Position-dependent diffusion of light in disordered waveguides

Summary form only given. Diffusion is a statistical description of random walk of a classical particle, and the diffusion constant D₀ is the only parameter in the diffusion equation. For light as well as for other kinds of waves, this is an approximation, because the interference of partial waves is ignored [1]. Such interference is essential to Anderson localization. Proper account of the interference effects in random samples of finite size [2] and/or with absorption [3] results in spatial variation of the diffusion coefficient D(r) in the self consistent theory (SCT) of localization.To observe position-dependent diffusion, disordered waveguide structures were fabricated with the silicon on insulator wafer (see Fig. 1a). The patterns were written by electron beam lithography and etched in an inductive coupled reactive ion etcher. The waveguides contained 2D random arrays of air holes that scattered light, and the scattering length was varied by the hole size and filling fraction. The waveguide walls were made of photonic crystals that had complete bandgap in 2D, so that light could not escape laterally. However, light will leak out of the plane while being scattered by the air holes. This vertical leakage can be described by an effective absorption or dissipation. The relevant parameters are the diffusive absorption length ȟa0 and the transport mean free path . The localization length ȟ is determined by and the waveguide width W. Light from a CW laser source was injected into the waveguide from one end, and transported through the random medium. Spatial distribution of light intensity on the sample surface was imaged onto a camera by an objective lens. After entering the random medium, light is attenuated due to competing effects of backscattering and dissipation. I(y, z) was integrated along the transverse y-direction to determine the variation of intensity along the axial z-direction (parallel to the waveguide axis).Fig. lb shows th- measured light intensity I(z) inside the ensembles of random waveguides of different width W (blue). The values of ξ and ξασ are obtained by fitting the most diffusive sample (W = 60 μm, longest ξ) with SCT (red dashed line) [2,3]. Using these values, SCT successfully predicts I(z) for all other samples. D(z) corresponding to red curves in Fig. lb are plotted in Fig. lc, showing a suppression of diffusion in the middle of the sample with increase ξασ/ξ (decrease of W) as predicted by SCT.