Abstract

The switching of optical solitons in a nonlinear fiber coupler is investigated by an approximate variational method that includes the effect of dispersive radiation. The solutions of the variational equations are in good agreement with full numerical solutions of the coupled nonlinear Schrödinger equations that govern the fiber coupler. In particular, it is found that to obtain an accurate description of the switching process and of the evolution of the solitary waves in each core of the fiber coupler one must include the effect of the dispersive radiation shed as the solitary waves evolve. As a consequence, the present variational method gives much improved results compared with other variational methods that do not include dispersive radiation.

© 1997 Optical Society of America

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References

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  1. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
    [CrossRef]
  2. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 13, 904–906 (1988).
    [CrossRef] [PubMed]
  3. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
    [CrossRef] [PubMed]
  4. M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
    [CrossRef]
  5. F. Kh. Abdullaev, R. M. Abrarov, and S. A. Darmanyan, “Dynamics of solitons in coupled optical fibers,” Opt. Lett. 14, 131–133 (1989).
    [CrossRef] [PubMed]
  6. Y. S. Kivshar, and B. A. Malomed, “Interaction of soliton in tunnel-coupled optical fibers,” Opt. Lett. 14, 1365–1367 (1989).
    [CrossRef] [PubMed]
  7. N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
    [CrossRef] [PubMed]
  8. J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
    [CrossRef]
  9. N. Akhmediev and J. M. Soto-Crespo, “Propagation dynamics of ultrashort pulses in nonlinear fiber couplers,” Phys. Rev. E 49, 4519–4529 (1994).
    [CrossRef]
  10. G. D. Peng and A. Ankiewicz, “Fundamental and second order soliton transmission in nonlinear directionalfiber couplers,” Int. J. Nonlinear Opt. Phys. 1, 135–150 (1992).
    [CrossRef]
  11. P. L. Chu, G. D. Peng, and B. A. Malomed, “Analytical solution to soliton switching in nonlinear twin-core fibers,” Opt. Lett. 18, 328–330 (1993).
    [CrossRef] [PubMed]
  12. P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analyticalresults,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
    [CrossRef]
  13. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  14. W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödingerequation,” Phys. Rev. E 51, 1484–1492 (1995).
    [CrossRef]
  15. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödingerequation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  16. W. L. Kath and N. F. Smyth, “Effect of radiation oncoupled solitons in nonlinear optical fibers,” to be submitted to Phys.Rev. E.
  17. W. L. Kath and N. F. Smyth, “Radiational instabilityof solitary waves in nonlinear optical fibers,” to be submitted to PhysicaD.
  18. D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: asingular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
    [CrossRef]
  19. E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-modefibers,” Phys. Rev. A 40, 4455–4466 (1989).
    [CrossRef] [PubMed]
  20. M. J. Miksis and L. Ting, “A numerical method for long time solutions of integrodifferential systemsin multi-phase flow,” Comput. Fluids 16, 327–340 (1988).
    [CrossRef]
  21. B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
    [CrossRef]

1995 (1)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödingerequation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

1994 (1)

N. Akhmediev and J. M. Soto-Crespo, “Propagation dynamics of ultrashort pulses in nonlinear fiber couplers,” Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

1993 (4)

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

P. L. Chu, G. D. Peng, and B. A. Malomed, “Analytical solution to soliton switching in nonlinear twin-core fibers,” Opt. Lett. 18, 328–330 (1993).
[CrossRef] [PubMed]

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analyticalresults,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[CrossRef]

1992 (3)

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödingerequation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

G. D. Peng and A. Ankiewicz, “Fundamental and second order soliton transmission in nonlinear directionalfiber couplers,” Int. J. Nonlinear Opt. Phys. 1, 135–150 (1992).
[CrossRef]

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

1989 (3)

1988 (3)

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

1978 (2)

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: asingular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Abdullaev, F. Kh.

Abrarov, R. M.

Akhmediev, N.

N. Akhmediev and J. M. Soto-Crespo, “Propagation dynamics of ultrashort pulses in nonlinear fiber couplers,” Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

G. D. Peng and A. Ankiewicz, “Fundamental and second order soliton transmission in nonlinear directionalfiber couplers,” Int. J. Nonlinear Opt. Phys. 1, 135–150 (1992).
[CrossRef]

Chu, P. L.

Darmanyan, S. A.

Fornberg, B.

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

Friberg, S. R.

Gordon, J. P.

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Kath, W. L.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödingerequation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Kaup, D. J.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: asingular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Kivshar, Y. S.

Malomed, B. A.

Miksis, M. J.

M. J. Miksis and L. Ting, “A numerical method for long time solutions of integrodifferential systemsin multi-phase flow,” Comput. Fluids 16, 327–340 (1988).
[CrossRef]

Newell, A. C.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: asingular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Peng, G. D.

Romagnoli, M.

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

Sfez, B. G.

Silberberg, Y.

Smith, P. S.

Smyth, N. F.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödingerequation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Soto-Crespo, J. M.

N. Akhmediev and J. M. Soto-Crespo, “Propagation dynamics of ultrashort pulses in nonlinear fiber couplers,” Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

Stegeman, G. I.

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-modefibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Ting, L.

M. J. Miksis and L. Ting, “A numerical method for long time solutions of integrodifferential systemsin multi-phase flow,” Comput. Fluids 16, 327–340 (1988).
[CrossRef]

Trillo, S.

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Wabnitz, S.

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-modefibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Weiner, A. M.

Whitham, G. B.

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

Wright, E. M.

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-modefibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Comput. Fluids (1)

M. J. Miksis and L. Ting, “A numerical method for long time solutions of integrodifferential systemsin multi-phase flow,” Comput. Fluids 16, 327–340 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Int. J. Nonlinear Opt. Phys. (1)

G. D. Peng and A. Ankiewicz, “Fundamental and second order soliton transmission in nonlinear directionalfiber couplers,” Int. J. Nonlinear Opt. Phys. 1, 135–150 (1992).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (5)

Opt. Quantum Electron. (1)

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (1)

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

Phys. Rev. A (2)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-modefibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (3)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödingerequation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

N. Akhmediev and J. M. Soto-Crespo, “Propagation dynamics of ultrashort pulses in nonlinear fiber couplers,” Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: asingular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Other (2)

W. L. Kath and N. F. Smyth, “Effect of radiation oncoupled solitons in nonlinear optical fibers,” to be submitted to Phys.Rev. E.

W. L. Kath and N. F. Smyth, “Radiational instabilityof solitary waves in nonlinear optical fibers,” to be submitted to PhysicaD.

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Figures (5)

Fig. 1
Fig. 1

Transmission coefficient T at z=1 for a nonlinear coupler as a function of the input energy E(0) normalized by the critical energy E0c. Here K=π/2. Solid curve, numerical solution; longer-dashed curve, present approximate equations; shorter-dashed curve, approximate equations by the method of Anderson.13

Fig. 2
Fig. 2

Amplitude of pulses as a function of z at t=0 for parameter values A=1.5 and K=π/2 with P=0.0 and ψ=0.0 at z=0. (a) Pulse u, (b) pulse v. Solid curves, numerical solution; longer-dashed curves, present approximate equations; shorter-dashed curves, approximate equations by the method of Anderson.13 (c) Pulse u. Solid curve, numerical solution; longer-dashed curve, present approximate equations; shorter-dashed curve, present approximate equations without dispersive radiation damping (i.e., α=0).

Fig. 3
Fig. 3

Amplitude of pulses as a function of z at t=0 for parameter values A=4.0 and K=π/2 with P=0.0 and ψ=0.0 at z=0. Solid curves, numerical solution; longer-dashed curves, present approximate equations; shorter-dashed curves, approximate equations by the method of Anderson.13 (a) Pulse u, (b) pulse v.

Fig. 4
Fig. 4

Pulse profiles at z=10 as a function of t for A=4 and K=π/2 with P=0.0 and ψ=0.0 at z=0. Solid curves, numerical solution. (a) |u|. Longer-dashed curve, |η cos P sech t/w| from the present approximate equations; shorter-dashed curve, |η cos P sech t/w+ig cos P| from the present approximate equations. (b) |v|. Longer-dashed curve, |η sin P sech t/w| from the present approximate equations; shorter-dashed curve, |η sin P sech t/w+ig sin P| from the present approximate equations.

Fig. 5
Fig. 5

Pulse profiles at z=15 as a function of t for A=4 and K=π/2 with P=0.0 and ψ=0.0 at z=0. Solid curves, numerical solution. (a) |u|. Longer-dashed curve, |η cos P sech t/w| from the present approximate equations; shorter-dashed curve, |η cos P sech t/w+ig cos P| from the present approximate equations. (b) |v|. Longer-dashed curve, |η sin P sech t/w| from the present approximate equations; shorter-dashed curve, |η sin P sech t/w+ig sin P| from the present approximate equations.

Equations (39)

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i uz+122ut2+|u|2u+Kv=0,
i vz+122vt2+|v|2v+Ku=0.
u(0, t)=A sech At,v(0, t)=0.
L=½i(u*uz-uuz*)-½|ut|2+½|u|4+K(u*v+uv*)+½i(v*vz-vvz*)-½|vt|2+½|v|4,
u=η cos P sech twexp(iσ-iψ/2)+ig cos P exp(iσ-iψ/2),
v=η sin P sech twexp(iσ+iψ/2)+ig sin P exp(iσ+iψ/2)
L=-Ldt,
L=-2η2w[σ-1/2ψ cos(2P)]+πgwη+πgηw-πηwg-g2l[σ-1/2ψ cos(2P)]-13η2w+13η4w[cos2(2P)+1]+g2η2w[cos2(2P)+1]+K(2η2w+lg2)sin(2P)cos ψ
(ηw)=l2π×g{η2[cos2(2P)+1]-w-2},
g=-23π×η{1/2η2[cos2(2P)+1]-w-2},
σ-1/2 cos(2P)ψ
=-1/2w-2+1/2η2[cos2(2P)+1]+K sin(2P)cos ψ,
ddz[(2η2w+lg2)cos(2P)]
=-2K(2η2w+lg2)sin(2P)sin ψ,
(2η2w+lg2)ψ=-4/3η4w cos(2P)+2K(2η2w+lg2)×cot(2P)cos ψ,
ddz(2η2w+lg2)=0
ddz-(|u|2+|v|2)dt=0,
ddz-[|ut|2+|vt|2-|u|4-|v|4-2K(uv*-u*v)]dt
=0,
dHdz=ddzη2w-η4w[cos2(2P)+1]-3K(2η2w+lg2)sin(2P)cos ψ=0.
ηˆ3+12Kηˆ cos ψˆ+2H0=0.
sin(2Pˆ)=3K cos ψˆηˆ2,
wˆ-2=ηˆ2-9K22ηˆ2,
ηˆ4+272K2+H0ηˆ2-9K22ηˆ21/2=0.
i uz+122ut2=-Kv,
i vz+122vt2=-Ku.
ddzl/2(|u|2+|v|2)dt=Im(u*ut+v*vt)|t=l/2.
ddz(2η2w+lg2)=-2r ddz0z r(ζ)[π(z-ζ)]1/2dζ,
r2=ηˆ8M(3+2S2M2)[2η2w-2ηˆ2wˆ+lg2]
M=ηˆwˆ,
S=sin(2Pˆ).
g=-23πη12η2[cos2(2P)+1]-w-2-2αg,
α=ηˆ(3+2S2M2)8rMddz0z r[π(z-ζ)]1/2dζ.
l=3π2wˆ5214ηˆ2[cos2(2Pˆ)+1]+K sin(2Pˆ)cos ψˆ2.
α=η(3+2S2M2)8rMddz0z r[π(z-ζ)]1/2dζ,
l=3π2w5214η2[cos2(2P)+1]+K sin(2P)cos ψ2.
A2K452/361/36.962.
T=E(z)E(0)=|u(z, 0)|2|u(0, 0)|2.
E0c=452/361/3K

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