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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 12347–12352
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Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion

A.V. Yulin, D.V. Skryabin, and A.G. Vladimirov  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 12347-12352 (2006)
http://dx.doi.org/10.1364/OE.14.012347


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Abstract

We study temporal modulational instability of spatial discrete solitons in waveguide arrays with group velocity dispersion (GVD). For normal GVD we report existence of the strong ‘neck’-type instability specific for the discrete solitons. For anomalous GVD the instability leads to formation of the mixed discrete-continuous spatio-temporal quasi-solitons. Feasibility of experimental observation of these effects in the arrays of silicon-on-insulator waveguides is discussed.

© 2006 Optical Society of America

1. Introduction

We should also mention here that the MIs of spatially extended, i.e. non-localized, super-modes of the waveguide arrays induced by the discrete diffraction have been recently observed experimentally [19

19. J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J.S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92163902 (2004). [CrossRef] [PubMed]

] and previously studied theoretically [20

20. D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 16, 446–448 (1991). [CrossRef] [PubMed]

]. Papers [21

21. S. Darmanyan, I. Relke, and F. Lederer, “Instability of continuous waves and rotating solitons in waveguide arrays,” Phys. Rev. E 55, 7662–7668 (1997). [CrossRef]

, 22

22. M. Stepic, L. Hadzieski, and M. Skoric, “Stability of one-dimensional array solitons,” Phys. Rev. E 65, 026604 (2002). [CrossRef]

] studied the same case of diffraction induced MI, but for the supermodes consisting from the temporal solitons. Spatial MI of spatial surface solitons in optical lattices has been recently reported [23

23. Y.V. Kartashov and L. Torner, “Surface soliton arrays,” Eur. Opt. Soc. Topical Meeting on Nonlinear Optics, TOM 6 (Paris, 2006).

]. However, none of the known to us studies explored the problems of GVD induced instabilities of the spatial discrete solitons.

2. Model

We model an array of dielectric waveguides by a set of coupled NLS equations

iζUn12β2τ2Un+κ(Un+1+Un12Un)+γUn2Un=0,n=1,2N,
(1)

with periodic boundary conditions U N+1=U 1 and U 0=UN . Here n enumerates the waveguides, τ and ζ are the time and coordinate along the waveguide, respectively. γ=2πn 2/() is the nonlinearity parameter, where S is the effective mode area and n 2 is the Kerr coefficient. β 2 is the GVD coefficient. κ=π/(2lc ) is the coupling parameter and lc is the coupling length. In order to put Eqs. (1) into dimensionless form we divide them by some fixed length l. Then in terms of dimensionless propagation distance z=ζ/l and dimensionless time t=τβ2l Eqs. (1) take the form

izAn12st2An+C(An+1+An12An)+An2An=0, ,

with C=πl/(2lc ), s=sign(β 2), and An=Unγl. .

3. Stability analysis

Let us consider time-independent discrete soliton solutions of Eqs. (1) [20

20. D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 16, 446–448 (1991). [CrossRef] [PubMed]

]. These solutions having the form Un =aneiqz can be found numerically, see Figs. 1(a,d,g). It is known that the discrete solitons are dynamically stable for β 2=0 and the Vakhitov-Kolokolov stability criterion qQ>0, where Q=n=1N|an|2, is satisfied for them [24

24. E.W. Laedke, K.H. Spatschek, and S.K. Turitsyn, “Stability of Discrete Solitons and Quasicollapse to Intrinsically Localized Modes,” Phys. Rev. Lett. 73, 1055–1059 (1994). [CrossRef] [PubMed]

]. As it is shown below the instabilities can arise as soon as GVD is taken into account.

To study stability with respect to time-dependent perturbations, we make the following ansatz

An=[an+εn,+eiωtiλz+εn,*eiλ*ziωt]eiqz.
(2)

Here ω is the perturbation frequency. The linearized equations for the amplitudes of small perturbations ε n can be transformed into the operator form

λε=L̂0ε+12sω2L̂1ε,
(3)
Fig. 1. MI of discrete solitons. The first row corresponds to C=7, the second to C=15 and the third one to C=30, respectively. q=10 and N=51 for all the panels. The right column shows transverse profiles of the discrete solitons. The middle column presents the frequency dependence of the MI growth rate (Iml>0) in the anomalous GVD regime (s<0). The right column shows all the unstable eigenvalues in the case of the normal GVD (s>0). Letters ‘N’ and ‘S’ mark the ‘neck’ and ‘snake’ instabilities, respectively.

where ε⃗=(ε 1,+,ε 1,- …, ε N,+,ε N,-) T and 1 is the diagonal N×N matrix: 1=diag(1, -1, …, 1, -1). The matrix 0 has the form

[q~2γa12γa12C0C0γa12q~+2γa120C0CC0q~2γa22γa22000Cγa22q~+2γa2200C000q~2γaN2γaN20C00γaN2q~+2γaN2],
(4)

where =q+2C. MI of discrete solitons manifests itself through a growth of perturbations in a certain range of frequencies ω. This means that in this range there exists an eigenvalue of the problem (3) such that Im(λ)>0. According to the classical results on MI of the bright solitons in the continuous NLS model [10

10. V.E. Zakharov and A.M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Zh. Eksp. Teor. Fiz.65, 997–1011 (1973) [Sov. Phys. JETP 38, 494 (1974).

] the eigenvectors of the corresponding eigenvalue problem can be either symmetric or antisymmetric with respect to the reflection about the soliton center. The instabilities associated with symmetric and antisymmetric eigenvectors are usually referred to as a ‘neck’-instability and ‘snake’-instability, respectively [10

10. V.E. Zakharov and A.M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Zh. Eksp. Teor. Fiz.65, 997–1011 (1973) [Sov. Phys. JETP 38, 494 (1974).

, 11

11. Y.S. Kivshar and D.E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 118–195 (2000). [CrossRef]

, 12

12. A.V. Buryak, P. Di Trapani, D.V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002). [CrossRef]

]. We will adopt the same terminology in our stability analysis of the discrete solitons. If the soliton is centered at n=n 0, then the eigenvectors of Eq. (3) can be either symmetric on the replacement of n 0+m with n 0-m (as the soliton itself) or antisymmetric.

Fig. 2. The left column shows patterns of the ‘neck’ instability for anomalous GVD (s< 0) for 3 consequential values of the propagation distance z: C=7. The middle column shows patterns of the ‘neck’ instability for normal GVD (s>0): C=7. The right column shows patterns of the ‘snake’ instability for normal GVD (s>0): C=30. q=10 and N=51 for all the panels.

Assuming that ω≪1 we write the following asymptotic expansions for the eigenvalue and corresponding eigenvector: λ=ωλ 1+ω 2 λ 2 + … and x⃗=x⃗ 0 + 1 ωx 1 +ω 2 x⃗ 2 + …. Substituting these expansions into Eqs. (3), we get in the third order of the perturbation theory the solvability condition λ12=sQ/(2qQ). According to this condition the long wavelength instability of the ‘neck’-type takes place for s<0 (anomalous GVD). This result is not surprising because essentially the same instability persists even for κ=C=0. The middle column in Fig. 1 shows the ‘neck’ instability growth rates for different values of C. The low frequency part of the instability growth rates is approximately described by the above analytical expression. Direct numerical modeling of Eqs. (1) with the initial condition corresponding to a discrete soliton shows in this case formation of the regular trains of spatio-temporal quasi-solitons, which are discrete in space and continuous in time, see the left column in Fig. 2. Thus development of MI for the anomalous GVD case is qualitatively similar to the MI of the spatial solitons in the continuous NLS equation with saturable nonlinearity [25

25. N. Akhmediev and J.M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a selffocusing medium,” Phys. Rev. A 47, 1358–1364 (1993). [CrossRef] [PubMed]

].

In the case of normal GVD (s>0) we have found complex instability spectra consisting from multiple sidebands, see the rightmost column in Fig. 1. These type of spectra appear to be specific to the discrete solitons. For the relatively small coupling strength, i.e. sufficiently far from the continuous limit, the dominant instability is of the ‘neck’-type. This instability leads to the break-up of the initial soliton to the localized lumps of light, which disperse with further propagation, see the middle column in Fig. 2. Contrary, for the continuous 2D NLS with the normal GVD the anti-symmetric ‘snake’-like instability dominates dynamics of 1D bright solitons [10

10. V.E. Zakharov and A.M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Zh. Eksp. Teor. Fiz.65, 997–1011 (1973) [Sov. Phys. JETP 38, 494 (1974).

, 26

26. B. Deconinck, D.E. Pelinovsky, and J.D. Carter, “Transverse instabilities of deep-water solitary waves,” Proc. of the Royal Soc. A 462, 2039–2061 (2006). [CrossRef]

]. More, recent studies [26

26. B. Deconinck, D.E. Pelinovsky, and J.D. Carter, “Transverse instabilities of deep-water solitary waves,” Proc. of the Royal Soc. A 462, 2039–2061 (2006). [CrossRef]

] have demonstrated that the symmetric ‘neck’-type eigenvectors also can be unstable in the continuous hyperbolic 2D NLS, with their growth rate been below the one for the ‘snake’ MI. As we have found in our model the ‘neck’ instability dominates the dynamics of the discrete solitons for small to moderate values of the coupling coefficient, see Fig. 1. Only for strong coupling, when the system becomes quasi-continuous, the ‘snake’ instability starts to be dominant over the ‘neck’ one. This leads to the break up of the discrete soliton in the snake-like fashion, see the right most column in Fig. 2. Note, that the dominant MI band of the ‘neck’-type found for small frequencies ω close to zero is associated with a pair of complex eigenvalues λ. At the same time the dominant band of the ‘snake’ MI and the strongest peak of the ‘neck’- MI at the relatively large frequencies ω have purely imaginary eigenvalues λ.

4. Physical estimates

As a guideline for physical estimates we consider parameters typical for SOI waveguides. In particular for a channel waveguide with width 480nm and thickness 220nm [16

16. P. Dumon, G. Priem, L.R. Nunes, W. Bogaerts, D. van Thourhout, P. Bienstman, T.K. Liang, M. Tsuchiya, P. Jaenen, S. Beckx, J. Wouters, and R. Baets, “Linear and nonlinear nanophotonic devices based on silicon-oninsulator wire waveguides,” Jap. J. of Appl. Phys. 45, 6589–6602 (2006). [CrossRef]

] GVD at 1.5µm is anomalous and its value is ≃ 580ps/nm/km. For width below 400nm or above 640nm GVD becomes normal. The coupling length for spacing around 400nm can be estimated at 200µm [17

17. F.N. Xia, L. Sekaric, and Y.A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express 14, 3872–3886 (2006). [CrossRef] [PubMed]

]. The kerr coefficient n 2 for silicon is ~ 6×10-14cm2/W. For our power estimates we take the effective area S≃0.3µm2. Then one can show that C=7, 15, 30 used in numerical modeling give the following values for the dimensionless unit of z: 0.88mm, 2mm, 3.8mm. One unit of the time t corresponds to 22fs, 34fs, 47fs and unit of the peak power – to 1.7W, 0.8W, 0.4W, respectively. Considering that in the best SOI waveguides the loss is few dB/cm and remembering about two-photon and free carrier absorption, our power estimates should be scaled up. In particular, MI in a single SOI waveguide reported in [18

18. M.A. Foster, A.C. Turner, J.E. Sharping, B.S. Schmidt, M. Lipson, and A.L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006). [CrossRef] [PubMed]

] has been observed for 10W of pump power. Note, that though more detailed account of the above absorption mechanisms is desirable in future research, it can be forecasted, that they will simply proportionally suppress the instability growth rate, without qualitative changes in the effect itself. The above time estimates show that MI can be observed already with pico-second pump pulses, when the role of the free carrier absorbtion and dispersion is negligible.

5. Summary

We have analyzed modulational instability of bright discrete solitons in the waveguide arrays with group velocity dispersion. In the case of normal GVD we have found multiple instability bands. For weak to moderate strength of coupling the discrete solitons exhibit the ‘neck’ instability leading to breakup of the solitons into a train of dispersive pulses. Only for strong coupling, i.e., in the quasi-continuous limit, this instability is getting gradually suppressed by the ‘snake’ instability known for the 2D continuous NLS model [10

10. V.E. Zakharov and A.M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Zh. Eksp. Teor. Fiz.65, 997–1011 (1973) [Sov. Phys. JETP 38, 494 (1974).

]. In the case of anomalous GVD the expected neck type instability leads to formation of composite discrete-continuous spatio-temporal quasi-solitons.

References and links

1.

D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

2.

F. Wise and P. Di Trapani, “The hunt for light bullets - Spatio-temporal solitons,” Opt. Phot. News (February, 2002), pages 29–32.

3.

D. Cheskis, S. Bar-Ad, R. Morandotti, J.S. Aitchison, H.S. Eisenberg, Y. Silberberg, and D. Ross, “Strong Spatiotemporal Localization in a Silica Nonlinear Waveguide Array,” Phys. Rev. Lett. 91, 223901 (2003). [CrossRef] [PubMed]

4.

P. St.J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

5.

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994). [CrossRef] [PubMed]

6.

A.B. Aceves, G. Fibich, and B. Ilan, “Gap-soliton bullets in waveguide gratings,” Physica D 189, 277–286 (2004). [CrossRef]

7.

B.B. Baizakov, B.A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70053613 (2004). [CrossRef]

8.

S. Droulias, K. Hizanidis, J. Meier, and D. Christodoulides, “X - waves in nonlinear normally dispersive waveguide arrays,” Opt. Express 13, 1827–1832 (2005). [CrossRef] [PubMed]

9.

A.A. Sukhorukov and Y.S. Kivshar, “Slow-light optical bullets in arrays of nonlinear Bragg-grating waveguides,” http://xxx.lanl.gov/abs/physics/0605194.

10.

V.E. Zakharov and A.M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Zh. Eksp. Teor. Fiz.65, 997–1011 (1973) [Sov. Phys. JETP 38, 494 (1974).

11.

Y.S. Kivshar and D.E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 118–195 (2000). [CrossRef]

12.

A.V. Buryak, P. Di Trapani, D.V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002). [CrossRef]

13.

S. Minardi, J. Yu, G. Blasi, A. Varanavicius, G. Valiulis, A. Berzanskis, A. Piskarskas, and P. Di Trapani “Red solitons: Evidence of spatiotemporal instability in χ(2) spatial soliton dynamics,” Phys. Rev. Lett. 91, 123901 (2003). [CrossRef] [PubMed]

14.

S.D. Jenkins, D. Salerno, S. Minardi, G. Tamosauskas, T.A.B. Kennedy, and P. Di Trapani, “Quantum-noise-initiated symmetry breaking of spatial solitons,” Phys. Rev. Lett. 95, 203902 (2005). [CrossRef] [PubMed]

15.

S.P. Gorza, P. Emplit, and M. Haelterman, “Observation of the snake instability of a spatially extended temporal bright soliton,” Opt. Lett. 31, 1280–1282 (2006). [CrossRef] [PubMed]

16.

P. Dumon, G. Priem, L.R. Nunes, W. Bogaerts, D. van Thourhout, P. Bienstman, T.K. Liang, M. Tsuchiya, P. Jaenen, S. Beckx, J. Wouters, and R. Baets, “Linear and nonlinear nanophotonic devices based on silicon-oninsulator wire waveguides,” Jap. J. of Appl. Phys. 45, 6589–6602 (2006). [CrossRef]

17.

F.N. Xia, L. Sekaric, and Y.A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express 14, 3872–3886 (2006). [CrossRef] [PubMed]

18.

M.A. Foster, A.C. Turner, J.E. Sharping, B.S. Schmidt, M. Lipson, and A.L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006). [CrossRef] [PubMed]

19.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J.S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92163902 (2004). [CrossRef] [PubMed]

20.

D.N. Christodoulides and R.I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 16, 446–448 (1991). [CrossRef] [PubMed]

21.

S. Darmanyan, I. Relke, and F. Lederer, “Instability of continuous waves and rotating solitons in waveguide arrays,” Phys. Rev. E 55, 7662–7668 (1997). [CrossRef]

22.

M. Stepic, L. Hadzieski, and M. Skoric, “Stability of one-dimensional array solitons,” Phys. Rev. E 65, 026604 (2002). [CrossRef]

23.

Y.V. Kartashov and L. Torner, “Surface soliton arrays,” Eur. Opt. Soc. Topical Meeting on Nonlinear Optics, TOM 6 (Paris, 2006).

24.

E.W. Laedke, K.H. Spatschek, and S.K. Turitsyn, “Stability of Discrete Solitons and Quasicollapse to Intrinsically Localized Modes,” Phys. Rev. Lett. 73, 1055–1059 (1994). [CrossRef] [PubMed]

25.

N. Akhmediev and J.M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a selffocusing medium,” Phys. Rev. A 47, 1358–1364 (1993). [CrossRef] [PubMed]

26.

B. Deconinck, D.E. Pelinovsky, and J.D. Carter, “Transverse instabilities of deep-water solitary waves,” Proc. of the Royal Soc. A 462, 2039–2061 (2006). [CrossRef]

OCIS Codes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 29, 2006
Manuscript Accepted: November 30, 2006
Published: December 11, 2006

Citation
A Yulin, Dmitry V. Skryabin, and A Vladimirov, "Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion," Opt. Express 14, 12347-12352 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12347


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References

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
  2. F. Wise and P. Di Trapani, "The hunt for light bullets - Spatio-temporal solitons," Opt. Photon. News (February, 2002), pp 29-32.
  3. D. Cheskis, S. Bar-Ad, R. Morandotti, J.S. Aitchison, H.S. Eisenberg, Y. Silberberg, and D. Ross, "Strong Spatiotemporal localization in a Silica Nonlinear Waveguide Array," Phys. Rev. Lett. 91, 223901 (2003). [CrossRef] [PubMed]
  4. P. St.J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
  5. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994). [CrossRef] [PubMed]
  6. A. B. Aceves, G. Fibich, and B. Ilan, "Gap-soliton bullets in waveguide gratings," Physica D 189, 277-286 (2004). [CrossRef]
  7. B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70053613 (2004). [CrossRef]
  8. S. Droulias, K. Hizanidis, J. Meier, and D. Christodoulides, "X - waves in nonlinear normally dispersive waveguide arrays," Opt. Express 13, 1827-1832 (2005). [CrossRef] [PubMed]
  9. A. A. Sukhorukov and Y. S. Kivshar, "Slow-light optical bullets in arrays of nonlinear Bragg-grating waveguides," http://xxx.lanl.gov/abs/physics/0605194.
  10. V. E. Zakharov and A. M. Rubenchik, "Instability of waveguides and solitons in nonlinear media," Zh. Eksp. Teor. Fiz. 65, 997-1011 (1973) [Sov. Phys. JETP 38, 494 (1974)].
  11. Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Phys. Rep. 331, 118-195 (2000). [CrossRef]
  12. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002). [CrossRef]
  13. S. Minardi, J. Yu, G. Blasi, A. Varanavicius, G. Valiulis, A. Berzanskis, A. Piskarskas, P. Di Trapani "Red solitons: Evidence of spatiotemporal instability in χ(2) spatial soliton dynamics," Phys. Rev. Lett. 91, 123901 (2003). [CrossRef] [PubMed]
  14. S. D. Jenkins, D. Salerno, S. Minardi, G. Tamosauskas, T. A. B. Kennedy, P. Di Trapani, "Quantum-noise-initiated symmetry breaking of spatial solitons," Phys. Rev. Lett. 95, 203902 (2005). [CrossRef] [PubMed]
  15. S. P. Gorza, P. Emplit, M. Haelterman, "Observation of the snake instability of a spatially extended temporal bright soliton," Opt. Lett. 31, 1280-1282 (2006). [CrossRef] [PubMed]
  16. P. Dumon, G. Priem, L. R. Nunes, W. Bogaerts, D. van Thourhout, P. Bienstman, T. K. Liang, M. Tsuchiya, P. Jaenen, S. Beckx, J. Wouters and R. Baets, "Linear and nonlinear nanophotonic devices based on silicon-oninsulator wire waveguides," Jpn. J. of Appl. Phys. 45, 6589-6602 (2006). [CrossRef]
  17. F. N. Xia, L. Sekaric, and Y. A. Vlasov, "Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators," Opt. Express 14, 3872-3886 (2006). [CrossRef] [PubMed]
  18. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006). [CrossRef] [PubMed]
  19. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004). [CrossRef] [PubMed]
  20. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 16, 446-448 (1991). [CrossRef] [PubMed]
  21. S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662-7668 (1997). [CrossRef]
  22. M. Stepic, L. Hadzieski, M. Skoric, "Stability of one-dimensional array solitons," Phys. Rev. E 65, 026604 (2002). [CrossRef]
  23. Y. V. Kartashov and L. Torner, "Surface soliton arrays," Eur. Opt. Soc. Topical Meeting on Nonlinear Optics, TOM 6 (Paris, 2006).
  24. E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete Solitons and Quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1059 (1994). [CrossRef] [PubMed]
  25. N. Akhmediev and J. M. Soto-Crespo, "Generation of a train of three-dimensional optical solitons in a selffocusing medium," Phys. Rev. A 47, 1358-1364 (1993). [CrossRef] [PubMed]
  26. B. Deconinck, D. E. Pelinovsky, and J. D. Carter, "Transverse instabilities of deep-water solitary waves," Proc. of the Royal Soc. A 462, 2039-2061 (2006). [CrossRef]

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