PHYSICAL REVIEW LETTERS 126, 027403 (2021) Quantifying the Robustness of Topological Slow Light Guillermo Arregui ,1 Jordi Gomis-Bresco ,1,* Clivia M. Sotomayor-Torres ,1,2 and Pedro David Garcia 1,† 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain (Received 31 July 2020; accepted 15 December 2020; published 14 January 2021) The backscattering mean free path ξ, 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 ξ. 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 ng which is the slowdown factor of the group velocity with respect to the speed of light in vacuum. These two figures of merit, ξ and ng , 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. DOI: 10.1103/PhysRevLett.126.027403 Slowing the speed of a light pulse down to human pace confirm that this limitation may be overcome by exploiting (meters per second) requires complex interference effects photonic topological effects that purely arise from engi- [1] which manifest as a flat dispersion relation ν ¼ νðkÞ, neering the lattice geometry. where k is the conserved wave vector and νðkÞ the Topological photonics has emerged very recently as a frequency. The group velocity vg of this slow light is competitive approach for robust light transport [13], some- determined by the derivative of the flat band and the thing extremely appealing for technological applications. In slowdown factor is given by the group index as ng ¼ c=vg , time-invariant topological insulators based on the quantum- where c is the speed of light in vacuum. ng is the figure of spin Hall effect [14] and the valley-Hall effect [15], merit for slow light and it determines the enhancement topology emanates from the breaking of particular spatial factor for diverse applications such as optical nonlinearities symmetries. In such implementations, reciprocity imposes [2], optical switching [3], pulse delay [4], quantum optics the existence of the counterpropagating mode at −k. This [5], optical storage [6], and optical gain [7]. A strategy to time-reversed edge state carries the opposite value of a bring slow light to the nanoscale exploits optical resonan- binary degree of freedom that plays the role of a pseudospin ces built up by nanostructuring a dielectric or semicon- [14]. In this case, the key open question is whether or not ductor material with low absorption, such as silicon at backscattering is reduced and the answer will depend on telecom wavelengths. Flat bands arise naturally in these whether or not the existing structural disorder preserves the systems based on the periodic modulation of the refractive pseudospin value. Recent ground breaking experiments index at optical or near infrared wavelengths [8,9] for reported robustness in terms of a certain lack of structural which the group index diverges as ng ∝ ð∂ν=∂kÞ−1 in the back reflection when precisely shaped local defects were ideal situation. However, in real devices there is a limitation introduced in different topological waveguides [16]. to the maximum ng achievable due to slight deviations of However, this claimed robustness still needs to be the fabricated parameters compared to the designed values. systematically quantified and compared with state-of-the Even fluctuations in the nanometer range [10] give rise to art conventional (nontopological) ones. Here, we engineer backscattering of the guided light, inducing a strong slow light in a valley-Hall waveguide [15] to calculate its interference [11], a photonic manifestation of Anderson backscattering length ξ versus disorder and ng and we localization in low dimensions [12]. Imperfection limits compare the results to those of a conventional photonic slow light in conventional photonic waveguides to maxi- waveguide. mum values around ng ≈ 100 for very short waveguides For our analysis, we focus on the parity-symmetry with lengths L ≃ 5 μm, much lower than the ng values breaking valley-Hall approach [15]. When applied to observed in atomic systems [1] but still sufficiently large to photonic slabs, the topological edge states at an interface explore weak light-matter interaction leading to cavity- between valley-Hall crystals of opposite K-valley pseudo- quantum electrodynamic phenomena [5]. In this Letter, we spin lay below the light line of the slab and are decoupled 0031-9007=21=126(2)=027403(6) 027403-1 © 2021 American Physical Society PHYSICAL REVIEW LETTERS 126, 027403 (2021) opening a band gap between these two bands [15] (see Supplemental Material [19]). Figure 1(c) shows the calcu- lated dispersion relation of the waveguide and in Fig. 1(d) we plot the group index of the topological edge mode with vanishing vg at the cutoff frequency ν ¼ 0.2955ðc=aÞ, corresponding to ν ¼ 177 THz for a ¼ 500 nm. For refer- ence, we use a standard photonic crystal waveguide obtained by leaving out a row of pillars in a triangular lattice of silicon pillars surrounded by air with a pillar diameter d ¼ 0.4a and the same lattice unit as the topo- logical waveguide, as shown in Fig. 1(f). The conventional guided mode presents an ideally vanishing group velocity when approaching the cutoff frequency of the waveguide at ν ¼ 0.3372ðc=aÞ, as shown in Figs. 1(g) and 1(h). Here, the cutoff of the topological and conventional waveguide lie on the X and Γ point, respectively. The ideal spatial field-intensity distributions in both waveguides and at frequencies corresponding to ng ¼ 300 in both cases are FIG. 1. Slow light in topological waveguides. (a) Valley plotted in Figs. 1(e) and 1(i) for reference, which shows a topological waveguide (illustration) formed at the interface of two valley crystals with different topological invariants as similar level of light confinement. Using silicon pillars proposed in Ref. [15]. (b) Dielectric function distribution (black surrounded by air instead of the usual air holes in silicon corresponds to silicon and gray to air). (c) Dispersion relation enables us to flatten the dispersion relation of both ν ¼ νðkÞ, and (d) calculated group index of the interface topological and conventional guided modes successfully topological edge state (inset displays the unit cell). For reference, without the need of local perturbations as in Ref. [22] or we design a conventional photonic waveguide obtained by progressive interfaces as done in Ref. [23]. The parameters leaving a row of pillars from a triangular lattice (f) with a to obtain flat bands in the valley-Hall waveguide are dispersion relation and a group index plotted in (g) and (h), detailed in the Supplemental Material [19]. respectively. The black point in the group index curves denotes The backscattering length ξ is the average ballistic the frequency at which each waveguide has a ng ≈ 300. The propagation distance along the waveguide in the absence electromagnetic field intensities calculated at these frequencies of any other major loss mechanism [17]. A slow-light and for perfect waveguides are plotted in (e) and (i). waveguide becomes virtually useless when L ≫ ξ. Interestingly, ξ is linked to the density of optical states from the radiation continuum. In principle, these edge of the waveguide, ρ, at least for a weak perturbation states have no intrinsic out-of plane losses and only [17,24], as ξ ∝ ρ−2 ¼ ð∂ν=∂kÞ2 . Intuitively, a larger den- fabrication imperfection can induce coupling to radiating sity of optical states induces a larger probability of modes. Even in that situation, in-plane backscattering is scattering, thus reducing the value of ξ. As ng ∝ largely the dominant loss mechanism at large values of ng ð∂ν=∂kÞ−1 ¼ ρ, both the group index and the backscatter- [17] so it is enough to consider the system as two dimen- ing length are intrinsically linked to each other via the ρ, at sional to capture the physics of slow-light backscattering, least in conventional photonic crystal waveguides [25]. something not possible with other implementations of Despite substantial theoretical work on ξ in nontopological topological photonics [18]. Here, we set up two- electronic [26–28] and photonic transport [17,25,29], this dimensional simulations instead of the more computation- parameter has only been explored recently in topological ally expensive three-dimensional slab. Figure 1(a) displays waveguides [30] although ignoring ng . Besides this, only an illustration of a section of a fully two-dimensional intrinsic out-of plane losses of topological guided modes in valley-Hall waveguide formed at the interface of two valley nondisordered photonic crystal slabs have been analyzed crystals. The valley crystals are created with a triangular [31]. We calculate ξ in perturbed topological and conven- lattice where the unit cell is formed by two circular silicon tional photonic crystal waveguides as pillars surrounded by air with different diameters d1 ¼ 0.4a and d2 ¼ 0.2a, where a is the lattice constant with di- x electric constant εSi ¼ 12 and εair ¼ 1, respectively. The − ¼ hln½IðνÞi; ð1Þ ξðνÞ two-dimensional layout of the waveguide is plotted in Fig. 1(b). For d1 ¼ d2, the system preserves the C6v where I is the finite-element solution of the electromagnetic symmetry and supports a symmetry-protected gapless band field intensity emitted by a dipole at frequency ν, x is the structure between the first and second lowest energy bands distance from the dipole position along the waveguide and for transverse magnetic polarized light. For different pillar the brackets indicate the statistical ensemble average over diameters, the spatial inversion symmetry is broken thus different configurations of positional disorder. In Eq. (1), 027403-2 PHYSICAL REVIEW LETTERS 126, 027403 (2021) the field intensity I is normalized by its (maximum) value at value. However, strong backscattering interference pre- the source position which is the same in each calculation. vents light transport in the conventional waveguide, as Our simulation domain has a length L ¼ 200a in the x revealed by the different examples plotted in Fig. 2(b). As direction, eleven unit cells on each side of the waveguide in ξ is a statistical parameter, this requires an ensemble- the y direction and it is all surrounded by perfectly matched average calculation of many (ideally all) different disorder layers to mimic an open system. In some cases, when the configurations. In numerical experiments, as the one calculated value of ξ exceeds L ¼ 200a, we extend the performed here, ξ is easily obtained by computing the simulation domain to L ¼ 500a which also allowed us to position-dependent field excited by the dipole source in double-check some of the obtained results. To simulate the the conditions described above for several structural effect of fabrication imperfection, we randomize the posi- configurations with the same nominal amount and type tion of the pillars around their ideal value according to a of disorder. Figures 2(c) and 2(d) show the intensity normal distribution which standard deviation σ is our pattern after ensemble averaging the electromagnetic field measure of disorder (more details in the Supplemental excited by this emitter over twenty different realizations, Material [19]). It is important to remark the fact that the the envelope of which decays exponentially from the backscattering length is a universal parameter in one- position of the source with a sufficiently well-averaged dimensional transport [32,33] and any mesoscopic- slope, as plotted along the waveguide axis in Figs. 2(e) transport observable in these systems depends only on ξ, and 2(f) for ng ¼ 300. regardless of the microscopic details of the medium The calculated ξ vs ng is plotted in Fig. 3(a) for the considered, i.e., regardless of it being silicon pillars topological (solid-red circles) and conventional waveguide surrounded by air or air inclusions in silicon. (open-black circles). ξ is calculated for a fixed amount of The electromagnetic field intensity excited by a dipole positional disorder σ ¼ 0.001a, which corresponds to source oscillating at the leftmost edge of the waveguide is fluctuations of ≈0.5 nm for a ¼ 500 nm. This is a realistic shown in Fig. 2. The oscillation frequency is chosen such measure of the residual imperfection resulting from a state- that both the topological (left) and the conventional (right) of-the art fabrication process [10,34]. At such a disorder waveguide would enable slow light transmission with level, the topological waveguide is more robust than ng ¼ 300 in the absence of any imperfection. The figure standard conventional slow-light waveguides when the full shows calculations corresponding to varying configura- phase is randomized, thus mimicking the effect of imper- tions of positional disorder for which the positions of the fection in real systems. Topological waveguides suffer pillars are randomized with a fixed σ ¼ 0.001a. As shown much less backscattering than conventional ones [see in Fig. 2(a), the excited Bloch modes in the topological Fig. 3(a)] with a ξ comparable to the waveguide length for waveguide are just slightly perturbed, which reveals a large group index values. Even at large values of the rather weak backscattering for this large group index group index, ng ≃ 1000, the interface edge mode of the FIG. 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 ν ¼ 0.2955ðc=aÞ, 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 σ ¼ 0.001a. (b) Calculations for a conventional photonic waveguide where the dipole emitter is excited at a frequency ν ¼ 0.3372ðc=aÞ. In (a) and (b), the dipole emission frequency corresponds to a group index of ng ¼ 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 σ ¼ 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. 027403-3 PHYSICAL REVIEW LETTERS 126, 027403 (2021) FIG. 3. Backscattering length versus group index in topological waveguides. (a) ξ vs ng 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 (σ ¼ 0.001a). The shaded area indicates a waveguide of length L ¼ 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 104 different configurations of positional disorder with σ ¼ 0.001a. The calculated frequency is formalized to the frequency of the unperturbed structures ν0 . (d) Standard deviation of the distribution of the frequency shifts △ν as calculated in (b) and (c) vs the strength of the perturbation. From the relation σ v ¼ ϕσ, we quantify the effective impact of disorder in the waveguides with the parameter ϕ. topological waveguide is slightly perturbed when compared protection observed here at very large values of ng , the main to the strong backscattering suffered by the conventional result of this Letter. one, as shown in the different configurations plotted in The calculated ξ vs disorder is plotted in Fig. 4 where Fig. 2(a). Here, it is relevant to discard mechanisms that the value of the group index is kept fixed at ng ¼ 100 (a) could affect in a different manner both systems and yield and ng ¼ 500 (b). ξ has a different dependence on the the data plotted in Fig. 3(a), rather than the topological disorder strength σ on both waveguides. This is clear in protection. Small geometrical perturbations in the pillars of Figs. 4(a) and 4(b), where the slope of this dependence is the waveguides result in positive or negative energy shifts markedly different for the topological and conventional △ν of the dispersion relation of a guided mode [35], waveguides and becomes even more different when the where △ν ≪ ν0 . Here, the ν0 is the unperturbed guided- value of ng increases. To quantify this dependence, we fit mode cutoff frequency, i.e., the frequency for which the the calculated values as ξ ∼ σ −β . We plot the dependence group index is maximum. To quantify this effect, we of the exponent β in Fig. 4(c) on ng . This dependence has calculate the cutoff frequency of the guided mode in both been studied in terms of the Lyapunov exponent γ, which systems after perturbing the waveguides by introducing a under certain hypotheses fulfilled here quantifies the fixed positional disorder σ ¼ 0.001a. Figures 3(b) and 3(c) exponential decay of the eigenfunctions in a disordered plot the distributions of this cutoff frequency, ν, for 104 system [36]. γ is the inverse of ξ, the localization length. different configurations normalized by ν0. For the same The dependence of γ on the amount of disorder γ ∼ σ β has amount of positional disorder the impact, in terms of the been analyzed in perturbed periodic photonic structures standard deviation of the distribution σ ν , is stronger on the [37] where a value of β ∼ 2 is obtained in the bulk of the topological edge state than on the conventional guided dispersion relation. Here, we recover this dependence in mode, as σ ν is clearly narrower for the former. We evaluate both waveguides for low values of ng . Near the cutoff the dependence of σ v ¼ ϕσ on positional disorder strength frequency, the parameter β is expected to converge to for a wide range of values in both waveguides, where the 2=3 as observed in Refs. [37,38]. Here, β displays a very parameter ϕ quantifies the effective impact of disorder in different behavior in both waveguides close to the cutoff both cases. As plotted in Fig. 3(d), the topological wave- frequency. While β ∼ 2=3 for the conventional waveguide guide suffers a larger effective impact of disorder than the [37], β ∼ 3=2 for the topological one. We also extract a conventional waveguide, i.e., disorder shifts the energy of critical level of disorder σ c, above which light in the its unperturbed cutoff frequency in a more pronounced way, conventional waveguide backscatters less than the topo- which underlines even more the potential of the topological logical one. σ c is obtained at the crossing between the 027403-4 PHYSICAL REVIEW LETTERS 126, 027403 (2021) This work was supported by the European Commission H2020 FET Proactive project TOCHA (No. 824140). 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