Abstract

We consider a model of a nonlinear planar waveguide with a sinusoidal modulation of the refractive index in the transverse direction, which gives rise to a system of parallel troughs that may serve as channels that trap solitary beams (spatial solitons). The model can also be considered as an asymptotic one describing a dense planar array of parallel nonlinear optical fibers, with the modulation representing the corresponding effective Peierls–Nabarro potential. By means of the variational approximation and by direct simulations we demonstrate that the one-soliton state trapped in a channel has no existence threshold and is always stable. In contrast with this a stationary state of two beams trapped in two adjacent troughs has an existence border, which is found numerically. Depending on the values of the parameters, the two-soliton states are found to be dynamically stable over an indefinitely long or a finite but large distance. We consider the possibility of switching the beam from a channel where it was trapped into an adjacent one by a localized spot attracting the beam through the cross-phase modulation. The spot can be created between the troughs by a focused laser beam shone transversely to the waveguide. By means of the perturbation theory and numerical method we demonstrate that the switching is possible, provided that the spot’s strength exceeds a certain threshold value.

© 1999 Optical Society of America

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References

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  1. S. Maneuf and F. Reynaud, “Quasi-steady-state self-trapping of first, second, and third order sub-nanosecond soliton beams,” Opt. Commun. 65, 325–328 (1988); J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
    [CrossRef] [PubMed]
  2. A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et. auto-confinement de Faisceasux laser par non linearte optique de Kerr,” Opt. Commun. 55, 201–206 (1985); S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear plasma waveguide,” Opt. Commun. 65, 193–198 (1988).
    [CrossRef]
  3. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
    [CrossRef]
  4. Q. Y. Li, C. Pask, and R. A. Sammut, “Simple model for spa-tial optical solitons in planar waveguides,” Opt. Lett. 16, 1083–1085 (1991).
    [CrossRef] [PubMed]
  5. A. W. Snyder and A. P. Sheppard, “Collision, steering, and guidance with spatial solitons,” Opt. Lett. 18, 482–484 (1993).
    [CrossRef] [PubMed]
  6. F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990); J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial optical soliton interactions,” Opt. Lett. 16, 15–17 (1991).
    [CrossRef] [PubMed]
  7. M. Shalaby and A. Barthelemy, “Experimental spatial soliton trapping and switching,” Opt. Lett. 16, 1472–1474 (1991).
    [CrossRef] [PubMed]
  8. S. Fan, P. R. Villeuneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3, 4–7 (1998).
    [CrossRef] [PubMed]
  9. A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995); V. V. Afanasjev, P. L. Chu, and Yu. S. Kivshar, “Breathing spatial solitons in non-Kerr media,” Opt. Lett. 22, 1388–1390 (1997); C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. PRLTAO 78, 4749–4752 (1997).
    [CrossRef]
  10. B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–453 (1993); Z. Chen, M. Segev, T. H. Goskun, and D. N. Christodoulides, “Observation of incoherently coupled photorefractive spatial soliton pairs,” Opt. Lett. 21, 1436–1437 (1996); D. N. Christodoulides, T. H. Goskun, M. Mitchell, and M. Segev, “Theory of incoherent focusing in biased photorefractive media,” Phys. Rev. Lett. PRLTAO 78, 646–649 (1997); W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. PRLTAO 80, 3240–3243 (1998).
    [CrossRef] [PubMed]
  11. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, UK, 1995).
  12. Y. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
    [CrossRef]
  13. A. Shipulin, G. Onishchukov, and B. Malomed, “Suppression of soliton jitter by a copropagating support structure,” J. Opt. Soc. Am. B 14, 3393–3402 (1997).
    [CrossRef]
  14. M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of wave equation in the media with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–785 (1973).
    [CrossRef]
  15. Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
    [CrossRef]

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

S. Fan, P. R. Villeuneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3, 4–7 (1998).
[CrossRef] [PubMed]

1997

1993

A. W. Snyder and A. P. Sheppard, “Collision, steering, and guidance with spatial solitons,” Opt. Lett. 18, 482–484 (1993).
[CrossRef] [PubMed]

Y. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

1991

1989

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

1973

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of wave equation in the media with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–785 (1973).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Barthelemy, A.

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Campbell, D. K.

Y. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Fan, S.

Haus, H. A.

Joannopoulos, J. D.

Kivshar, Y. S.

Y. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of wave equation in the media with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–785 (1973).
[CrossRef]

Li, Q. Y.

Malomed, B.

Malomed, B. A.

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Onishchukov, G.

Pask, C.

Sammut, R. A.

Shalaby, M.

Sheppard, A. P.

Shipulin, A.

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Snyder, A. W.

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of wave equation in the media with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–785 (1973).
[CrossRef]

Villeuneuve, P. R.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. E

Y. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Phys. Rev. Lett.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Radiophys. Quantum Electron.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of wave equation in the media with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–785 (1973).
[CrossRef]

Rev. Mod. Phys.

Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Other

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995); V. V. Afanasjev, P. L. Chu, and Yu. S. Kivshar, “Breathing spatial solitons in non-Kerr media,” Opt. Lett. 22, 1388–1390 (1997); C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. PRLTAO 78, 4749–4752 (1997).
[CrossRef]

B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, and G. Salamo, “Self-trapping of optical beams in photorefractive media,” J. Opt. Soc. Am. B 10, 446–453 (1993); Z. Chen, M. Segev, T. H. Goskun, and D. N. Christodoulides, “Observation of incoherently coupled photorefractive spatial soliton pairs,” Opt. Lett. 21, 1436–1437 (1996); D. N. Christodoulides, T. H. Goskun, M. Mitchell, and M. Segev, “Theory of incoherent focusing in biased photorefractive media,” Phys. Rev. Lett. PRLTAO 78, 646–649 (1997); W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. PRLTAO 80, 3240–3243 (1998).
[CrossRef] [PubMed]

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

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990); J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial optical soliton interactions,” Opt. Lett. 16, 15–17 (1991).
[CrossRef] [PubMed]

S. Maneuf and F. Reynaud, “Quasi-steady-state self-trapping of first, second, and third order sub-nanosecond soliton beams,” Opt. Commun. 65, 325–328 (1988); J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
[CrossRef] [PubMed]

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et. auto-confinement de Faisceasux laser par non linearte optique de Kerr,” Opt. Commun. 55, 201–206 (1985); S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear plasma waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Shapes of the one-soliton solution to Eq. (4) at =1 with (a) k=0.2 and (b) k=5.0. The shape of the soliton predicted by the variational approximation is shown by the dashed curves.

Fig. 2
Fig. 2

Solitary beam’s energy flux F versus propagation constant k for the numerically found stationary one-soliton solutions.

Fig. 3
Fig. 3

Existence border for the stationary states, in the form of two solitary beams trapped in adjacent troughs, on the plane (, k), as obtained from the numerical solution of Eq. (4). The two-soliton states exist above the curve.

Fig. 4
Fig. 4

Dynamical evolution of the state with two solitary beams trapped in two adjacent troughs at =-4 and k=2. The propagation distance is z=12.

Fig. 5
Fig. 5

Same as in Fig. 4 for =-10 and k=10. The propagation distance is z=12.

Fig. 6
Fig. 6

Example of the stationary two-soliton state taken very close to the stability border: =-4, k=0.7. Only the half of the symmetric two-soliton state at x>0 is shown; at x<0, U(x)=U(-x).

Fig. 7
Fig. 7

Results of the simulations of Eq. (1) with the initial condition (17), where the regularized delta function is taken as per identity (18). Shown is |u(x)| versus x at the points z=1, z=4, and z=8. The values of the parameters are =1, k=2 [k determines the function a0(x) in the initial condition (17)], μ=30, Δ=100, and x0=1/2 (the filled circle shows the location of the attracting spot amenable for the switching). The smooth curve shows the initial shape of the solitary beam [i.e., a0(x)] at z=0. The numerically generated curves were smoothed in order to remove some roughness induced by the numerical scheme that was used.

Equations (20)

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iuz+12uxx+ cos(qx)u+|u|2u=0,
u(x, z)=exp(ikz)U(x)
iuz+12uxx+ cos(2πx)u+|u|2u
=-μδ(x-x0)δ(z)u,
12U+[ cos(2πx)-k]U+U3=0.
L=14-+dx{(U)2+[2k- cos(2πx)]U2-U4}.
U=A sech(ηx),
η2-2π2 [2π2 cosh(π2/η)-3η sinh(π2/η)]η2 sinh2(π2/η)=2k,
A2=14η2+6k-6π2η sinh(π2/η).
Δϕ(x, z)ϕ(x, z=+0)-ϕ(x, z=-0)=μδ(x-x0).
a(x, z)=2k+c2 sech[2k+c2(x-cz-ξ)],
P=i-+ux*udx,
P=Mc,Ekin=P2/2M,M=22k+c2.
P=-+a2(x)Δϕ(x)dx=μ-+a2(x)δ(x-x0)dx-2µa(x0)a(x0).
Ekin=μ2(2k)5/2 sinh2(2kx0)sech6(2kx0).
W(ξ)=-π2sinh(π/2k)cos(2πξ)
ΔW=2π2sinh(π/2k).
μthr2=2π2(2k)5/2cosh6(2kx0)sinh(π/2k)sinh2(2kx0).
u(x, z=0)=a0(x)exp[iμδ˜(x-x0)],
δ˜(x-x0)Δ/π exp[-Δ(x-x0)2],

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