Abstract

The interaction of two colored solitons was analyzed in the framework of a particle-like model, derived from a soliton perturbation theory. From “energy” considerations, a soliton capture threshold and the re-coloring of the escaping solitons were derived. The results were compared to the spectral boundaries of a second order soliton as well as to previous reports. The capture of colored solitons was shown to be impractical without additional means. This particle-like model was further generalized to apply also for non-equal intensity colored solitons. Detailed calculations—beyond the particle-like approximation, exhibited additional mechanisms, namely dissipation and friction-like forces, which served as sources for the relaxation of the solitons oscillations within the captured state, thus enhancing the capture phenomenon.

© 2004 Optical Society of America

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References

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Elec. Lett. (1)

M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, �??Infinite-distance soliton transmission with soliton controls in time and frequency domains,�?? Elec. Lett. 28, 1099-1100 (1992).
[CrossRef]

IEEE J. of Quant. Elec. (1)

D. Arbel and M. Orenstein �??Self stabilization of dense soliton trains is passively mode-locked ring laser,�?? IEEE J. of Quant. Elec. 35, 977-982 (1999)
[CrossRef]

IEEE Phot. Tech. Lett. (1)

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, �??Cascade of ultrafast soliton-dragging and trapping logic gates,�?? IEEE Phot. Tech. Lett. 4, 1043-1046 (1992)
[CrossRef]

J. Eng. Math. (1)

C.R. Menyuk, �??Application of multiple-length-scale method to the study of optical fiber transmission,�?? J. Eng. Math. 36, 113-136 (1999).
[CrossRef]

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

J. Selected Areas In Com. (1)

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, �??All-optical network consortium �?? ultrafast TDM networks,�?? J. Selected Areas In Com. 14, 999-1012 (1996).
[CrossRef]

OFC 2001 (1)

A. Agarwal, L. Wang, Y. Su, and P. Kumar, �??All-optical erasable storage buffer based on parametric nonlinearity in fiber,�?? OFC 2001, Th-H5 (2001)

Opt. Eng. (1)

M.F. Mahmood, W.W. Zachary, and T.L. Gill, �??Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,�?? Opt. Eng. 35, 1844-1846 (1996)
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (2)

D.J. Kaup and B.A. Malomed, �??Soliton trapping and daughter waves in the Manakov model,�?? Phys. Rev. A 48, 599-604 (1993)
[CrossRef] [PubMed]

B.A. Malomed, �??Polarization dynamics and interactions of solitons in a birefringent optical fiber,�?? Phys. Rev. A 43, 410-423 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (2)

N. C. Panoiu, I. V. Mel�??nikov, D. Mihalache, C. Etrich, and F. Lederer, �??Soliton generation in optical fibers for dual-frequency input,�?? Phys. Rev. E 60, 4868-4876 (1999).
[CrossRef]

B.A. Malomed and R.S. Tasgal , �??Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,�?? Phys. Rev. E 58, 2564-2575 (1998)
[CrossRef]

Phys. Scripta (1)

M. Karlsson, D Anderson, A Höök, and M. Lisak, �??A variational approach to optical soliton collisions,�?? Phys. Scripta 50, 265-270 (1994).
[CrossRef]

Rev. Mod. Phys. (1)

H.A. Haus and W.S. Wong, �??Solitons in optical communications,�?? Rev. Mod. Phys. 68, 423-444 (1996)
[CrossRef]

Other (7)

E. Feigenbaum and M. Orenstein, �??Mutual Capture of Colored Solitons Assisted by Matched Modulator,�?? Submitted

E. Feigenbaum, �??Multi-colored optical storage rings �?? solitons interaction,�?? Thesis (EE Department - Technion, Israel, 2003)

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons �?? The inverse scattering method, (Plenum Press, New York ,1984), 68-79

G.P. Agrawal, Nonlinear Fiber Optics, 2�??nd Ed, (Academic Press, NY, 1995), Chap. 2,7.

Y.R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), 40.

M.N. Islam, Ultrafast Fiber Switching Devices and Systems, (Cambridge, GB, 1992).

H.A. Haus, �??Lecture 11�?? in Optical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz, ed. (Springer, NY, 1999).

Supplementary Material (3)

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Figures (9)

Fig. 1.
Fig. 1.

Perturbation calculation of escaping colored solitons. p0=0.145×2π, τ0=5. (a) Center (τ) of escaping soliton (blue) vs. the center of non-perturbed one (black). (b) Solitons carrier (red) vs. center accumulated perturbation (i.e. difference of the curves depicted at figure 1(a)) (blue).

Fig. 2.
Fig. 2.

The soliton trace in the energy plane. The red and blue curves are for p0=-0.137×2π and -0.16×2π respectively with τ0=0.

Fig. 3.
Fig. 3.

Simulation (XPM coupled NLSEs) of equal intensity colored solitons interaction. The intensity envelope of one soliton as the two solitons propagate simultaneously in the fiber for: (a) (1MB) Escape (p0=-0.20×2π), (b) (1MB) Intermediate (p0=-0.15×2π) and (c) (0.83MB) Capture (p0=-0.07×2π). τ0=0.

Fig. 4.
Fig. 4.

Solitons re-coloring vs. initial frequency (τ0=0).

Fig. 5.
Fig. 5.

Trajectories of re-coloring in the energy plane.

Fig. 6.
Fig. 6.

Temporal amplitude calculated by XPM simulations for one of the captured solitons (contour) and the soliton center calculated using Eq. (9) (bold curve). τ0=0, θ1= θ2=0, p0=-0.07×2π.

Fig. 7.
Fig. 7.

Comparison of solitons capture using full fledged NLSE simulation (contours) and the perturbation calculations (bold curves) curves: (a) soliton center, (b) center frequency. p0=0.05×2π, θ12=0, τ0=0.

Fig. 8.
Fig. 8.

A comparison of a second order soliton spectrum at its temporal propagation peaks and the combined spectrum of two co-centered colored solitons at capture threshold. The two colored solitons parameters: Dτ=0, W=1, Dp=2×0.193×2π, Dθ=0.

Fig. 9.
Fig. 9.

Re-coloring (pf/ p0 vs. p0) within particle-like model predictions compared to full fledged simulation. W2=1.5, W1=1, τ0=0.

Equations (51)

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z u = j { 1 2 β " 2 T + δ u 2 } u ,
u = W sech { ε W ( T τ ) } exp { j ( pT + θ ) } ,
z τ = β " p ,
z θ = 1 2 ( δ W 2 + β " p 2 ) ,
z Δ W = S W ,
z Δ τ = S τ + β " Δ p ,
z Δ p = S p ,
z Δ θ = S θ + δ W ΔW ,
S m Im dT { f m * s ( z , T ) exp { j 2 W 2 z } } .
z W = S W ,
z τ = S τ + β " p ,
z p = S p ,
z θ = S θ + δW ( S W dz ) + 1 2 ( δ W 2 + β " p 2 ) .
u 1 = W 1 sech { ε W 1 ( T τ 1 ) } exp { j ( p 1 T + θ 1 ) } ,
u 2 = W 2 sech { ε W 2 ( T τ 2 ) } exp { j ( p 2 T + θ 2 ) } ,
z u 1 = j ( 1 2 β " 2 T + δ ( u 1 2 + 2 u 2 2 ) ) ,
z u 2 = j ( 1 2 β " 2 T + δ ( u 2 2 + 2 u 1 2 ) ) .
δ ( u 2 ) 2 u 1 * + δ ( u 1 ) 2 u 2 * .
s XPM = 2 δ u 2 2 u 1 .
S W XPM = 0 ,
S τ XPM = 0 ,
S p XPM = 2 δ ε 2 W 1 2 W 2 2 dT { tanh ( ε W 1 ( T τ 1 ) ) ·
· sech 2 ( ε W 1 ( T τ 1 ) ) sech 2 ( ε W 2 ( T τ 2 ) ) } ,
S θ XPM = 2 δ ε W 1 W 2 2 dT { ( 1 ε W 1 ( T τ 1 ) tanh ( ε W 1 ( T τ 1 ) ) ) ·
· sech 2 ( ε W 1 ( T τ 1 ) ) sech 2 ( ε W 2 ( T τ 2 ) ) } ,
F ( ) = ( 2 δ β " ε W 3 ) + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( ξ ) } ,
z ( β " p ) = F ( 2 τ ) = F ( 2 τ ) ,
z τ = β " p ,
2 z τ = F ( 2 τ ) .
Δ τ ( z : + ) = εW p 0 2 ,
2 z τ ( 16 15 δ 2 W 4 ) F L = 2 F L τ .
E k = 1 2 ( β " ) 2 p 2 .
E p = 2 ( β " ε W ) 2 1 ( 2 ε W τ ) coth { 2 ε W τ } sinh 2 { 2 ε W τ } .
E p min = 2 3 ( β " ε W ) 2 .
p 0 TH = 4 3 ε W .
RC = p f p 0 = 1 4 3 ( ε W ) 2 p 0 2 ,
E k 0 = 1 1 RC 2 E p 0 .
2 z τ = 2 F L τ V z τ .
z ( β " p k ) = 2 ( δ W k W 3 k ) 2 + { tanh ( ε W k ξ ) sech 2 ( ε W k ξ ) sech 2 ( ε W 3 k ( ξ ( τ 3 k τ k ) ) ) } ,
z τ k = β " p k ; k∈ 1,2 .
m 2 m 1 t 2 r 1 t 2 r 2 = + { tanh { ε W 1 ξ } sech 2 { ε W 1 ξ } sech 2 { ε W 2 ( ξ ) } } , + { tanh { ε W 2 ξ } sech 2 { ε W 2 ξ } sech 2 { ε W 1 ( ξ ) } } , = W 2 W 1 .
m k = W k
z ( β " p k ) = F m k ,
F = 2 β " δ ε ( W 1 W 2 ) 2 + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( W 2 W 1 ξ ε W 2 ) } .
μ 2 t r = F .
E K = 1 2 μ ( t r ) 2 = 2 W 1 W 2 W 1 + W 2 ( β " p 0 ) 2 ,
E p = 2 β " δ ε ( W 1 W 2 ) 2 Δ τ + dr { + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( W 2 W 1 ξ ε W 2 r ) } } .
E pmin = 2 β " δ ε W 1 2 W 2 + { tanh ( ξ ) sech 2 ( ξ ) tanh ( W 2 W 1 ξ ) } .
p 0 TH = 1 3 ε ( W 1 + W 2 ) ; W 2 W 1 .
p o TH ( W 1 , W 2 ) = 1 2 { p o TH ( W 1 , W 1 ) + p o TH ( W 2 , W 2 ) } .
Dp = ( D p 0 ) 2 4 3 ε 2 ( W 1 + W 2 ) 2 ; W 1 W 2 ,

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