Abstract

We consider a system of cubic complex Ginzburg–Landau equations governing copropagation of two waves with opposite signs of the dispersion in a nonlinear optical fiber in the presence of gain and losses. The waves are coupled by cross-phase modulation and stimulated Raman scattering. A special exact solution for a bound state of bright and dark solitons is found (unlike the well-known exact dark-soliton solution to the single complex Ginzburg–Landau equation, which is a sink, this compound soliton proves to be a source emitting traveling waves). Numerical simulations reveal that the compound soliton remains stable over ∼10 soliton periods. Next, we demonstrate that a very weak seed noise, added to an initial state in the form of a dark soliton in the normal-dispersion mode and nothing in the anomalous-dispersion one, gives rise to a process of the generation of a bright pulse in the latter mode, while the dark soliton gets grayer and eventually disappears. Thus this scheme can be used for an effective transformation of a dark soliton into a bright one, which is of interest by itself and may also find applications in photonics.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).
  2. W. Zhao and E. Bourkoff, “Propagation properties of dark solitons,” Opt. Lett. 14, 703–705 (1989); Y. Chen and J. Atai, “Absorption and amplification of dark solitons,” Opt. Lett. 16, 1933–1935 (1991); M. Lisak, D. Anderson, and B. A. Malomed, “Dissipative damping of dark solitons in optical fibers,” Opt. Lett. OPLEDP 16, 1936–1937 (1991).
    [CrossRef] [PubMed]
  3. J. P. Hamaide, Ph. Emplit, and M. Haelterman, “Dark-soliton jitter in amplified optical transmission systems,” Opt. Lett. 16, 1578–1580 (1991); Yu. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, “Gordon–Haus effect on dark solitons,” Opt. Lett. 19, 19–21 (1994).
    [CrossRef] [PubMed]
  4. W. Zhao and E. Bourkoff, “Interactions between dark solitons,” Opt. Lett. 14, 1371–1373 (1989).
    [CrossRef] [PubMed]
  5. M. Nakazawa and K. Suzuki, “10 Gb/s Pseudorandom dark soliton data transmission over 1200 km,” Electron. Lett. 31, 1076–1077 (1995).
    [CrossRef]
  6. V. V. Afanasjev, P. L. Chu, and B. A. Malomed, “Dark soliton generation in fused coupler,” Opt. Commun. 137, 229–232 (1997).
    [CrossRef]
  7. M. Nakazawa and K. Suzuki, “Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit shifting with a Mach–Zehnder interferometer,” Electron. Lett. 31, 1084–1085 (1995).
    [CrossRef]
  8. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Optical solitary waves induced by cross-phase modulation,” Opt. Lett. 13, 871–873 (1988).
    [CrossRef] [PubMed]
  9. M. Lisak, A. Höök, and D. Anderson, “Symbiotic solitary wave pairs sustained by cross-phase modulation in optical fibers,” J. Opt. Soc. Am. B 7, 810–814 (1990).
    [CrossRef]
  10. L. M. Hocking and K. Stewartson, “On the nonlinear response of marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. London, Ser. A 326, 289–313 (1972); N. R. Pereira and L. Stenflo, “Nonlinear Schroedinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
    [CrossRef]
  11. K. Nozaki and N. Bekki, “Exact solution of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
    [CrossRef]
  12. H. Sakaguchi, “Instability of the hole solution in the complex Ginzburg–Landau equation,” Prog. Theor. Phys. 85, 417–421 (1991).
    [CrossRef]
  13. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
    [CrossRef]
  14. R. I. Joseph and D. N. Christodoulides, “Bright–dark solitary wave pairs resulting from stimulated Raman scattering and loss,” Opt. Lett. 17, 1101–1103 (1992).
    [CrossRef] [PubMed]
  15. B. A. Malomed, A. Schwache, and F. Mitschke, “Soliton lattice and gas in passive fiber-ring resonators,” Fiber Integr. Opt. 17, 267–277 (1998).
    [CrossRef]
  16. B. A. Malomed, “Nonsteady waves in distributed dynamical systems,” Physica D 8, 353–359 (1983).
    [CrossRef]
  17. B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg–Landau equation,” Phys. Rev. A 42, 6009–6014 (1990); W. Saarloos and P. C. Hohenberg, “Pulse and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990); H. Sakaguchi and B. A. Malomed, “Grain boundaries in two-dimensional travelling wave patterns,” Physica D PDNPDT 118, 250–260 (1998).
    [CrossRef] [PubMed]

1998 (1)

B. A. Malomed, A. Schwache, and F. Mitschke, “Soliton lattice and gas in passive fiber-ring resonators,” Fiber Integr. Opt. 17, 267–277 (1998).
[CrossRef]

1997 (1)

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, “Dark soliton generation in fused coupler,” Opt. Commun. 137, 229–232 (1997).
[CrossRef]

1995 (2)

M. Nakazawa and K. Suzuki, “Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit shifting with a Mach–Zehnder interferometer,” Electron. Lett. 31, 1084–1085 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, “10 Gb/s Pseudorandom dark soliton data transmission over 1200 km,” Electron. Lett. 31, 1076–1077 (1995).
[CrossRef]

1993 (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

1992 (1)

1991 (1)

H. Sakaguchi, “Instability of the hole solution in the complex Ginzburg–Landau equation,” Prog. Theor. Phys. 85, 417–421 (1991).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

1984 (1)

K. Nozaki and N. Bekki, “Exact solution of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

1983 (1)

B. A. Malomed, “Nonsteady waves in distributed dynamical systems,” Physica D 8, 353–359 (1983).
[CrossRef]

Afanasjev, V. V.

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, “Dark soliton generation in fused coupler,” Opt. Commun. 137, 229–232 (1997).
[CrossRef]

Anderson, D.

Bekki, N.

K. Nozaki and N. Bekki, “Exact solution of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

Bourkoff, E.

Christodoulides, D. N.

Chu, P. L.

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, “Dark soliton generation in fused coupler,” Opt. Commun. 137, 229–232 (1997).
[CrossRef]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Höök, A.

Joseph, R. I.

Lisak, M.

Malomed, B. A.

B. A. Malomed, A. Schwache, and F. Mitschke, “Soliton lattice and gas in passive fiber-ring resonators,” Fiber Integr. Opt. 17, 267–277 (1998).
[CrossRef]

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, “Dark soliton generation in fused coupler,” Opt. Commun. 137, 229–232 (1997).
[CrossRef]

B. A. Malomed, “Nonsteady waves in distributed dynamical systems,” Physica D 8, 353–359 (1983).
[CrossRef]

Mitschke, F.

B. A. Malomed, A. Schwache, and F. Mitschke, “Soliton lattice and gas in passive fiber-ring resonators,” Fiber Integr. Opt. 17, 267–277 (1998).
[CrossRef]

Nakazawa, M.

M. Nakazawa and K. Suzuki, “Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit shifting with a Mach–Zehnder interferometer,” Electron. Lett. 31, 1084–1085 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, “10 Gb/s Pseudorandom dark soliton data transmission over 1200 km,” Electron. Lett. 31, 1076–1077 (1995).
[CrossRef]

Nozaki, K.

K. Nozaki and N. Bekki, “Exact solution of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

Sakaguchi, H.

H. Sakaguchi, “Instability of the hole solution in the complex Ginzburg–Landau equation,” Prog. Theor. Phys. 85, 417–421 (1991).
[CrossRef]

Schwache, A.

B. A. Malomed, A. Schwache, and F. Mitschke, “Soliton lattice and gas in passive fiber-ring resonators,” Fiber Integr. Opt. 17, 267–277 (1998).
[CrossRef]

Stegeman, G. I.

Suzuki, K.

M. Nakazawa and K. Suzuki, “10 Gb/s Pseudorandom dark soliton data transmission over 1200 km,” Electron. Lett. 31, 1076–1077 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, “Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit shifting with a Mach–Zehnder interferometer,” Electron. Lett. 31, 1084–1085 (1995).
[CrossRef]

Trillo, S.

Wabnitz, S.

Wright, E. M.

Zhao, W.

Electron. Lett. (2)

M. Nakazawa and K. Suzuki, “10 Gb/s Pseudorandom dark soliton data transmission over 1200 km,” Electron. Lett. 31, 1076–1077 (1995).
[CrossRef]

M. Nakazawa and K. Suzuki, “Generation of a pseudorandom dark soliton data train and its coherent detection by one-bit shifting with a Mach–Zehnder interferometer,” Electron. Lett. 31, 1084–1085 (1995).
[CrossRef]

Fiber Integr. Opt. (1)

B. A. Malomed, A. Schwache, and F. Mitschke, “Soliton lattice and gas in passive fiber-ring resonators,” Fiber Integr. Opt. 17, 267–277 (1998).
[CrossRef]

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

J. Phys. Soc. Jpn. (1)

K. Nozaki and N. Bekki, “Exact solution of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

Opt. Commun. (1)

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, “Dark soliton generation in fused coupler,” Opt. Commun. 137, 229–232 (1997).
[CrossRef]

Opt. Lett. (3)

Physica D (1)

B. A. Malomed, “Nonsteady waves in distributed dynamical systems,” Physica D 8, 353–359 (1983).
[CrossRef]

Prog. Theor. Phys. (1)

H. Sakaguchi, “Instability of the hole solution in the complex Ginzburg–Landau equation,” Prog. Theor. Phys. 85, 417–421 (1991).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Other (5)

L. M. Hocking and K. Stewartson, “On the nonlinear response of marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. London, Ser. A 326, 289–313 (1972); N. R. Pereira and L. Stenflo, “Nonlinear Schroedinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

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

W. Zhao and E. Bourkoff, “Propagation properties of dark solitons,” Opt. Lett. 14, 703–705 (1989); Y. Chen and J. Atai, “Absorption and amplification of dark solitons,” Opt. Lett. 16, 1933–1935 (1991); M. Lisak, D. Anderson, and B. A. Malomed, “Dissipative damping of dark solitons in optical fibers,” Opt. Lett. OPLEDP 16, 1936–1937 (1991).
[CrossRef] [PubMed]

J. P. Hamaide, Ph. Emplit, and M. Haelterman, “Dark-soliton jitter in amplified optical transmission systems,” Opt. Lett. 16, 1578–1580 (1991); Yu. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, “Gordon–Haus effect on dark solitons,” Opt. Lett. 19, 19–21 (1994).
[CrossRef] [PubMed]

B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg–Landau equation,” Phys. Rev. A 42, 6009–6014 (1990); W. Saarloos and P. C. Hohenberg, “Pulse and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990); H. Sakaguchi and B. A. Malomed, “Grain boundaries in two-dimensional travelling wave patterns,” Physica D PDNPDT 118, 250–260 (1998).
[CrossRef] [PubMed]

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 (6)

Fig. 1
Fig. 1

Results of numerical simulations of Eqs. (1) and (2), with the initial condition taken in the form of the exact solutions (6) and (7), the values of the parameters being taken as per Eq. (8). The graphs show the evolution of (a) |u(z, τ)|2 and (b) |v(z, τ)|2.

Fig. 2
Fig. 2

Evolution of the initial configuration in the form of the dark soliton in the normal-dispersion mode, as per Eq. (7), and random noise in the anomalous-dispersion mode, with the maximum intensity 10-4, in the case specified by Eq. (9). The graphs show the evolution of (a) |u(z, τ)|2 and (b) |v(z, τ)|2.

Fig. 3
Fig. 3

Snapshots of (a) |u(τ)|2 and (b) |v(τ)|2, showing the final state, at z=50 (solid curve) in the case corresponding to Fig. 2 compared with the initial state (dashed curve).

Fig. 4
Fig. 4

Same as in Fig. 2 with the seed-noise maximum intensity 10-3.

Fig. 5
Fig. 5

Same as in Fig. 2 with the seed-noise maximum intensity 5×10-3.

Fig. 6
Fig. 6

Same as in Fig. 2 for the parameter values given in Eq. (10) with the seed-noise maximum intensity 10-3.

Equations (13)

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

iuz+12uττ+(|u|2+2|v|2)u
=iγ0u+iγ1uττ-iγ2|u|2u-i|v|2u,
ivz-12Duττ+(|v|2+2|u|2)v
=iγ0v+iγ1vττ-iγ2|v|2v+i|u|2v,
v(z, τ)B exp(iqz±iωτ),B=const.,
B2=2γ1q-D2γ1-γ2D,ω2=2 1-γ2q2γ1-γ2D
δγ1-B20.
u=A[cosh(μτ)]-(1+iα) exp(ikz),
v=B tanh(μτ)[cosh(μτ)]-iβ exp(iqz),
μ=0.7068,α=1.1751,β=2.7277,
A=1.1002,B=0.6175,k=0.8629,q=2.2398.
γ1=D=γ2==1
γ1=D=2,γ2==1.

Metrics