Abstract
The unique properties of light underpin the visions of photonic quantum technologies, optical interconnects and a wide range of novel sensors, but a key limiting factor today is losses due to either absorption or backscattering on defects. Recent developments in topological photonics have fostered the vision of backscatteringprotected waveguides made from topological interface modes, but, surprisingly, measurements of their propagation losses were so far missing. Here we report on measurements of losses in the slowlight regime of valleyHall topological waveguides and find no indications of topological protection against backscattering on ubiquitous structural defects. We image the light scattered out from the topological waveguides and find that the propagation losses are due to Anderson localization. The only photonic topological waveguides proposed for materials without intrinsic absorption in the optical domain are quantum spinHall and valleyHall interface states, but the former exhibit strong outofplane losses, and our work, therefore, raises fundamental questions about the realworld value of topological protection in reciprocal photonics.
Main
Planar nanostructures built with highindex dielectric materials using topdown nanofabrication techniques have enabled the precise control of the spatial and spectral properties of electromagnetic fields at optical frequencies, stimulating the development of integrated photonic devices such as quantum light sources^{1}, programmable photonics^{2}, nanolasers^{3} and optical communication technology^{4}. Although highly optimized performance can be achieved with dedicated aperiodic structures^{5}, periodic structures—that is, photonic crystals—offer simple building blocks that readily scale to larger architectures and allow tailoring the dispersion relation of light. In addition, deliberately introduced regions that break the periodic symmetry allows building highqualityfactor (Q) optical cavities^{6} or waveguides that slow light by several orders of magnitude^{7}. Introducing such defects can greatly enhance the interaction of light with material degrees of freedom^{1,8,9}.
Longrange translational symmetries can also be effectively destroyed by random structural disorder. This disorder results in extrinsic scattering events, incurring substantial propagation losses^{10} with detrimental consequences for applications. For example, photonic quantum technologies rely on encoding information in fragile quantum states, which are extremely sensitive to losses, and optical interconnects aim to reduce the energy consumption in integrated information technology; here transmission loss directly translates into energy loss^{1,11,12}. Improvements in nanoscale fabrication down to nanometre tolerances can reduce the magnitude of structural disorder, but stochastic deviations from the designed structures are inherent to any fabrication method and can never be completely eliminated. Since the scattering crosssection is unfortunately often enhanced at the spectral and spatial regions targeted for device operation, such residual disorder is a primary obstacle to the application of photonic crystals^{11}. A wellknown case is that of slow light in photoniccrystal waveguides, where the disorder ultimately limits the maximum slowdown by the localization of the light field induced by multiple coherent backscattering^{13,14}.
A possible solution to this problem has sought inspiration from solidstate systems, for which the quantum Hall effect offers unidirectional propagation, that is, completely suppressed backscattering, by breaking timereversal symmetry. These quantum states are related to the underlying wavevectorspace topology of the Bloch eigenstates, which can be equally explored and exploited for photoniccrystal structures, indicating a deep analogy between solidstate quantum states and classical waves^{15,16}. This naturally led to the development of topological photonics^{17,18} and to the demonstration of oneway robust electromagnetic waveguides^{19}, that is, photonic topological insulator (PTI) waveguides. Although early attempts relied on real magnetic fields and nonreciprocal magnetooptical materials to generate nontrivial topologies, further realizations were achieved by effective magnetic fields through timemodulated media^{20}. However, such approaches to combat backscattering have seen only microwavedomain implementations that use intrinsically lossy materials^{21} or complex active schemes of difficult practical implementation^{22}. PTIs that instead rely on breaking spatial symmetries to emulate pesudospins akin to that in quantum spinHall and quantum valleyHall (VH) solidstate topological insulators have been predicted^{23,24} and demonstrated^{25,26}. The resulting interface states, although not unidirectional owing to reciprocity, can, in principle, exhibit robustness to a certain class of perturbations^{27}. Both quantum spinHall and quantum VH interface states in highindex dielectric photoniccrystal slabs have been observed at telecom wavelengths, but the former support states above the light line^{28} that are intrinsically lossy^{29}, making VH topological interface states particularly attractive to test and exploit topological protection against backscattering.
Length scales of disorder in photoniccrystal structures
Much of the existing work on disorder in topological photonics has explored topological protection against defects on scales relevant to their electronic counterpart. Topological quantum states of electrons can travel unhindered along paths prone to crystallographic defects such as vacancies, interstitials or dislocations^{30}, all of which are on the scale of one to a few crystal unit cells. Such latticescale disorder has been mimicked in PTIs, with the most paradigmatic case being that of sharp Z or Ωshaped bends. Unlike in conventional linedefect photoniccrystal waveguides (Fig. 1a and ref. ^{31}), suppressed backreflection through sharp bends over a large bandwidth has been demonstrated for topological interface states^{26,32}, enabling flexibly shaped photonic circuits like ring cavities^{33}. Nanophotonic waveguides are, however, prone to nanometrescale roughness in the etched sidewalls (Fig. 1b). Such structural disorder occurs at a scale considerably smaller than the unit cell and is consistently present across the entire crystal, thus questioning how the notions of topological protection can be directly transferred to the interface states in PTIs. Recent numerical works have addressed this question, but the studies are limited to effective disorder models in twodimensional crystals^{34}, semianalytical models^{35} or singleevent incoherent scattering theory^{36}. A more accurate modelling of the propagation losses including coherent multiple scattering has been applied to conventional monomode slowlight waveguides^{37,38}, but such studies are still lacking for topological interface states. Ultimately, the subtle interplay between outofplane losses and backscattering in photoniccrystal slab waveguides calls for experiments that directly compare the propagation losses of topological and conventional slowlight waveguides subject to equivalent disorder and taking into account the group index. Here we address this by fabricating and characterizing a set of suspended silicon VH PTI waveguides^{39} that support both a topologically protected and a topologically trivial guided mode with nearly identical group indices. We characterize waveguides with and without sharp bends (Fig. 1c,d). As shown in previous work^{32}, the difference between the two guided modes is profoundly evident when introducing four sharp turns in the waveguide path. They effectively suppress the transmittance in the trivial mode but leave the transmission through the topological mode essentially unaffected (Fig. 1e). This is a striking demonstration of topological protection, but it offers no evidence of the applicability of PTIs for protecting against backscattering from fabrication imperfections. From simple experiments, for example, comparing the transmittance of short and long waveguides (Fig. 1f), it is clear that the propagation loss of the topological mode studied here is nonzero. However, assessing topological protection against backscattering requires the precise extraction and modelling of the propagation losses to disentangle outofplane radiation losses from backscattering, the careful study of which has so far been absent from the literature, to the best of our knowledge.
Design of a VH slowlight waveguide
The VH PTI waveguides explored here rely on a photonic crystal that emulates graphene with two equilateral triangular holes arranged in a honeycomb structure, the unit cell of which is shown in the inset of Fig. 2a. For identical triangles s_{1} = s_{2}, the crystal dispersion exhibits a Dirac cone for transverseelectriclike modes at the K point in the Brillouin zone. Breaking inversion symmetry such that s_{1} ≠ s_{2} lifts the degeneracy and opens a bandgap (Fig. 2a). The nontrivial geometrical structure in momentum space of the wave functions of air and dielectric bands of the crystal results in nonvanishing Berry curvatures^{40} around the K and K′ points (Fig. 2b). When two such crystals with inverted symmetry are interfaced, the bulkedge correspondence theorem^{24} ensures the existence of two degenerate counterpropagating states localized at the domain wall and exhibiting a linear dispersion around the projection of K in the waveguide direction. The particular geometry explored here (Fig. 2c) uses a bearded interface^{18} between two mutually inverted VH crystals. Compared with the commonly investigated zigzag interface^{26}, the bearded interface obeys a composition of mirror and translation symmetry, that is, glide symmetry. This enforces a degeneracy at the edge of the Brillouin zone^{41}, leading to the existence of two guided modes (Fig. 2d). The superior transmission through sharp bends at wavelengths in the lowenergy band relative to those in the highenergy band (Fig. 1e) indicates that the former is topological (Supplementary Section 6), and shows that protection to backreflection at sharp 120° bends extends far from the K valley. The dispersion diagram is computed from geometric contours extracted from scanning electron microscopy (SEM) images (Supplementary Section 1.2) and shows that the fabricated structure is single moded (Supplementary Section 7). Figure 2d also shows the group index as a function of wavelength. A group index of n_{g} ≈ 30 is achieved in the topological band, a value for which backscattering typically dominates outofplane radiation losses^{38}, making the waveguide an ideal testbed to study the aforementioned topological protection to backscattering.
Characterization of optical propagation losses
We characterize the dispersive propagation losses of suspended silicon photoniccrystal waveguides fabricated with a slab thickness of 220 nm. This is achieved by measuring the optical transmission of suspended photonic circuits where waveguides of varying lengths (L) from 250a_{0} to 1,750a_{0}, with a_{0} = 512 nm denoting the lattice constant, are embedded. SEM images of the characteristic devices are shown in Fig. 3a. The circuits comprise input and output freespace broadband grating couplers, strip silicon waveguides to direct light into the region of interest (Fig. 3b) and intermediate waveguides^{42} (Fig. 3c) to couple to the VH interface modes (Fig. 3d) with high efficiency (87%; Supplementary Section 2.3). Figure 3e shows a highmagnification SEM image, which reveals the presence of roughness along the sidewalls. In principle, the roughness could be measured and used to calculate the scattering, but, in practice, such a procedure is experimentally unfeasible and numerically intractable^{34,43}. Therefore, we instead benchmark our VH waveguides against conventional linedefect W1 waveguides fabricated on the same chip such that the structural disorder has practically equivalent statistics. We extract an average propagation loss as low as 0.47 ± 0.04 dB cm^{–1} over a 40 nm bandwidth in the nondispersive region of the W1 waveguide (Supplementary Section 3.2). This constitutes a recordlow value for suspended silicon photonics and shows that our nanofabrication^{5} provides an ideal testing ground for measuring VH waveguides with the lowest level of roughness realized to date, to the best of our knowledge.
The circuit transmittance for a single VH device for each waveguide length is shown in Fig. 3f. We convolute the raw spectra with a Gaussian kernel (standard deviation, σ = 2.5 nm) to simultaneously remove Fabry–Pérot fringes resulting from reflections at the grating couplers and account for possible systematic structuretostructure variations (Supplementary Section 2.1). Inside the transmission band, we observe that the loss is largest at around λ = 1,515 nm, which is a clear spectral indication of the n_{g} peak shown in Fig. 2d. We account for the stochastic nature of the sidewall roughness by studying the averaged quantities obtained from measurements done over three nominally identical circuits for each waveguide length. We find that the ensembleaveraged transmission intensity can be described by an exponential spatial decay, with an attenuation coefficient \(\alpha (\lambda )\equiv {\ell }_{\mathrm{L}}^{1}(\lambda )\). This leads to the following damping law:
where additional losses in the circuit are cast into T_{0} (Supplementary Section 2.1). Characteristic fits to the ensembleaveraged data are shown in Fig. 3g. Although such Beer–Lambertlike attenuation has been theoretically shown to fail for particular periodic monomode waveguides^{44}, the moderate values of n_{g} explored here and the stateoftheart nanofabrication process justify the model. The same arguments also support the use of three devices per length since the variance of the stochastically distributed transmission (which depends on the loss pathway, group index and waveguide length^{38}) is low as confirmed by the data subsets with an increasing number of nominally identical devices (Supplementary Section 2.2).
Groupindex dependence
The propagation loss over a wide wavelength range is shown in Fig. 4a. It exhibits prominent dispersion across the maximum group index at around λ = 1,515 nm. As a consequence of strong dispersion, the extracted propagation loss depends on the width of the filtering kernel; therefore, the values shown in Fig. 4a constitute a lower bound to the propagation losses. Broadly speaking, the dependence of propagation loss on wavelength reflects the physics underlying several distinct scattering mechanisms^{45}. The losses of a propagating mode in a photoniccrystal waveguide can be classified into intrinsic losses, \({\ell }_{\mathrm{i}}^{1}(\lambda )\), scattering into radiation modes in the cladding, \({\ell }_{{{{\rm{out}}}}}^{1}(\lambda )\), and inter or intramodal scattering into other slab or waveguide modes. Intrinsic losses include absorption material losses, which can be neglected in crystalline silicon at telecom wavelengths, except for twophoton absorption in ultrahighQ cavities^{46}, and intrinsic radiation losses when operating above the light line^{29}. In the case of a monomode photoniccrystal waveguide with vertical sidewalls operated at wavelengths within the bulk bandgap, all the sources of inter and intramodal scattering except for backscattering, \({\ell }_{\mathrm{s}}^{1}(\lambda )\), are strongly suppressed. This holds provided that the disorder levels are perturbative and that the size of the photonic crystal in the direction perpendicular to the waveguide axis (here 16) substantially exceeds the Bragg length. All such conditions are satisfied for the VH waveguides explored here, and the remaining loss lengths add up reciprocally to the propagation length as
Both contributions are generally dispersive and may independently vary based on geometry, material properties, disorder and wavelength. As a consequence, the precise scaling with n_{g} is far from trivial^{38}. Nevertheless, based on the experimental observations and perturbation theory^{45}, we model \({\ell }_{\mathrm{L}}^{1}(\lambda )\) by
where n_{g}(λ) is the theoretical group index (Fig. 2d). The coefficients β and γ describe the loss due to backscattering and outofplane radiation, respectively. Furthermore, to account for the observed spectral shifts of about 20 nm between the calculated n_{g} and observed loss peak, we introduce an additional model parameter, Δλ, describing a linear spectral shift between theory and experiment. The fit is shown in Fig. 4a and agrees well with the experiment, which shows that equation (3) describes the measured losses well, irrespective of the transition between the topological and trivial modes. The fitted coefficients are β = 0.151 ± 0.003 dB cm^{–1} and γ = 0.390 ± 0.070 dB cm^{–1}. The band structure is shifted by Δλ = 20.22 ± 0.05 nm, which accounts for minor deviations in average dimensions between the model and samples, and this shift is henceforth applied to all the theoretical quantities, placing the degeneracy point at 1,517 nm and the K valley of the topological mode at 1,587 nm. We note that the measured propagation losses are not minimal at the K valley. For reference, the propagation loss was also measured in both zigzag and bearded interface waveguides using a different unit cell^{26} (Supplementary Section 3.3). The minimal propagation loss does not coincide with the location of the K valley for any of the measured waveguides. The calculated intrinsic radiation losses (Methods) of the topological mode above the light line is included for reference in Fig. 4a and shows good agreement with the measured propagation loss. We obtain the propagation loss as a function of group index (Fig. 4b) and observe that the losses of the two modes coincide within the statistical uncertainty. In addition, we use the fitted γ and β to infer that similar to conventional W1 photoniccrystal waveguides^{38}, backscattering losses exceed the outofplane radiation losses even at very low group indices, which include the vicinity of the K valley. We conclude that the topological interface mode incurs equivalent propagation losses as the trivial mode in the slowlight regime, and thus, we observe no topological protection against fabrication disorder.
Observation of coherent backscattering
We now turn to exploring the physics of backscattering in the topological waveguide beyond the ensemble behaviour. We image the vertically scattered far fields from singlewaveguide realizations (L = 1,750a_{0}) using a tunable laser and a nearinfrared camera. These measurements use a different sample with propagation losses comparable with those reported in Fig. 3 (Supplementary Section 3.2). Since it is challenging to distinguish between the topological and the trivial mode from farfield measurements on straight waveguides (Supplementary Section 4), we employ sharply bent waveguides, which act as highly efficient (Supplementary Section 5) filters that only allow the topological mode to pass^{39}. Figure 5a shows a microscopy image of a device, overlaid with the scattered farfield at a wavelength well within the topological band. In addition to scattering at the interface between the strip waveguide and intermediate waveguide as well as at the two vertices of the bend, we observe spatially varying scattering in a finite region of the waveguide close to the second corner. This is a clear fingerprint of strong coherent backscattering leading to complex interference patterns in the nearfield of the propagating mode and projected into the farfield by outofplane radiation losses^{13}. Coherent backscattering ultimately leads to spectral and spatial localization of the light field and onedimensional Anderson localization^{14}. In a single realization, this corresponds to the formation of random optical cavities with distinct spatial patterns (Fig. 5b) and Q ≈ 2 × 10^{5} (Fig. 5c). The observation of random Andersonlocalized cavities with high Q corroborates that coherent backscattering is the dominant source of loss at the imaged wavelengths, that is, the inverse loss length (Fig. 4) may be interpreted as the inverse localization length^{47}. Finally, we step the laser wavelength across a wide wavelength range around the degeneracy point and study the transition from the topological to the trivial mode. The spectrospatial map (Fig. 5d) depicts the acquired intensities after the second corner and along the axis of the waveguide. It reveals the presence of multiple spectral resonances associated with spatially localized farfield patterns, whose spatial extent generally increases with wavelength, as expected from the behaviour of the ensembleaveraged loss length (Fig. 4). The modes labelled A–E correspond to the images in Fig. 5a,b and are not visible in the transmission spectrum of the circuit, which further evidences their localized nature. In addition, a closeup view of the bend (Fig. 5e) unveils the topological or trivial nature of the propagating modes. For all the wavelengths below λ = 1,554 nm, we observe a single scattering spot at the first corner (Fig. 5e) and no emission from the waveguide (Fig. 5d), indicating wavelengths within the trivial band, consistent with previous observations for this waveguide geometry^{32}. At the resonant wavelength of mode E, light is also strongly scattered at the first corner. This is consistent with our numerical simulations of highgroupindex wavelengths near the degeneracy, even in the topological band (Supplementary Section 5). At the wavelengths of the modes labelled A–D, the radiation losses in both corners are not only suppressed but also evenly distributed, confirming the existence of localized modes within the topological band.
Conclusion and outlook
The experiments presented here—the dependence of the measured loss length on the group index as well as the scattered light observed via farfield imaging—establish a consistent picture of the transmission and scattering of slow light in VH PTI interface modes: backscattering dominates over outofplane losses and is sufficiently strong to induce random cavities with high Q. Additionally, we observe no difference between the dependence of loss length on group index for the topological and trivial modes.
Obviously, our experiments do not rule out the existence of backscattering resilience in other timereversalinvariant PTIs with different symmetries, unitcell geometries, interfaces or disorder levels. Even if structural disorder eventually destroys the crystal symmetry behind nontrivial topologies, backscattering might still be suppressed for limited disorder^{34}. To approach that regime, we have employed a bearded VH interface, which has been theoretically shown to be more robust than other types of interface^{35}. In addition, our recordlowloss W1 waveguides show that we are probing some of the lowest levels of disorder realized in silicon photonics so far. Even so, we do not observe any signature of reduced backscattering and our results therefore cast doubts on whether any topological protection against backscattering from nanoscale roughness is possible in an alldielectric platform, that is, without breaking timereversal symmetry or dynamic modulation techniques.
We hope that our work will motivate further research to consider robustness against a realworld disorder, for example, in developing new magnetooptic materials to break timereversal symmetry at optical frequencies^{48} or studying the mechanisms behind Anderson localization^{49} in systems with valleymomentum locking. The interplay between disorder and topology has surprising consequences, such as topological Anderson insulators^{50}, and our work takes the first steps into research at the nexus between Anderson localization, topology and silicon photonics.
Methods
Sample fabrication
The measurements are performed on two samples (sample 1 and sample 2). The data in Figs. 1 and 5 are taken from sample 1 and the data shown in Figs. 2–4 are from sample 2. Both samples are fabricated from the same silicononinsulator substrate with a nominally 220nmthick silicon device layer. The fabrication process is detailed elsewhere^{5} with some minor modifications. Sample 1 is fabricated using a highresolution electronbeam lithography process, the details of which may be found in another work^{51}. Sample 2 is fabricated using a modified process, which introduces a silicon–chromium hard mask^{52}.
Optical spectral measurements
The transmission of each device is measured using a confocal freespace optical setup with crosspolarized and spatially offset excitation and collection achieved via orthogonal freespace grating couplers. The broadband optical characterization is performed using a fibrecoupled supercontinuum coherent whitelight source (NKT Photonics SuperK EXTREME EXR15) focused onto the input grating coupler using a longworkingdistance apochromatic microscope objective (Mitutoyo Plan Apo NIR 20X, numerical aperture = 0.4, 10 mm effective focal length). The input power (typically 120 μW at the sample surface) is controlled using a halfwave plate and a polarizing beamsplitter and the excitation polarization selected with a halfwave plate. Light coupled out from the chip is collected using the same microscope objective, split via a 50:50 beamsplitter and filtered in polarization and space, respectively, using a linear polarizer and singlemode fibre aligned to the output grating coupler. The light is then sent to an optical spectrum analyser (Yokogawa AQ6370D, 2 nm resolution bandwidth) for retrieving the spectrum. When shown dimensionless, the transmittance spectra are normalized to reference measurements on a silver mirror (Thorlabs PF1003P01) substituted in place of the fabricated sample (Supplementary Section 2.1). For imaging the vertically scattered fields, we employ the same freespace optical setup but instead use a fibrecoupled tunable diode laser (Santec TSL710) for excitation and focus the scattered light into a nearinfrared/visible camera (Aval Global ABA013VIR) using a longfocallength (f = 200 mm) planoconvex lens. The intense direct reflection from the input grating coupler is filtered out using a linear polarizer perpendicular to the waveguide axis in the collection path. The camera also serves the purpose of imaging the sample surface using a nearinfrared lightemitting diode (λ = 1.2 μm).
Numerical modelling
We employ a finiteelement method using commercially available software (COMSOL Multiphysics versions 5.6 and 6.0) for the numerical calculations. Silicon is modelled as a lossless dielectric with a refractive index of 3.48 and air with a refractive index of 1.00. We simulate the optical eigenmodes of the perfect photoniccrystal waveguides by terminating them with perfectly matched layers in the y and z directions and imposing Floquet boundary conditions on both facets of the supercell along the waveguide axis (x). The real part of the eigenfrequencies ω_{k} are used for the band structures (Fig. 1) and the imaginary part, to extract the intrinsic losses above the light cone (Fig. 3a, green). The latter data are obtained as \({\ell }_{{{{\rm{i,dB}}}}}^{1}=4.34\times \left(2{{\mathrm{Im}}}\,({\omega }_{{{{\bf{k}}}}})/ {v}_{{{{\rm{g}}}}} \right)\), where v_{g} is the group velocity. The transmission simulations (Supplementary Sections 2.3 and 5) are solved as frequencydomain problems using the fundamental strip waveguide mode for both input and output ports. All the simulations use the symmetry relative to the midplane of the slab and solve for transverseelectriclike electromagnetic fields. This relies on the assumption of vertical sidewalls, which we observe, as well as the absence of scattering into transversemagneticlike modes.
Data availability
The data that support the figures in this manuscript are available from the corresponding authors upon request.
Code availability
The code that support the figures in this manuscript are available from the corresponding authors upon request.
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Acknowledgements
We gratefully acknowledge A. N. Babar and T. A. S. Weis for assistance with the nanofabrication and valuable discussions and M. L. Korsgaard for assistance with the device design. We gratefully acknowledge financial support from the Villum Foundation Young Investigator Programme (grant no. 13170), Innovation Fund Denmark (grant nos. 017500022 (NEXUS) and 205400008 (SCALE)), the Danish National Research Foundation (grant no. DNRF147 (NanoPhoton)), Independent Research Fund Denmark (grant no. 013500315 (VAFL)), the European Research Council (grant no. 101045396 (SPOTLIGHT)) and the European Union’s Horizon 2021 research and innovation programme under Marie SkłodowskaCurie Action (grant no. 101067606 (TOPEX)).
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C.A.R., G.A. and S.S. designed and developed the experiment. C.A.R., G.A., A.V., M.A. and B.V.L. performed the numerical design and analysis of the structures and device components. M.A. developed the nanofabrication process. C.A.R. fabricated the samples. C.A.R. and G.A. carried out the measurements and data analysis. C.A.R., G.A., A.V. and S.S. prepared the manuscript with input from all the authors. S.S. conceived, initiated and supervised the project with cosupervision by G.A. and R.E.C.
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Rosiek, C.A., Arregui, G., Vladimirova, A. et al. Observation of strong backscattering in valleyHall photonic topological interface modes. Nat. Photon. 17, 386–392 (2023). https://doi.org/10.1038/s4156602301189x
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DOI: https://doi.org/10.1038/s4156602301189x
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