Localization in silicon nanophotonic slow-light waveguides

Slowing down light on a chip can lead to the development of optical buffers1, filters2,3 and memory elements4 useful for optical interconnects and for resonantly enhanced chip-based nonlinear optics. Several approaches to slow light rely on the phenomenon of light interference in a sequence of coupled resonators; however, light interference is also responsible, in disordered structures, for the localization of light, an effect particularly prominent in one-dimensional devices. Until now, the length of the waveguides investigated has been less than the localization length. Here we report the first observation of light localization in compact silicon nanophotonic slow-light waveguides consisting of long sequences of coupled resonators. Our results show that disorder limits how much light can be slowed, and that localization leads to spatially concentrated and locally trapped light in a quasi-one-dimensional waveguide at wavelengths near the band edge. Optical slow-wave structures, like their microwave counterparts, consist of a chain or network of repeated unit cells in which light propagates by tunnelling from one unit cell to its nearest neighbours2. Each unit cell could consist, for example, of a microring resonator, a defect resonator in a photonic crystal, or a microsphere. This underlying physical principle of nearest-neighbour coupling can be used to derive an analytical description of the waveguide dispersion, similar to the tight-binding theory used in solid-state physics. Another model uses matrices to describe the interactions between adjacent resonators, or the entire slowwave structure and this model is especially well suited to understanding the effects of disorder. In computing the fields in a slow-wave structure, which consists of a concatenation of unit cells, the field amplitude is described by a column vector u, which lists the fields in the individual unit cells that comprise the structure. The evolution of u is described by a matrix equation i du/dt 1⁄4 Mu, where the coupling matrix M is typically band tri-diagonal in form, because the field in each unit cell couples only to the fields in its nearest neighbours. Consequently, the dispersion relationship has the familiar tightbinding form, v/V 1⁄4 1 þ 2k cos(KR), where v is the frequency, k is the coupling coefficient, R is the centre-to-centre distance between unit cells, K is the Bloch wavenumber and Dv; 2kV is the half-width of the waveguiding band around the centre frequency V (see Supplementary Information). Slow light is particularly observed near the band edges V+Dv (see Fig. 1a), where the group velocity, defined as the slope of the dispersion curve vg ; dv/dK, goes to zero. However, in practice, the unit cells cannot be exactly identical because of fabrication tolerances. As shown in the inset to Fig. 1a, a closer examination of the edge shows the existence of a band tail, and the slope of the dispersion curve (and hence vg) is no longer zero . To understand why the group velocity does not go to zero, it is necessary to consider the connection between group velocity and the density of states. In a weakly disordered one-dimensional slow-wave structure consisting of N unit cells with periodicity R, the group velocity vg is inversely proportional to the density of states, and is given by21