Abstract

The propagation of spatial soliton arrays in waveguides with nonlinear boundaries was studied theoretically. We found equilibrium states of the soliton arrays in a waveguide by employing soliton perturbation theory. The propagation of the array was shown to be accompanied by oscillations of the solitons’ positions and phases. The oscillation modes of the system were analyzed analytically and numerically, revealing the presence also of nonmechanical oscillations associated with the soliton phases.

© 2002 Optical Society of America

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References

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  1. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  2. M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
    [CrossRef]
  3. J. Scheuer and M. Orenstein, “Interactions and switching of spatial soliton pairs in the vicinity of a nonlinear interface,” Opt. Lett. 24, 1735–1737 (1999).
    [CrossRef]
  4. D. Arbel and M. Orenstein, “Self-stabilization of dense soliton trains in a passively mode-locked ring laser,” IEEE J. Quantum Electron. 35, 977–982 (1999).
    [CrossRef]
  5. J. M. Arnold, “Complex Toda lattice and its application to the theory of interacting optical solitons,” J. Opt. Soc. Am. A 15, 1450–1458 (1998).
    [CrossRef]
  6. V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
    [CrossRef]
  7. V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).
  8. D. J. Kaup and N. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
    [CrossRef]
  9. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  10. D. Anderson and M. Lisak, “Variational approach to incoherent two-soliton interactions,” Phys. Scr. 33, 193–196 (1986).
    [CrossRef]
  11. A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809–1827 (1989).
    [CrossRef] [PubMed]
  12. A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828–1840 (1989).
    [CrossRef] [PubMed]
  13. E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
    [CrossRef]

1999 (2)

D. Arbel and M. Orenstein, “Self-stabilization of dense soliton trains in a passively mode-locked ring laser,” IEEE J. Quantum Electron. 35, 977–982 (1999).
[CrossRef]

J. Scheuer and M. Orenstein, “Interactions and switching of spatial soliton pairs in the vicinity of a nonlinear interface,” Opt. Lett. 24, 1735–1737 (1999).
[CrossRef]

1998 (2)

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

J. M. Arnold, “Complex Toda lattice and its application to the theory of interacting optical solitons,” J. Opt. Soc. Am. A 15, 1450–1458 (1998).
[CrossRef]

1997 (1)

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

1994 (1)

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

1990 (1)

1989 (2)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828–1840 (1989).
[CrossRef] [PubMed]

1986 (1)

D. Anderson and M. Lisak, “Variational approach to incoherent two-soliton interactions,” Phys. Scr. 33, 193–196 (1986).
[CrossRef]

1983 (1)

1978 (1)

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

1977 (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828–1840 (1989).
[CrossRef] [PubMed]

Alvarado-Méndez, E.

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

Anderson, D.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

D. Anderson and M. Lisak, “Variational approach to incoherent two-soliton interactions,” Phys. Scr. 33, 193–196 (1986).
[CrossRef]

Arbel, D.

D. Arbel and M. Orenstein, “Self-stabilization of dense soliton trains in a passively mode-locked ring laser,” IEEE J. Quantum Electron. 35, 977–982 (1999).
[CrossRef]

Arnold, J. M.

Diankov, G. L.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Evstatiev, E. G.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Gerdjikov, V. S.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Gordon, J. P.

Haus, H. A.

Höök, A.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Karlsson, M.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Karpman, V. I.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

Kaup, D. J.

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

Lai, Y.

Lisak, M.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

D. Anderson and M. Lisak, “Variational approach to incoherent two-soliton interactions,” Phys. Scr. 33, 193–196 (1986).
[CrossRef]

Maslov, E. M.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828–1840 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828–1840 (1989).
[CrossRef] [PubMed]

Newell, N. C.

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

Orenstein, M.

J. Scheuer and M. Orenstein, “Interactions and switching of spatial soliton pairs in the vicinity of a nonlinear interface,” Opt. Lett. 24, 1735–1737 (1999).
[CrossRef]

D. Arbel and M. Orenstein, “Self-stabilization of dense soliton trains in a passively mode-locked ring laser,” IEEE J. Quantum Electron. 35, 977–982 (1999).
[CrossRef]

Sánchez-Mondragón, J. J.

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

Scheuer, J.

Torres-Cisneros, G. E.

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

Torres-Cisneros, M.

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

Uzunov, I. M.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Vysloukh, V.

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Arbel and M. Orenstein, “Self-stabilization of dense soliton trains in a passively mode-locked ring laser,” IEEE J. Quantum Electron. 35, 977–982 (1999).
[CrossRef]

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

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

Opt. Lett. (2)

Opt. Quantum Electron. (1)

E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. 30, 687–696 (1998).
[CrossRef]

Phys. Rev. A (2)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828–1840 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (1)

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Phys. Scr. (2)

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical solitons collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

D. Anderson and M. Lisak, “Variational approach to incoherent two-soliton interactions,” Phys. Scr. 33, 193–196 (1986).
[CrossRef]

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

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

Sov. Phys. JETP (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

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Figures (7)

Fig. 1
Fig. 1

Soliton array propagating in a nonlinear waveguide.

Fig. 2
Fig. 2

Propagation of a two-soliton array, a, in a homogeneous medium and, b, in a waveguide with n01=1.45, n02=1.499, and n21=n22=0.0625 (all coordinates are normalized).

Fig. 3
Fig. 3

a, BPM simulation of the antimode oscillation mode; b, positions and phase difference of the solitons according to the BPM (solid curves) and to the particle model (circles) for the anti-mode (all coordinates are normalized).

Fig. 4
Fig. 4

a, BPM simulation of the in-mode oscillation mode; b, positions and phase difference of the solitons according to the BPM (solid curves) and the particle model (circles) for the in mode (all coordinates are normalized).

Fig. 5
Fig. 5

a, BPM simulation of the oscillatory phase mode; b, positions and phase difference of the solitons according to the BPM (solid curves) and the particle model (circles) for the oscillatory phase mode (all coordinates are normalized).

Fig. 6
Fig. 6

Positions of, a, the right and, b, the left soliton, c, phase difference according to the full model (solid curves) and the approximate solution (dotted curves and circles) for the oscillatory phase mode (all coordinates are normalized).

Fig. 7
Fig. 7

Propagation of three-soliton and four-soliton arrays with arbitrary boundary conditions in a waveguide (all coordinates are normalized).

Equations (37)

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2iβk0E(z, x)z+2E(z, x)x2-k02{β2-[n02(x)+2n0(x)n2(x)|E(z, x)|2]},E(z, x)=0,
n0(x)=n01-d<x<0n02x<-d;x>0, n2(x)=n21-d<x<0n22x<-d;x>0.
iAz+2Ax2+2A|A|2=WA,
W=n012-n022-2n02n22n01n21-1|A|2x<-d;x>00-d<x<0.
A=A0 sech[A0(x-x1)]+exp(iΔψ)A0 sech[A0(x-x2)],
Δx¨=-32A03 cos(Δψ)exp(-A0|Δx|),
Δψ¨=-32A04 sin(Δψ)exp(-A0|Δx|),
x¯=p-1 - xAA*dx.
d2x¯dz2=-2p-1 -WxAA*dx.
A(x, z)=A0 sech{A0[x-x¯(z)]}exp[ivx/2+η(z)].
d2x¯dz2=-x¯UL(x¯),
UL(x¯)=2p-1 Δ|A(x¯)|2-(α-1-1)|A(x¯)|4dx¯=Δ[1-(αS0)-1]tanh(A0x¯)+Δ(3αS0)-1 tanh3(A0x¯),
x¨n=-A0Δ sech2(A0xn)[1-sech2(A0xn)/αS0]-16A03 cos(φn-φn-1)exp(-A0|xn-xn-1|),
φ¨n=16A04 sin(φn-φn-1)exp(-A0|xn-xn-1|),
x¨i=16A03 cos(φi-φi+1)exp(-A0|xi-xi+1|)-16A03 cos(φi-φi-1)exp(-A0|xi-xi-1|),
φ¨i=16A04 sin(φi-φi+1)exp(-A0|xi-xi+1|)+16A04 sin(φi-φi-1)exp(-A0|xi-xi-1|).
φi-φi-1=π,2in,
xi-xi-1=const.2in.
Δ sech2(A0ΔxSI)1-sech2(A0ΔxSI)/αS0
=16A02exp(-A0|ΔxSS|),
x1=x01+Δx1,
x2=x02+Δx2,
Δψ=ψ2-ψ1=Δψ0+δψ,
d2Δx2dz2=-(k1+k2)Δx2+k2Δx1-12k2δψ2,
d2Δx1dz2=-(k1+k2)Δx1+k2Δx2+12k2δψ2,
k1=-2A02Δ sech2(A0x02)tanh(A0x02)×1-2 sech2(A0x02)αS0,
k2=16A04 exp(-A0|x01-x02|).
d2δψdz2=-2k2δψ.
w1=2k2+k1,
w2=k1.
δψ=δψ0 cos2k2z,
w1=k1+k2,
w2={3k2+k1+{[(3k2+k1)2-8k2k1]1/2}1/2/2,
w3=[3k2+k1-{[(3k2+k1)2-8k2k1]1/2}1/2/2,
δψ¨21=-k2(δψ23+2δψ21),
δψ¨23=-k2(2δψ23+δψ21).
w4=3k2,w5=k2,

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