Abstract

The process of modulational instability of nonlinear plane waves is investigated in media with periodically and randomly varying nonlinearity. The wave behavior is analyzed in the parametric instability region under the joint action of periodic modulation and the self-steepening effect, and a recurrence phenomena is observed for the nonlinear wave in the parametric resonance region. Stochastic parametric resonance is predicted in the case of a random media. The increment value of stochastic parametric resonance obtained numerically is in good agreement with the theoretical estimate.

© 1997 Optical Society of America

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References

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  1. E. R. Tracy and H. H. Chen, Phys. Rev. A 37, 815 (1988).
    [CrossRef] [PubMed]
  2. N. A. Akhmediev, N. N. Kulagin, and V. M. Eleonsky, Sov. Phys. JETP 62, 894 (1985).
  3. S. Trillo and S. Wabnitz, in Nonlinearity with Disorder, F. Kh. Abdullaev, A. R. Bishop, and S. Pnevmatikos, eds. (Springer-Verlag, Berlin, 1992).
  4. A. Hasegawa, Opt. Lett. 9, 288 (1984).
    [CrossRef] [PubMed]
  5. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]
  6. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, Opt. Lett. 18, 1499 (1993).
    [CrossRef]
  7. F. Kh. Abdullaev, Pisma JTP 20, 25 (1994) (in Russian).
  8. J. P. Gordon, J. Opt. Soc. Am. B 9, 9 (1992).
    [CrossRef]
  9. R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
    [CrossRef]
  10. L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1973).
  11. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1990).
  12. F. Kh. Abdullaev and S. A. Darmanyan, Optical Solitons (Springer-Verlag, Berlin, 1993).
  13. V. I. Klyatzkin, Stochastic Differential Equations and Waves in Random Media (Nauka, Moscow, 1980) (in Russian).
  14. F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 50, 1552 (1994).
    [CrossRef]
  15. F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
    [CrossRef]
  16. D. W. McLaughlin, Physica D 3, 335 (1983).
    [CrossRef]
  17. R. Badii, T. Schneider, and M. P. Soerensen, Phys. Rev. B 35, 297 (1987).
    [CrossRef]

1995 (1)

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

1994 (2)

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 50, 1552 (1994).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

1993 (1)

1992 (1)

J. P. Gordon, J. Opt. Soc. Am. B 9, 9 (1992).
[CrossRef]

1988 (1)

E. R. Tracy and H. H. Chen, Phys. Rev. A 37, 815 (1988).
[CrossRef] [PubMed]

1987 (2)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

R. Badii, T. Schneider, and M. P. Soerensen, Phys. Rev. B 35, 297 (1987).
[CrossRef]

1985 (1)

N. A. Akhmediev, N. N. Kulagin, and V. M. Eleonsky, Sov. Phys. JETP 62, 894 (1985).

1984 (1)

1983 (1)

D. W. McLaughlin, Physica D 3, 335 (1983).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 50, 1552 (1994).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Akhmediev, N. A.

N. A. Akhmediev, N. N. Kulagin, and V. M. Eleonsky, Sov. Phys. JETP 62, 894 (1985).

Badii, R.

R. Badii, T. Schneider, and M. P. Soerensen, Phys. Rev. B 35, 297 (1987).
[CrossRef]

Bauer, R. G.

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

Bischoff, S.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

Caputo, J. G.

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 50, 1552 (1994).
[CrossRef]

Chen, H. H.

E. R. Tracy and H. H. Chen, Phys. Rev. A 37, 815 (1988).
[CrossRef] [PubMed]

Christiansen, P. L.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

Eleonsky, V. M.

N. A. Akhmediev, N. N. Kulagin, and V. M. Eleonsky, Sov. Phys. JETP 62, 894 (1985).

Flytzanis, N.

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 50, 1552 (1994).
[CrossRef]

Gordon, J. P.

J. P. Gordon, J. Opt. Soc. Am. B 9, 9 (1992).
[CrossRef]

Hasegawa, A.

Kulagin, N. N.

N. A. Akhmediev, N. N. Kulagin, and V. M. Eleonsky, Sov. Phys. JETP 62, 894 (1985).

Matera, F.

McLaughlin, D. W.

D. W. McLaughlin, Physica D 3, 335 (1983).
[CrossRef]

Mecozzi, A.

Melnikov, L. A.

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

Romagnoli, M.

Schneider, T.

R. Badii, T. Schneider, and M. P. Soerensen, Phys. Rev. B 35, 297 (1987).
[CrossRef]

Settembre, M.

Soerensen, M. P.

R. Badii, T. Schneider, and M. P. Soerensen, Phys. Rev. B 35, 297 (1987).
[CrossRef]

Sørensen, M. P.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

Tracy, E. R.

E. R. Tracy and H. H. Chen, Phys. Rev. A 37, 815 (1988).
[CrossRef] [PubMed]

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

J. P. Gordon, J. Opt. Soc. Am. B 9, 9 (1992).
[CrossRef]

Opt. Commun. (2)

R. G. Bauer and L. A. Melnikov, Opt. Commun. 115, 190 (1995).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

E. R. Tracy and H. H. Chen, Phys. Rev. A 37, 815 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. Badii, T. Schneider, and M. P. Soerensen, Phys. Rev. B 35, 297 (1987).
[CrossRef]

Phys. Rev. E (1)

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 50, 1552 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Physica D (1)

D. W. McLaughlin, Physica D 3, 335 (1983).
[CrossRef]

Sov. Phys. JETP (1)

N. A. Akhmediev, N. N. Kulagin, and V. M. Eleonsky, Sov. Phys. JETP 62, 894 (1985).

Other (6)

S. Trillo and S. Wabnitz, in Nonlinearity with Disorder, F. Kh. Abdullaev, A. R. Bishop, and S. Pnevmatikos, eds. (Springer-Verlag, Berlin, 1992).

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1973).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1990).

F. Kh. Abdullaev and S. A. Darmanyan, Optical Solitons (Springer-Verlag, Berlin, 1993).

V. I. Klyatzkin, Stochastic Differential Equations and Waves in Random Media (Nauka, Moscow, 1980) (in Russian).

F. Kh. Abdullaev, Pisma JTP 20, 25 (1994) (in Russian).

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Figures (7)

Fig. 1
Fig. 1

Region of modulational instability for the first parametric resonance in the (Ωa) plane (A=1). Solid curves correspond to the case α=0, dashed curves correspond to α=0.5, and dotted curves correspond to α=1.0: (a) negative group dispersion and (b) positive group dispersion.

Fig. 2
Fig. 2

Space–time evolution of |u|2 for ν=+1, q=v=1×10-4, A=1.0, 0=0.1, Ω=4.1888, α=0, and a=30.83355.

Fig. 3
Fig. 3

Space–time evolution of |u|2 for ν=-1, q=v=10-4, A=1.0, 0=0.1, Ω=4.1888, a=38,88, and δ=0.2.

Fig. 4
Fig. 4

Space–time evolution of |u|2 when α=0.5, a=31.1168, and the other parameters are as in Fig. 2.

Fig. 5
Fig. 5

(a) Space–time evolution of the steady-state solution in a random media. ν=+1, q=10-4, v=0, A=1.0, Ω= 4.1888, σ2= 0.04992, α=0, and 140 realizations are used. (b) Plot of log(max(|u|2)-A) versus x (solid curve). The dashed curve is the theoretical predicted growth rate.

Fig. 6
Fig. 6

(a) Space–time evolution of the steady-state solution in a random media. ν=+1, q=10-2, v=0, A=1.0, Ω=4.1888, σ2=0.1, α=0, and 40 realizations are used. (b) The associated plot of log(max(|u|2)-A) versus x (solid curve). The dashed curve is the theoretical predicted growth rate.

Fig. 7
Fig. 7

Space–time evolution of the steady-state solution in the positive group-dispersion domain (ν=-1), when q=10-4, v=0, A=1.0, σ2=0.04992, 0=0.1, and Ω=4.1888.

Equations (41)

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iux+νuττ+2(x)|u|2u=-iα(|u|2u)τ.
u=A exp(iϕ),ϕ=2A20x(x)dx.
u=[A+ψ(x, τ)]exp(iϕ),|ψ|A.
iψx+νψττ+2A2(x)(ψ+ψ*)=-iαA2(2ψτ+ψτ*).
ψ(x, τ)=C(x)exp(iΩτ)+B*(x)exp(-iΩτ),
ibx-3αA2Ωb-νΩ2c=0,
icx-νΩ2b+4(x)A2b-αA2Ωc=0,
b˜xx+Ω2[Ω2-4νA2+α2A4+4ν˜(x)A2]b˜=0,
b˜xx+ω02[1+h cos(ax)]b˜=0,
h=4A20νΩ2-4A2ν+α2A4,
ω02=Ω2(Ω2-4A2ν+α2A4).
Ω2=2A2ν-α24A2+4A4ν-α2A242+a241/2.
|δ|<|hω0|2=20A2ΩΩ2-4A2ν+α2A4.
ΔΩa20νΩ(Ω04+a2-Ω02)Ω04+a2,
gmax0A2.
gmax80A4a1-α2A24.
Ωa2-2A2-α2A421/2.
Ωa2(4A2+α2A4)1/2.
iEx-β22Ett+γf(z)(|E|)2E=0,
f(z)1+2Γlπsin 2πzl.
b˜xx+ω02[1+h˜(x)]b˜=0,
˜=0,˜(x)˜(x)=2σ2δ(x-x).
b˜x=y,
yx=-ω02b˜-ω02h˜(x)b˜.
b˜=b0 cos ω0x+y0ω0sin ω0x,
b˜x=-b0ω0 sin ω0x+y0 cos ω0x.
b˜2x=2b˜y,
b˜yx=y2-ω02b˜2,
y2x=-2ω02b˜y+2ω04h2σ2b˜2.
b˜(x=0)=b0=C0+B0,
b˜x(x=0)=y0=-i(αA2Ωb0+νΩ2c0),
c(x=0)=c0=C0-B0.
b˜2exp(ω02σ2h2x)b1+σ2b24+b34ω02+exp-12ω02h2σ2xb22ω0-ω0h2σ22b1-3σ2h2b316ω0sin2ω0x-b34ω02+σ2b24×cos2ω0x,
y2ω02 exp(ω02h2σ2x)b1+b34ω02+σ2h2b24+exp-12ω02h2σ2x-14(ω02σ2b2+b3)×cos(2ω0x)+12ω03h2σ2b1-ω0b2-18h2σ2ω0b3sin(2ω0x).
b1=b02,b2=2b0y0,b3=2y02-2ω02b02.
b2=1Ω4(-y2-2iαA2Ωb˜y+α2Ω2A4b˜2)exp(-4iαA2Ωx).
P(λ)=π2σ2[(λ-ξ)2+η2]2cosh2π(λ-ξ)2η,
η(x)=η0(1+κx)-1/4,κ=4πσ2μη04,μ=12815.
ψ=q cos(Ωτ)+iv sin(Ωτ).
Tt2 log||γ,=ψsatq,
|u|2=A2+2ARe(ψ)+|ψ|2=A2+ARe[(c+b)exp(iΩτ)+(b*-c*)exp(-iΩτ)]+|c|22+|b|22-12Re[(c2-b2)exp(2iΩτ)].

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