Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion

A.V. Yulin; D.V. Skryabin; A.G. Vladimirov

doi:10.1364/OE.14.012347

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

Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion

A.V. Yulin, D.V. Skryabin, and A.G. Vladimirov

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A.V. Yulin, D.V. Skryabin, and A.G. Vladimirov, "Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion," Opt. Express 14, 12347-12352 (2006)

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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.

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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.

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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.

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i \partial_{ζ} U_{n} - \frac{1}{2} β_{2} \partial_{τ}^{2} U_{n} + κ (U_{n + 1} + U_{n - 1} - 2 U_{n}) + γ {∣ U_{n} ∣}^{2} U_{n} = 0, n = 1, 2 \dots N,

A_{n} = [a_{n} + ε_{n}, + e^{i ω t - i λ z} + ε_{n, -}^{*} e^{i λ^{*} z - i ω t}] e^{iqz} .

λ \vec{ε} = {\hat{L}}_{0} \vec{ε} + \frac{1}{2} s ω^{2} {\hat{L}}_{1} \vec{ε},

[\begin{matrix} \tilde{q} - 2 γ a_{1}^{2} & - γ a_{1}^{2} & - C & 0 & \dots & - C & 0 \\ γ a_{1}^{2} & - \tilde{q} + 2 γ a_{1}^{2} & 0 & C & \dots & 0 & C \\ - C & 0 & \tilde{q} - 2 γ a_{2}^{2} & - γ a_{2}^{2} & \dots & 0 & 0 \\ 0 & C & γ a_{2}^{2} & - \tilde{q} + 2 γ a_{2}^{2} & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & ⋮ & ⋮ \\ - C & 0 & 0 & 0 & \dots & \tilde{q} - 2 γ a_{N}^{2} & - γ a_{N}^{2} \\ 0 & C & 0 & 0 & \dots & γ a_{N}^{2} & - \tilde{q} + 2 γ a_{N}^{2} \end{matrix}],

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