Attenuation Coefficient of Single-Mode Periodic Waveguides

It is widely accepted that, on ensemble average, the transmission $T$ of guided modes decays exponentially with the waveguide length $L$ due to small imperfections, leading to the important figure of merit defined as the attenuation-rate coefficient $\alpha =-⟨\ln (T)⟩/L$. In this Letter, we evidence that the exponential-damping law is not valid in general for periodic monomode waveguides, especially as the group velocity decreases. This result, that contradicts common beliefs and experimental practices aiming at measuring $\alpha$, is supported by a theoretical study of light transport in the limit of very small imperfections, and by numerical results obtained for two waveguide geometries that offer contrasted damping behaviors.

Figure 1

Dispersion relation of the waveguide geometries considered in the present Letter. The PhCW has a periodicity constant $a=420\mathrm{nm}$ and a membrane thickness $h=220\mathrm{nm}$. The segment dimensions of the GNW are $c_x=300\mathrm{nm}$, $c_y=260\mathrm{nm}$ and $c_z=300\mathrm{nm}$, and the period is $a=400\mathrm{nm}$, corresponding to a ${\mathrm{SiO}}_2$ 100-nm-gap length. The band edges ($n_g\to \infty$) of the PhCW and GNW are located at $\lambda =1.57$ and $1.33\mu \mathrm{m}$, respectively. Note that the ${\mathrm{SiO}}_2$ host medium of the GNW is not depicted in the inset.

Figure 2

Testing the validity of the exponential-damping law. (a) $⟨\ln (T^N)⟩$ as a function of the waveguide length $L$ for $n_g=30$. (b) and (c) The normalized local attenuation-rate ${\beta }_{\mathrm{loc}}/{\beta }_{\infty }$ from $L=5a$ ($\approx 2\mu \mathrm{m}$) to $1000a$ ($\approx 500\mu \mathrm{m}$) for several $n_g$, see the legend in (c). (b) PhCW. (c) GNW. The inset in (b) is an enlarged view essentially showing that the exponential damping is satisfied for PhCWs and that $\beta \approx {\beta }_{\infty }$ for all $n_g$. Note that ${\beta }_{\infty }$ indeed depends on $n_g$ and $\sigma$, ${\beta }_{\infty }\propto (\sigma n_g{)}^2$ [6, 8]. All plots are obtained for $\sigma =2\mathrm{nm}$. The rapid oscillations in (b) and (c) are artifacts resulting from an ensemble averaging over a finite number of ${10}^3$ independent realizations.

Figure 3

Definition of the main scattering coefficients involved in the 1D model. $t^N$ and $r^N$ denote the transmission and rear-reflection coefficients of a disordered waveguide of length $Na$, and ${\tau }_{N+1}$ and ${\rho }_{N+1}$ are the elementary scattering coefficients of the unit cell $N+1$. For short-range and uniform disorders, the ${\rho }_m$‘s (or ${\tau }_m$‘s) all belong to the same population and are all independent.

Figure 4

Modulus of the averaged rear-reflection coefficients $r^N$ as a function of $L=Na$ for the PhCW (red [medium gray]), the GNW (blue [dark gray]), and for $n_g=10$, 30 and 70. Note that $r^N$ is expected to be null on the ensemble average in the limit of small perturbations. The results hold for $\sigma =2\mathrm{nm}$ and are obtained by averaging over ${10}^4$ independent realizations of disorder.