H.

H. Haken, Synergetics, 2nd ed. (Springer-Verlag, Heidelberg, 1978).
[CrossRef]

Kossakowski, A.

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

Lugiato, L. A.

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

L. A. Lugiato, “Theory of optical Instability,” in Progress in Otpics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–216.
[CrossRef]

Mandel, P.

X.-G. Wu and P. Mandel, “Intracavity second-harmonic generation with a periodic modulation,” J. Opt. Soc. Am. B 2, 1678–1681 (1986).
[CrossRef]

N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–34 (1985).
[CrossRef]

P. Mandel and T. Erneux, “Laser-Lorenz equations with a time-dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).
[CrossRef]

P. Mandel and T. Erneux, “Amplitude self-modulation of intra-cavity second harmonic generation,” Opt. Acta 29, 7–21 (1982).
[CrossRef]

P. Mandel, “On the transition from absorptive to dispersive behavior of a laser due to nonlinear losses,” Phys. Lett. A83207–210 (1981).

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

P. Mandel and T. Erneux, Second Harmonic Generation in a Resonant Cavity, Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1983), pp. 132–141.

McNeil, K. J.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” I, Opt. Acta 27, 321–335 (1980); II, Opt. Acta 28, 211–225 (1981).
[CrossRef]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Bistability and photon antibunching in sub/second harmonic generation,” Opt. Commun. 28, 255 (1979).
[CrossRef]

K. J. McNeil, P. D. Drummond, and D. F. Walls, “Selfpulsing in second harmonic generation,” Opt. Commun. 27, 292 (1978).
[CrossRef]

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden, M. Ciftan, and H. R. Robl, eds. (Plenum, New York, 1984).

Peplowski, P.

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

Savage, C. M.

C. M. Savage and D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557–561 (1983).
[CrossRef]

Smith, S. D.

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

Walls, D. F.

C. M. Savage and D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557–561 (1983).
[CrossRef]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” I, Opt. Acta 27, 321–335 (1980); II, Opt. Acta 28, 211–225 (1981).
[CrossRef]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Bistability and photon antibunching in sub/second harmonic generation,” Opt. Commun. 28, 255 (1979).
[CrossRef]

K. J. McNeil, P. D. Drummond, and D. F. Walls, “Selfpulsing in second harmonic generation,” Opt. Commun. 27, 292 (1978).
[CrossRef]

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden, M. Ciftan, and H. R. Robl, eds. (Plenum, New York, 1984).

Wu, X.-G.

I, Opt. Acta (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” I, Opt. Acta 27, 321–335 (1980); II, Opt. Acta 28, 211–225 (1981).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Acta (2)

P. Mandel and T. Erneux, “Amplitude self-modulation of intra-cavity second harmonic generation,” Opt. Acta 29, 7–21 (1982).
[CrossRef]

C. M. Savage and D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557–561 (1983).
[CrossRef]

Opt. Commun. (2)

K. J. McNeil, P. D. Drummond, and D. F. Walls, “Selfpulsing in second harmonic generation,” Opt. Commun. 27, 292 (1978).
[CrossRef]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Bistability and photon antibunching in sub/second harmonic generation,” Opt. Commun. 28, 255 (1979).
[CrossRef]

Phys. Lett. (1)

P. Mandel, “On the transition from absorptive to dispersive behavior of a laser due to nonlinear losses,” Phys. Lett. A83207–210 (1981).

Phys. Lett. A (1)

S. T. Dembinski, A. Kossakowski, P. Peplowski, L. A. Lugiato, and P. Mandel, “Laser instability below threshold,” Phys. Lett. A 68, 20–22 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

P. Mandel and T. Erneux, “Laser-Lorenz equations with a time-dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).
[CrossRef]

Rep. Prog. Phys. (1)

E. Abraham and S. D. Smith, “Optical bistability and related devices,” Rep. Prog. Phys. 45, 815–885 (1982).
[CrossRef]

Other (6)

C. M. Bowden, M. Ciftan, and H. R. Robl, eds., Optical Bistability (Plenum, New York, 1981).
[CrossRef]

C. M. Bowden, H. M. Gibbs, and S. L. McCall, eds., Optical Bistability 2 (Plenum, New York, 1984).
[CrossRef]

H. Haken, Synergetics, 2nd ed. (Springer-Verlag, Heidelberg, 1978).
[CrossRef]

L. A. Lugiato, “Theory of optical Instability,” in Progress in Otpics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–216.
[CrossRef]

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden, M. Ciftan, and H. R. Robl, eds. (Plenum, New York, 1984).

P. Mandel and T. Erneux, Second Harmonic Generation in a Resonant Cavity, Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1983), pp. 132–141.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Steady-state solutions for the fundamental mode with δ1 = δ|| = δ = 1 and μ = 0.1.

Fig. 2
Fig. 2

Periodic solutions for δ1 = 1, δ = δ = 0.8, μ = 0.1, and A = 10. The upper trace is the second harmonic with maxima at 13.39, minima at 9.98, and a time average of 11.89. The lower trace is the fundamental mode with maxima at 8.77, minima at 3.43, and a time average of 5.39.

Fig. 3
Fig. 3

Hopf-bifurcation boundaries in the (μ, A) parameter plane for δ1 = 0.01 and δ = 0.25. D1 is the domain of stable steady states. The boundary between D1 and D2 is an amplitude instability, whereas the boundary between D1 and D3 is a phase instability. The two boundaries merge at A = 5211.4492 and μ = 0.9929.

Fig. 4
Fig. 4

Forward scan across the region of transition between normal and anomalous solutions. The static bifurcation occurs at  = 4.2. The parameters are δ1 = δ = δ = 1, μ = 0.1, and A(t) = 3 + 10−2t.

Fig. 5
Fig. 5

Forward scan across a steady and a Hopf bifurcation. The static bifurcations occur at A = 1 and A = 7.125. The parameters are δ1 = 1, δ = δ = 0.8, μ = 0.1, and A = 10−2t.

Fig. 6
Fig. 6

Backward scan across the region of transition between normal and anomalous solutions. The parameters are δ1 = δ1 = δ = 1, μ = 0.1, and A(t) = 8 – 10−2t.

Fig. 7
Fig. 7

Backward scan across a steady and a Hopf bifurcation. The parameters are δ1 = 1, δ = δ = 0.8, μ = 0.1, and A = 10 – 10−2t.

Equations (60)

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

A ( 1 ) = [ δ 1 + i Δ ( 1 ) ] A ( 1 ) + A * ( 1 ) A ( 2 ) + E , A ( 2 ) = [ 1 + i Δ ( 2 ) ] A ( 2 ) A 2 ( 1 )
A 1 = ( δ 1 + i Δ 1 ) A 1 + A 1 * A 2 + P ¯ , A 2 = ( 1 + i Δ 2 ) A 2 A 1 2 , P ¯ = ( δ + i Δ p ) P ¯ + D A 1 , D = δ ( D D 0 ) μ ( P ¯ A 1 * + P ¯ * A 1 ) .
A j = E j exp ( i ϕ j ) , P ¯ = P exp ( i ϕ p ) .
E 1 = δ 1 E 1 + E 1 E 2 cos ψ 2 + P cos ψ p , E 2 = E 2 E 1 2 cos ψ 2 , P = δ P + D E 1 cos ψ p , D = δ ( D D 0 ) 2 μ P E 1 cos ψ p
ψ 2 = E 1 2 2 E 2 2 E 2 sin ψ 2 2 P E 1 sin ψ p , ψ 2 = D E 1 2 + P 2 E 1 P sin ψ p E 2 sin ψ 2
ψ 2 = ϕ 2 2 ϕ 1 , ψ p = ϕ p ϕ 1 .
ϕ 1 = E 2 sin ψ 2 + P E 1 sin ψ p , ϕ 2 = E 1 2 E 2 sin ψ 2 , ϕ p = D E 1 P sin ψ p .
E j = 0 , P = 0 , D = D 0
E 1 2 I = E 2 , P = D E 1 / δ , D = A δ 1 δ / ( 1 + S I ) , Ψ 2 = π , Ψ p = 0 ,
I = 1 2 S { 1 δ 1 S + [ ( 1 δ 1 S ) 2 + 4 A S δ 1 ] 1 / 2 } .
S = 2 μ / δ δ , D 0 = A δ 1 δ .
A ¯ 1 ,
I 1 = α I 2 , I 2 = β ( A A 0 ) , P = D E 1 δ 1 cos Ψ p , D = D 0 2 μ 1 + α δ 1 δ I 2 , cos Ψ 2 = 1 / α E 2 , cos Ψ p = ( 1 + α 2 I 2 1 4 δ 2 ) 1 ,
α = 2 δ 1 δ 1 + δ , β 1 = 2 μ 1 + α δ 1 δ 1 δ δ α ( α 2 ) 4 δ 1 δ , A 0 = ( 1 + 1 2 δ 1 ) ( 1 + 1 2 δ ) .
ϕ 1 = 1 2 ϕ 2 = ϕ p ω ω = α 2 ( I 2 α 2 ) 1 / 2 .
δ > 1 / 2 , A ¯ A ¯ = δ ( 2 δ 1 + 1 ) δ 1 ( 2 δ 1 ) ( 1 + S δ + δ 1 2 δ 1 ) .
I 1 = 1 / α , I 2 = 1 / α 2 , ω = 0.
λ = 1 , δ , 1 / 2 { δ 1 δ ± [ ( δ 1 δ ) 2 + 4 A δ 1 δ ] 1 / 2 } .
n = 0 4 a n λ n = 0 , a 4 = 1 , a 3 = 1 + δ 1 + δ + δ + I 1 > 0 , a 2 = 2 μ I 1 + δ ( 1 + δ 1 + δ + I 1 ) + δ + δ 1 3 I 1 > 0 , a 1 = 2 μ I 1 ( 1 + 2 δ 1 + 2 I 1 ) + δ ( δ 1 + δ + 3 I 1 ) + 2 I 1 δ > 0 , a 0 = 4 μ I 1 ( δ 1 + 2 I 1 ) + 2 I 1 δ δ ¯ 0.
λ 2 + λ ( 1 + δ 1 + δ I 1 ) + δ + δ 1 + I 1 ( 1 2 δ ) = 0.
λ = 0 , 1 , 1 , δ , δ 1 , δ , δ 1 , δ .
λ = 4 μ δ 1 / δ ( δ 1 + δ ) ,
λ 1 = 0 , λ 2 = 1 δ 1 δ + I 1 .
2 δ 1 ( 1 δ ) < 2 δ 2 1 ,
A c = 1 + 2 δ 1 + δ δ 1 [ 1 + S ( 1 + δ 1 + δ ) ] ,
λ = ± i Ω, Ω 2 = ( 1 δ ) ( 1 + 2 δ 1 + 2 δ ) δ .
2 δ 1 ( 1 δ ) > 2 δ 2 1.
δ = δ δ , δ 1 δ .
( λ 2 + b λ + c ) ( λ 2 + Ω 2 ) = 0.
b = a 3 , c + Ω 2 = a 2 , b Ω 2 = a 1 , c Ω 2 = a 0 .
a 0 a 3 2 a 1 a 2 a 3 + a 1 2 = 0 ,
μ 2 α + μ β + γ = 0 .
E 1 = α 1 / 2 , E 2 = α 1 , P = α 3 / 2 , D = δ / α , Ψ 2 = π , Ψ P = 0.
δ D 0 = δ D 0 , c + x 2
A = A ¯ + A = A ¯ + x 2 / δ 1 δ 2 ,
E 1 ( t ) = E 1 ( τ , ) = α 1 / 2 + 2 e 1 ( τ ) + 0 ( 4 ) , E 2 ( t ) = E 2 ( τ , ) = α 1 + 2 e 2 ( τ ) + 0 ( 4 ) , P ( t ) = P ( τ , ) = α 3 / 2 + 2 p ( τ ) + 0 ( 4 ) , D ( t ) = D ( τ , ) = δ / α + 2 d ( τ ) + 0 ( 4 ) , Ψ 2 ( t ) = Ψ 2 ( τ , ) = π + ϕ 2 ( 1 , τ ) + 3 ϕ 2 ( 3 , τ ) + 0 ( 5 ) , Ψ p ( t ) = Ψ p ( τ , ) = ϕ p ( 1 , τ ) + 3 ϕ p ( 3 , τ ) + 0 ( 5 ) ,
ϕ 2 ( 1 , τ ) = β ( τ ) , ϕ p ( 1 , τ ) = β ( τ ) / 2 δ ,
ϕ 2 ( 3 , τ ) ( 2 α 1 ) ϕ p ( 3 , τ ) = f ( β ) , ϕ 2 ( 3 , τ ) 1 α ( δ + 1 α ) ϕ p ( 3 , τ ) = g ( β ) ,
β = β ( β 2 a + b x ) , a = 2 δ 1 2 δ 2 1 δ 2 μ δ + α 8 μ + 2 α δ 2 , b = ( 2 δ 1 ) 2 2 δ 2 1 2 α 8 μ + 2 α δ 2 > 0.
β 2 ( τ ) = b x [ ( a + b x β 2 ( 0 ) ) exp ( 2 b x τ ) a ] 1 .
X 1 = δ 1 X 1 + X 1 X 2 + Y 1 Y 2 + P 1 , Y 1 = δ 1 Y 1 + X 1 Y 2 X 2 Y 1 + P 2 , X 2 = X 2 X 1 2 + Y 1 2 , Y 2 = Y 2 2 X 1 Y 1 , P 1 = δ P 1 + X 1 D , P 2 = δ P 2 + Y 1 D , D = δ ( D D 0 ) 2 μ ( X 1 P 1 + Y 1 P 2 ) .
X 1 = δ 1 X 1 + X 1 X 2 + P 1 , X 2 = X 2 X 1 2 , P 1 = δ P 1 + X 1 D D = δ ( D D 0 ) 2 μ X 1 P 1 .
D 1 : Re λ ( 1 , 2 ) < 0 , Re λ ( 3 , 4 ) < 0 ; D 2 : Re λ ( 1 , 2 ) < 0 , Re λ ( 3 , 4 ) < 0 ; D 3 : Re λ ( 1 , 2 ) > 0 , Re λ ( 3 , 4 ) < 0 ;
A = A ( t ) = A 0 + υ t , A 0 < 1 , 0 < υ 1.
E 1 = E 2 = P = 0 , D ( t ) = D 0 ( t ) δ 1 υ ( 1 exp ( δ t ) ) .
D ( t ) δ 1 δ A ( t ) = D 0 ( t ) .
E j = e j + 0 ( 2 ) , P = p + 0 ( 2 ) , D = D 0 ( t ) + d + 0 ( 2 ) , Ψ 2 = π + ϕ 2 + 0 ( 2 ) Ψ p = ϕ p + 0 ( 2 ) .
e 1 = δ 1 e 1 + p , p = δ p + D 0 e 1 ,
e 2 = e 2 , d = δ d , ϕ 2 = 2 p e 1 ϕ p , ϕ p = e 2 ϕ 2 .
p ( t ) = π ( δ 1 δ ) 3 / 2 υ 1 / 2 e 1 ( 0 ) e Δ t [ A i ( τ 0 ) B i ( τ ) B i ( τ 0 ) A i ( τ ) ] ,
Δ = ( δ 1 + δ ) / 2 , τ = ( Δ 2 β ) 1 / 3 ( t + γ β ) , β = υ δ 1 δ Δ 2 γ = 1 + δ 1 δ ( A 0 1 ) Δ 2 .
t ρ ( t ) = [ t ρ ( t ) ] laser + [ t ρ ( t ) ] SHG ,
i ћ [ t ρ ( t ) ] laser = [ H L , ρ ] + i ћ ( A + F )
i ћ [ t ρ ( t ) ] SHG = [ H SHG , ρ ] .
H L = ћ p ω ( p ) b p + b p + ћ υ 1 α 1 + α 1 + ћ p { g p α 1 b p + g p * b p α 1 + } ,
H SHG = h υ 2 α 2 + α 2 + i 2 ћ k ( α 1 + α 1 + α 3 α 1 α 1 α 2 + ) .
[ i ( t + κ 1 ) υ 1 ] α 1 = p g p * b p + i k α 1 * α 2 , [ i ( t + κ 2 ) υ 2 ] α 2 = ( i k / 2 ) α 1 2 , [ i ( t + γ p ) ω p ] b p = g p d p α 1 , i ( t + γ p ) d p = i σ p γ p 2 ( g p * g p α 1 * c . c ) .
α 1 = α 1 exp ( i υ t ) , α 2 = α 2 exp ( 2 i υ t ) , g * b = a exp ( i υ t ) ,
α 1 = κ 1 α 1 i ( υ 1 υ ) α 1 i N a + k α 1 * α 2 , α 2 = κ 2 α 2 i ( υ 2 2 υ ) α 2 k 2 α 1 2 , a = γ a i ( ω υ ) a + i | g | 2 d α 1 , d = γ ( d σ ) + 2 i ( a α 1 * a * α 1 ) .
α 1 = 2 κ 2 k A 1 , α 2 = κ 2 k A 2 , a = i 2 κ 2 2 N k P ¯ , d = | g | 2 N κ 2 2 D , τ = κ 2 t , δ = γ / κ 2 , δ = γ / κ 2 , δ 1 = κ 1 / κ 2 Δ 1 = ( υ 1 υ ) / κ 2 , Δ 2 = ( υ 2 2 υ ) / κ 2 , Δ p = ( ω υ ) / κ 2 , μ = 4 | g | 2 / k 2

Metrics