Abstract

We study in detail stability of exact chirped solitary-pulse solutions in a model in which stabilization of the pulses is achieved by means of short segments of an extra lossy core, which is parallel coupled to the main one. We demonstrate that, in the model’s three-dimensional parameter space, there is a vast region in which the pulses are fully stable, for both signs of the group-velocity dispersion. These results open the way to a stable transmission of solitary optical pulses in the normal-dispersion region and thus to an essential expansion of the bandwidth offered by the nonlinear optical fibers for telecommunications in the return-to-zero regime. In the cases in which the pulses are unstable, we study the development of the instability, which may end by either blowing up or decaying to zero. In the case when the pulses are stable, we also simulate interactions between them, concluding that they always eventually merge into one pulse.

© 2000 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).
  2. E. Desurvire, Erbium-Doped Fiber Amplifibers (Wiley, New York, 1994).
  3. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, UK, 1995).
  4. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991); Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
    [CrossRef] [PubMed]
  5. A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997).
    [CrossRef]
  6. M. Matsumoto, “Theory of stretched-pulse transmission dispersion-managed fibers,” Opt. Lett. 22, 1238–1240 (1997); B. A. Malomed, “Jitter suppression guiding filters in combination with dispersion management,” Opt. Lett. 23, 1250–1252 (1998); A. Berntson and B. A. Malomed, “Dispersion management with filtering,” Opt. Lett. OPLEDP 24, 507–509 (1999).
    [CrossRef] [PubMed]
  7. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
    [CrossRef]
  8. L. M. Hocking and K. Stewartson, “On the nonlinear response of a 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]
  9. B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
    [CrossRef]
  10. B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
    [CrossRef]
  11. J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
    [CrossRef]
  12. H. G. Winful and D. T. Walton, “Passive mode locking through nonlinear coupling in a dual-core fiber laser,” Opt. Lett. 17, 1688–1690 (1992); D. T. Walton and H. G. Winful, “Passive mode locking with an active nonlinear directional coupler: positive group-velocity dispersion,” Opt. Lett. 18, 720–722 (1993).
    [CrossRef] [PubMed]
  13. A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
    [CrossRef]
  14. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
    [CrossRef]
  15. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991); P. V. Mamyshev and L. F. Mollenauer, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guidings,” Opt. Lett. 21, 1658–1660 (1996).
    [CrossRef] [PubMed]

1999

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

1998

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

1997

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997).
[CrossRef]

1996

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

1993

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

1991

Atai, J.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Chen, Y.

Cross, M. C.

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

Frantzeskakis, D. J.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997).
[CrossRef]

Hizanidis, K.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

Hohenberg, P. C.

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

Kodama, Y.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997).
[CrossRef]

Malomed, B. A.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

Maruta, A.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997).
[CrossRef]

Mitchell, D. J.

Nistazakis, H. E.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

Poladian, L.

Rowland, D. R.

Snyder, A. W.

Tsigopoulos, A.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

Winful, H. G.

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Fiber Technol. Mater. Devices Syst.

A. Hasegawa, Y. Kodama, and A. Maruta, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. Mater. Devices Syst. 3, 197–213 (1997).
[CrossRef]

Phys. Lett. A

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

Phys. Rev. E

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[CrossRef]

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Rev. Mod. Phys.

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

Other

L. M. Hocking and K. Stewartson, “On the nonlinear response of a 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]

M. Matsumoto, “Theory of stretched-pulse transmission dispersion-managed fibers,” Opt. Lett. 22, 1238–1240 (1997); B. A. Malomed, “Jitter suppression guiding filters in combination with dispersion management,” Opt. Lett. 23, 1250–1252 (1998); A. Berntson and B. A. Malomed, “Dispersion management with filtering,” Opt. Lett. OPLEDP 24, 507–509 (1999).
[CrossRef] [PubMed]

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

E. Desurvire, Erbium-Doped Fiber Amplifibers (Wiley, New York, 1994).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, UK, 1995).

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991); Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

H. G. Winful and D. T. Walton, “Passive mode locking through nonlinear coupling in a dual-core fiber laser,” Opt. Lett. 17, 1688–1690 (1992); D. T. Walton and H. G. Winful, “Passive mode locking with an active nonlinear directional coupler: positive group-velocity dispersion,” Opt. Lett. 18, 720–722 (1993).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991); P. V. Mamyshev and L. F. Mollenauer, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guidings,” Opt. Lett. 21, 1658–1660 (1996).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Regions of the stability of the zero solution and regions of the existence and stability of the exact solitary-wave solution in the (Γ, K) parameter plane in the dual-core model. Region I: The zero solution is unstable. Region II: The solitary-wave solution is stable. Region III: The zero solution is stable, while the solitary-wave solution is not, showing a decay into zero. Region IV: The solitary-wave solution does not exist. Note that, outside region I, which is bounded by the curves Γ=K2 (dotted curve) and Γ=1 (dotted–dashed curve), the zero solution is unstable [see relation (13)].

Fig. 2
Fig. 2

Evolution of a stable pulse in the case D=-18, k0=0, and (K, Γ)=(5,4), corresponding to region II in Fig. 1.

Fig. 3
Fig. 3

Evolution of an unstable pulse in the case D=-18, k0=0, and (K, Γ)=(1.5, 1.5), corresponding to region I in Fig. 1. In this case, an initially laminar propagation of the pulse ends with a blowup.

Fig. 4
Fig. 4

Evolution of an unstable pulse in the case D=-18, k0=0, and (K, Γ)=(6.2, 3.6), corresponding to region III in Fig. 1. In this case, the initially laminar propagation of the pulse ends with a decay into zero.

Fig. 5
Fig. 5

Peak power of the solitary-wave solution versus the coupling parameter K for Γ=3, D=-18, and k0=0. The thin dashed curve corresponds to an unphysical solitary-wave solution [the root of the cubic equation (8) gives |u0|2<0 in this case]. The heavy solid and dashed curves correspond to physically existing stable and unstable solutions (including both the solitary-pulse and the zero solutions), respectively. The upward and downward arrows indicate the direction of a transition from the unstable solution to a stable one.

Fig. 6
Fig. 6

Area of the stability region of the solitary-wave solution [in the plane (K, Γ)] versus D for k0=-1,0,0.5,1.

Fig. 7
Fig. 7

Stability region of the exact solitary-pulse solutions of the dual-core model in the three-dimensional parameter space (K, D, Γ). The stability region is bounded by the surface shown from two different directions.

Fig. 8
Fig. 8

Interaction of two solitons at K=5 and Γ=4. The initial normalized pulse separation and phase difference are ητ=8 and ϕ=0, respectively.

Fig. 9
Fig. 9

Collision distance as a function of the normalized initial pulse separation ητ, for several values of the initial phase difference: ϕ=0, π/4, π/2.

Fig. 10
Fig. 10

Merger of two chirped solitons into one in the case when the initial pulse and phase separations are ητ=8 and ϕ=π.

Equations (17)

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

iuz+(1/2)Dutt+|u|2u=iu+iutt,
iuz+12D-iutt+|u|2u-iu=Kv,
ivz+iΓv+k0v=Ku.
u=u0 exp(ikz)[sech(η t)](1+iμ),
v=v0 exp(ikz)[sech(η t)](1+iμ),
μ=-(3/4)D+(1/4)32+9D2
v0=(k0-k+iΓ)-1Ku0,
-(μD-2)δ3+[(1-μ2)D+4µ+k0(μD-2)]δ2
+(μD-2)(K2-Γ2)δ+[(1-μ2)D+4µ]
×Γ(Γ-K2)+(μD-2)k0Γ2=0.
η2=δ(1-Γ)+k0Γδ(2-Dμ)+Γ[D(1-μ2)+4µ],
u02=(3/4)μ(4+D2)η2.
u=u1 exp[i(kz-ωt)],v=v1 exp[i(kz-ωt)],
K2Γ(1-ω2)1+(2k0+Dω2)24(Γ-1+ω2)2.
1<Γ<K2.
u=u0 sechηt-τ21+iμ+u0 sechηt+τ21+iμ exp(iϕ),
v=v0 sechηt-τ21+iμ+v0 sechηt+τ21+iμ exp(iϕ).

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