Disorder-Induced Multiple Scattering in Photonic-Crystal Waveguides

In this Letter, we study slow-light transport in photonic-crystal waveguides in the presence of structural imperfections. In contrast with previous theoretical works that rely on perturbation theories, the present formalism takes into account multiple scattering and localization effects. It allows for a quantitative prediction of the main statistical transport coefficients, including averaged values as well as probability distributions. In particular, we evidence that, as the group velocity decreases, the attenuation probability distribution exhibits a rapid broadening that one should consider for designing slow-light devices.

Figure 1

Instance of a disorder realization for a PhCW with a finite length $L=\mathrm{Na}$. It is defined by a random sequence $U=[u_1,u_2\dots u_N]$ of $N$ independent elementary single-cell disorder realizations $u$, in which the hole radii of the two inner rows are randomly and independently varied. $R$ and $T$ are the backreflection and transmission of the fundamental Bloch mode.

Figure 2

Elementary single-cell scattering events for single-cell transports. The results are obtained for $5<n_g<250$, for $-3<\Delta \gamma <3\mathrm{nm}$, and for symmetrical hole perturbations, ${\gamma }_1={\gamma }_2=\gamma +\Delta \gamma$. (a) Backreflection $R_u=|r_u{|}^2$. (b) Out-of-plane loss $L_u=1-|r_u{|}^2-|t_u{|}^2$. The solid black curve (dotted red curve) corresponds to positive (negative) hole size perturbations. They should be superimposed within the framework of first-order perturbation theory. $r_u$ and $t_u$ are defined in the inset.

Figure 3

Effective mean free path. (a) Ensemble-averaged transmission $⟨\ln (T)⟩$ as a function of the waveguide length $L=\mathrm{Na}$ ($N=1,2,3,\dots ,200$) for realistic values of the group velocities, $n_g=20$, 50, and 100, and of the disorder levels, $\sigma =1$ and 2 nm. (b) $n_g$ dependence of the mean free path $\ell$ (the right vertical axis shows the averaged attenuation $\alpha =1/\ell$ in $\mathrm{dB}/\mathrm{cm}$) for several values of the disorder parameter $\sigma$. The dashed line indicates the $-2$ slope.

Figure 4

Probability density functions $P(T)$ for the transmission $T$. The results for various values of $\sigma$, $n_g$, and $L$ are all gathered at $⟨T⟩\approx 0.5$, i.e., for PhCW lengths nearly equal to the effective mean free paths. (a) Gaussian-like distributions are obtained in the presence of a substantial out-of-plane leakage for small $n_g$’s. The dashed curve is a log-normal distribution with $⟨\ln (T)⟩=\ln (0.5)$ and with the same standard deviation as the curve obtained for $n_g=13$. (b) For large $n_g$’s, backscattering largely prevails and uniform laws (showing almost equally likely outcomes over the interval $[01]$) are obtained. The histograms are calculated for $5\times {10}^6$ independent disorder realizations.