Gap Solitons in Waveguide Arrays

Bright and dark spatial gap solitons are demonstrated in waveguide arrays. These gap solitons travel across the array at zero transverse velocity, in complete analogy with stationary (immobile) temporal gap solitons. Furthermore, the launching configuration for observing these stationary gap solitons is shown to be the analog of an “ideal experiment” for observing stationary temporal gap solitons, never observed so far. A clear distinction is established between the family of Floquet-Bloch solitons in general and discrete solitons in particular, and the limiting case of gap solitons.

Figure 1

The band-gap diagram of a typical waveguide array, relating the propagation constant ($\beta$) to the Bloch wave number ($K$). The three highest bands are shown, with the shaded regions representing the gaps. The excited gap-soliton modes (at band No. 1 and band No. 2) are marked by circles and their propagation direction by arrows.

Figure 2

Side-coupling geometry, for exciting a pure FB mode in a waveguide array. The beam is injected into the array from the continuum region along its edge. (a) At low power, the shown beam is Bragg reflected from the array into the continuum. (b) At high power, part of the beam penetrates the array, while exciting a gap soliton, traveling at a slow transverse velocity, as compared to that of the input beam (marked by an arrow). The inset shows an enlarged plot of the modal shape of the excited gap soliton.

Figure 3

Head-on geometry, for exciting a pure FB mode in a waveguide array, by shaping the input beam such that it matches the modal shape of the desired FB mode. Band No. 2 modal shape is shown, for the marked FB mode in Fig. 1 (at $K=\pi /\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}$).

Figure 4

Experimental results of a bright gap-soliton measurement, excited exactly at the top edge of band No. 2 (marked in Fig. 1), using a shaped input beam. Shown are photographs of the beam at the array output, for low and high powers. At low power, the input beam diffracts into about twice its initial width. At high power, a bright gap soliton is formed, traveling across the array at zero transverse velocity. We note that high index-step samples were used in this measurement, which possess a slightly double-humped modal shape of band No. 2, unlike the simulations shown in previous figures, where a smaller index step was assumed, which possess a single-humped modal shape of band No. 2.

Figure 5

Experimental results of a dark gap-soliton measurement, excited exactly at the bottom edge of band No. 1 (marked in Fig. 1), using a shaped input beam. Shown are photographs of the beam at the array output, for low and high powers. At low power, the narrow notch at the input beam diffracts into about a three array-period wide notch. At high power, the beam experiences nonlinear defocusing, while the notch narrows, until a dark gap soliton of 1.5 array-periods width is formed, traveling across the array at zero transverse velocity.