Observation of Surface Gap Solitons in Semi-Infinite Waveguide Arrays

We report on the observation of surface gap solitons found to exist at the interface between uniform and periodic dielectric media with defocusing nonlinearity. We demonstrate strong self-trapping at the edge of a ${\mathrm{LiNbO}}_3$ waveguide array and the formation of staggered surface solitons with propagation constant inside the first photonic band gap. We study the crossover between linear repulsion and nonlinear attraction at the surface, revealing the mechanism of nonlinearity-mediated stabilization of the surface gap modes.

Figure 1

Linear propagation of a narrow low-power beam when only the edge waveguide of the array is excited. (a) Measured transverse output intensity profile ($P=0.1\mu \mathrm{W}$) and (b) corresponding theoretically calculated longitudinal propagation inside the sample. Inset in (b) shows the waveguide geometry. (c)–(e) Formation of the surface gap soliton at the array output 920, 1050, and 1550 s, respectively, after the input beam power is increased to $P=0.5\mathrm{mW}$. Grey shading marks the waveguide positions.

Figure 2

Measured surface localization time vs probe beam power. Solid curve: $A+B/(P-P_{\mathrm{th}})$ fit to experimental data (red dots). Vertical dashed line marks the threshold power ($P_{\mathrm{th}}=0.042\mathrm{mW}$). (a)–(c) Beam intensity profiles of decreasing width corresponding to the indicated points.

Figure 3

(a) Three-dimensional representation of the surface gap soliton observed near the threshold [Fig. 2a]. (b) Plane-wave interferogram demonstrating the staggered phase structure of the surface gap soliton.

Figure 4

(a) Example of the staggered surface mode calculated by use of the discrete nonlinear model (1) at $\gamma =8$. The discrete mode amplitudes are marked by + signs. (b) Normalized width of the localized surface state calculated numerically (solid curve) and measured experimentally (diamonds) as a function of the beam power. Vertical dashed line marks the threshold power ($P_{\mathrm{th}}=0.042\mathrm{mW}$).

Figure 5

Effective potential vs the collective coordinate $X$ (a) below and (b) above the threshold power. Integer values of $X$ correspond to the waveguide numbers. Black dot in (b) refers to the stationary solution shown in Fig. 4a.