Quantifying the Robustness of Topological Slow Light

The backscattering mean free path $\xi$, the average ballistic propagation length along a waveguide, quantifies the resistance of slow light against unwanted imperfections in the critical dimensions of the nanostructure. This figure of merit determines the crossover between acceptable slow-light transmission affected by minimal scattering losses and a strong backscattering-induced destructive interference when the waveguide length $L$ exceeds $\xi$. Here, we calculate the backscattering mean free path for a topological photonic waveguide for a specific and determined amount of disorder and, equally relevant, for a fixed value of the group index $n_g$ which is the slowdown factor of the group velocity with respect to the speed of light in vacuum. These two figures of merit, $\xi$ and $n_g$, should be taken into account when quantifying the robustness of topological and conventional (nontopological) slow-light transport at the nanoscale. Otherwise, any claim on a better performance of topological guided light over a conventional one is not justified.

Figure 1

Slow light in topological waveguides. (a) Valley topological waveguide (illustration) formed at the interface of two valley crystals with different topological invariants as proposed in Ref. [15]. (b) Dielectric function distribution (black corresponds to silicon and gray to air). (c) Dispersion relation $\nu =\nu (k)$, and (d) calculated group index of the interface topological edge state (inset displays the unit cell). For reference, we design a conventional photonic waveguide obtained by leaving a row of pillars from a triangular lattice (f) with a dispersion relation and a group index plotted in (g) and (h), respectively. The black point in the group index curves denotes the frequency at which each waveguide has a $n_g\approx 300$. The electromagnetic field intensities calculated at these frequencies and for perfect waveguides are plotted in (e) and (i).

Figure 2

Backscattering in topological slow-light waveguides. (a) Normalized electromagnetic-field intensity excited by a dipole emitter positioned on the left side of a disordered valley-Hall waveguide at a frequency $\nu =0.2955(c/a)$, where $a$ is the lattice constant and $c$ is the speed of light in vacuum. Three different configurations of randomized positions of the pillars are plotted with a standard deviation of $\sigma =0.001a$. (b) Calculations for a conventional photonic waveguide where the dipole emitter is excited at a frequency $\nu =0.3372(c/a)$. In (a) and (b), the dipole emission frequency corresponds to a group index of $n_g=300$. (c) and (d) show the ensemble-averaged electromagnetic field intensity excited by the dipole source at the same frequencies of (a) and (b). For this calculation, over twenty configurations of positional disorder of the pillars with $\sigma =0.001a$ were averaged. (e) and (f) illustrate the calculated ensemble-averaged electromagnetic field-intensity profile along the topological and conventional waveguide axis, respectively. The localization length is extracted from the inverse of the exponential decay slope.

Figure 3

Backscattering length versus group index in topological waveguides. (a) $\xi$ vs $n_g$ calculated in a valley (solid-red circles) and a conventional photonic crystal waveguide (open-black circles) for a fixed amount of disorder in the position of the pillars ($\sigma =0.001a$). The shaded area indicates a waveguide of length $L=100a$ and the lines are guides to the eye. (b) and (c) Distributions of the cutoff frequency calculated in a topological and a conventional waveguide for ${10}^4$ different configurations of positional disorder with $\sigma =0.001a$. The calculated frequency is formalized to the frequency of the unperturbed structures ${\nu }_0$. (d) Standard deviation of the distribution of the frequency shifts $\pm △\nu$ as calculated in (b) and (c) vs the strength of the perturbation. From the relation ${\sigma }_v=\varphi \sigma$, we quantify the effective impact of disorder in the waveguides with the parameter $\varphi$.

Figure 4

Backscattering length versus disorder in topological waveguides. Calculated backscattering length vs positional pillar disorder for a fixed group index value $n_g=100$ (a) and $n_g=500$ (b), in solid-red and open-black circles for the topological and the conventional waveguide, respectively. The dependence of the backscattering length vs the strength of disorder is fitted as $\xi \sim {\sigma }^{-\beta }$. Above a critical level of disorder ${\sigma }_c$ the conventional waveguide sufferers less backscattering than the topological one. ${\sigma }_c$ is estimated from the crossing of the fits. (c) The fitting parameter $\beta$ plotted for a wide range of values of the group index.