Discrete gap solitons in waveguide arrays with alternating spacings

We consider an array of waveguides with identical widths but alternating spacings using the discrete nonlinear Schrödinger model (tight-binding approximation). In the highly discrete (anticontinuous) limit when one of the spacings is infinite, the model reduces to an integrable chain of uncoupled dimers. From this limit, we identify the two fundamental, antisymmetric and symmetric, discrete gap solitons, which can be numerically continued to a continuum limit gap soliton at one band edge. Other composite solutions at the uncoupled limit disappear in bifurcations. Similarly to the case of waveguides with alternating widths and constant spacings, oscillatory instabilities appear for the fundamental solutions only for frequencies in the upper half of the gap. In contrast to the alternating-width case, there is no stability exchange between the two fundamental solutions: the symmetric solution is always unstable while the antisymmetric solution is always stable in the lower half of the gap. Thus, the Peierls-Nabarro barrier can vanish only in the continuum limit.

Figure 1

(Color online) Modulated array with couplings $V_1$ and $V_2$.

Figure 2

(Color online) Gap modes for $\Lambda =0$: (a) fundamental antisymmetric, (b) fundamental symmetric, and (c) twisted. Shaded and white regions correspond to couplings $V_1$ and $V_2$, respectively. $V_1=\gamma =1$ and $V_2=0.5$.

Figure 3

(Color online) Power versus frequency. Lower, middle, and upper lines correspond to the fundamental antisymmetric, symmetric, and twisted solutions, respectively. $V_1=\gamma =1$, $V_2=0.5$.

Figure 4

(Color online) $\text{Re}\left(\omega \right)$ and $\text{Im}\left(\omega \right)$ vs $\Lambda$: antisymmetric (top), symmetric (center), and twisted (bottom). $V_1=1$, $V_2=0.5$, $N\simeq 50$.

Figure 5

(Color online) Stability zones of the (a) fundamental antisymmetric, (b) fundamental symmetric, and (c) twisted discrete gap modes. Black parts are stable while the brighter parts correspond to stronger instabilities.