Robustness of gap-solitons in disordered photonic crystal waveguides

Soliton propagation in GaInP photonic-crystal line-defect waveguides (PhCW) have been recently demonstrated and described in terms of gap-solitons, which can propagate at frequencies where the linear modes are forbidden extending the bandwidth of PhCW-based devices, and offer potential for control of group-delay over sub-mm lengths at relatively low powers. However, these applications require to assess the impact of the broken symmetry of the lattice due to disorder caused by technological built-in imperfections. It is well known indeed that such disorder poses major limitations to the linear propagation in PhCW, especially when operating close to the cut-off frequency in order to achieve very slow light. Here we address how the unavoidable disorder affects the nonlinear propagation and in particular the control of group-delay of the soliton pulses, and the occurrence of Anderson localization, a universal mechanism due to wave interference, predicted for electrons, but occurring also for all wave phenomena including microwaves, acoustic, or coherent matter waves. According to the latter, electrons diffused by a disordered potential in doped semiconductors see a metal-to-insulator transition when the disorder is sufficiently high. Here we demonstrate an equivalent metal-to-insulator transition for nonlinear waves, namely Gap-Solitons, in disordered PhCW. Specifically, we demonstrate that for solitons the transition from localized to ballistic regime goes faster than the v_g² law for linear waves, being v_g the group velocity of the Bloch mode. By overcoming this scaling law, we find improved robustness to disorder of nonlinear waves with respect to the linear regime. Relying on a coupled-mode description, localization is described by introducing a new metric able to track the wavepacket center of mass. In particular, we define the barycenter ξ_k^b and its displacement δξ_k^{av for a single (k-th) realization of disorder (see Fig. 1a), while providing characterization of localization by statistical averaging of these quantities over many realizations, and identifying the averaged barycenter with the localization length l_loc defined in literature. First of all, by comparing the coupled mode results to 2D-FDTD simulations mimicking experiments in disordered PhCW, we demonstrate that, in linear regime, this barycenter method exactly yields the usual localization length. Secondly, we exploit our barycenter method for the study of Gap-Solitons and we demonstrate their improved robustness against disorder, which notably extend to PhCW the range of applicability of nonlinear waves in disordered structures, investigated so far only in fiber Bragg gratings or evanescently coupled waveguides.}