Exceptional Reduction of the Diffusion Constant in Partially Disordered Photonic Crystals

In real photonic crystals light is scattered by the imperfections of the periodic potential. We study experimentally the propagation of diffused light in silicon inverse opals and report an exceptionally reduced diffusion constant of $3.0\pm 0.7{\mathrm{m}}^2/\mathrm{s}$, in samples which are only partially disordered. Waves scattered both by the lattice planes and by their imperfections interfere and light is efficiently trapped in this hybrid scattering regime. Not only higher quality crystals, but also random materials present an order of magnitude bigger diffusion constant and hence weaker scattering.

Figure 1

Structural characterization of three different materials. The SEM images of the samples are shown, together with their Fourier transforms (inset) and linear autocorrelation function. The latter is calculated as the correlation between points at distance $r$ along a line on the SEM image. The periodicity of the crystals yields an oscillating spatial correlation (solid and dotted curve), which is damped on different length scales, depending on the amount of imperfections. The dash-dotted curve corresponds to the autocorrelation of the random sample.

Figure 2

Reflection measurements at normal incidence compared to the band diagram (a) of a highly ordered silicon inverse opal of type $M1$ (b), a more disordered one $M2$ (c), and the fine silicon powder representing material $M3$ (d). As for the periodic structures, one can recognize in the gray-highlighted regions different reflection peaks, at $\omega a/2\pi c=0.55$ and 0.8, in correspondence with two pseudogaps in the $\Gamma -L$ direction and at $\omega a/2\pi c=0.9$, relative to the predicted band gap.

Figure 3

Diffusive propagation of light at $\lambda =1500\mathrm{nm}$ ($a/\lambda =0.8$) through three silicon materials. The partially ordered sample (small circles) clearly exhibits the slowest diffusion. The solid curve is a fit with a solution to the diffusion equation.

Figure 4

Diffusion constant for inverse opal ($M2$) as a function of wavelength. On the same $x$ axes are plotted (triangles) the reflection spectra, collected on the same point and under the same incident angle and (small circles) the calculated density of states of the electromagnetic field in the photonic crystal.