Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides

1 Introduction

The recent progress in the field of photonic topological insulators (TIs) has led to the discovery of several schemes of inducing edge states at topological interfaces between photonic insulators [1], [2], [3], [4], [5], [6], [7], [8], [9]. Not only the presence of edge modes is guaranteed by the topology of the surrounding bulk insulators, but also in some cases the edge modes are immune to backscattering even when defects are present along the interface, such as geometrical disorder or interface bends [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. This is always the case for Chern insulators, a class of photonic insulators whose topology relies on time-reversal symmetry breaking, and induces non-reciprocal unidirectional transport on the topological boundary [6], [20], [21]. Conversely, immunity to backscattering is not a priori true for designs that preserve time-reversal symmetry, in which reciprocity necessarily holds implying that every edge mode always comes with its time-reversed image [1], [4], [5], [7], [22], [23]. In these time-reversal invariant cases, the question of robustness boils down to whether the two time-reversed states (often called photonic pseudo-spins) can be coupled to each other by the considered defect. In the literature, the two dominant time-reversal invariant topological designs, namely, spin-Hall (SP) designs based on six-fold rotational symmetry (C6) [8], [22], [23] and VH designs based on valley preservation [5], [18], [24], [25], have often been claimed as topologically protected, although they are not strictly immune to backscattering. The real level of robustness of the edge modes originating from these designs has been left largely unexplored, limited to field maps without any quantitative analysis and/or simulations for a specific realization of disorder.

Here, we report a quantitative assessment of the level of robustness of time-reversal invariant topological edge modes, based on a clearly defined universal metric: the Anderson localization length [26], [27], [28] of the mode along the topological interface [29]. Such a length is numerically computed by taking ensemble averages of the transmission coefficient through a disordered region of variable length along the interface, for many realizations of disorder [30]. This allows us to rigorously quantify and compare the robustness of C6 and VH edge modes to disorders of different possible types of interfaces.

We want to stress here that we have chosen to focus on topological insulators of a special type, i.e. based on locally resonant metamaterial crystals [31], [32]. These artificial materials are built from spatially local subwavelength resonant scattering inclusions (e.g. electrically small resonant dipoles), which are arranged periodically at a subwavelength scale, leading to deeply subwavelength edge modes [33]. The reason for this choice is that such compact systems are of large practical importance to manipulate electromagnetic waves along the subwavelength paths (from microwave engineering to plasmonics), however such slow-light propagation is usually particularly sensitive to defects and disorder. In this regime, other types of slow-light waveguides which are not based on topology have been demonstrated and their robustness is quantitatively explored, which allows us to not only to compare topological designs between each other, but also to determine whether their topological origin provides them with a superior robustness with respect to conventional designs, like the ones based on defect-lines [33] or chirality [30].

The paper is structured in to following Sections: Section 2, we have briefly described the designs of C6-based and VH-based metamaterial TIs along with the existence of topological edge modes between different interfaces. We have defined different kinds of disorder that can be introduced at such interfaces and define the basis of our statistical study. In Section 3, we have presented the obtained results for all types of interfaces and disorders. Finally, in Section 4, we have discussed how different disorders affect the transmission efficiency and how important the type of interface is, for the robustness of the waveguide.

2 Materials and methods

The schematics of the C6 and VH metamaterial TIs studied in this work are shown in Figure 1A and B, respectively where the colored disks represent the spatially local resonators composing the metamaterial crystals. These resonators are assumed here to be quarter-wavelength copper rods standing on a ground plane and oriented along z (the figure view is from the top), corresponding to a microwave implementation of locally resonant metamaterials, whose structure is effectively subwavelength in the xOy plane. This specific microwave implementation does not impact the generality of our study, since similar dispersion curves and topological behavior are obtained as long as one considers 2D arrangements of far-field coupled dipolar resonances [33], which can also be obtained at optical frequencies (with dielectric or plasmonic resonators), or in other physical platforms, like acoustics [34] or elastodynamics [35].

Figure 1:

The schematics and corresponding numerical band structures for metamaterial topological insulators (TIs) considered in this work.

Metamaterial TI based on (A) C6 symmetry and (B) valley conservation. In the top panels, the colored disks represent the spatially local resonators composing the metamaterial crystals. These resonators are assumed to be quarter-wavelength copper rods standing on a ground plane and oriented along z (the figure view is from the top), corresponding to a microwave implementation of local resonant metamaterials, whose structure is effectively subwavelength in the xOy plane. The bottom panels represent the associated numerical band structures obtained from full-wave numerical simulations.

First, we started with designing a C6 TI from the usual procedure [8], [22], i.e. by considering a honeycomb lattice of resonators (in red in Figure 1A), separated by a distance c=0.125λ₀, where λ₀ is the operating wavelength. Instead of looking at this lattice using a primitive rhombic unit cell with two resonators, we follow Ref. [8] and consider an extended unit cell containing six-resonators (dashed lines), viewing the honeycomb lattice as a triangular lattice of hexagonal units (with a period a=3c) (see Figure 1A). Therefore, as demonstrated in previous works [8], [22], by expanding (shrinking) this hexagonal cluster of resonators, thereby preserving the six-fold rotational (C6) symmetry of the lattice, it is possible to open a complete topological (trivial) band gap. This band gap is highlighted in orange in the band structures computed using the finite element method (FEM) which are represented in the bottom panel of Figure 1A for the cases 3c/a=0.8 (red dots, shrunken hexagons), and 3c/a=1.2 (blue dots, expanded hexagons). The well-known distinct topological nature of the metamaterial insulators with expanded and shrunken hexagonal clusters [8], [22] will be exploited in the following to form topological edge modes at an interface between these two media.

Second, we have also designed a VH metamaterial TI by starting again from a hexagonal lattice of resonators of equal resonance frequencies (i.e. equal height, in blue), and open a band gap by detuning the two resonators within the unit cell, increasing the first resonance frequency by a factor 1+β (yellow disks) and decreasing the second by a factor 1–β (orange disks), i.e. f_1,2=(1±β)f_c. Being time-reversed of each other, the two valleys of this insulator at K and K′ have opposite Berry curvatures, implying the existence of an edge mode at the interface between a crystal with opposite signs of the inversion symmetry breaking parameter β [we have referred them as type A (f₁<f₂) and B (f₂>f₁)] [5], [36]. This mechanism will form the basis of the emergence of VH edge modes in the following.

Next, we defined the possible interfaces for both C6-based and VH waveguides. For the first waveguide (C6), there are two possible interfaces: armchair and zigzag, as shown in the top panels of Figure 2A and B. In the bottom panels, we represent the dispersion of the edge modes propagating along the interface (red bands) within the band gap left by the bulk modes (grey). Importantly, the presence of the interface itself already breaks the C6 symmetry of the system, which implies that time-reversed topological edge modes can interact with each other and are gapped near Γ (panel A, bottom). As already known, in case of zigzag interface, the breaking of symmetry is less important [37] and the minigap closes almost completely (see Figure 2B). Note the appearance of a third flat band in panel B, which is unimportant for our purpose as it corresponds to a trivial edge mode that depends on this specific microwave implementation.

Figure 2:

Possible interfaces and associated edge mode dispersions between C6-based and VH-based metamaterial TIs.

(A) Armchair and (B) zigzag interfaces for the C6-based edge modes, (C) bridge, (D) armchair and (E) zigzag interfaces for VH-based edge modes. The bulk modes are shown in grey, and the edge modes in blue and red, depending on the sign of β.

For the VH waveguide there are three possible interfaces: bridge (Figure 2C), armchair (Figure 2D) and zigzag (Figure 2E). In the case of the bridge and zigzag interfaces, there exist an additional two possible connections, depending on whether the interface is taken next to the low-frequency resonators or the high frequency ones: A-B and B-A. For the bridge interface both interconnections give the same valley-protected edge modes (see Figure 2C), while for the zigzag interface B-A, the edge dispersion occurs at a lower frequency than A-B (see Figure 2D). Note, only the bridge and zigzag interface point in the direction ΓK, and therefore are consistent with the VH conservation scheme. The armchair interface, which goes along ΓM, is not expected to support particularly robust edge states, since they are constructed away from the K and K′ valleys. In the following Section 3, we have rigorously proved that it is the case.

Having introduced the two metamaterial crystals and their interfaces, we move forward to the study of their robustness against different sorts of disorder, which are defined in Figure 3. The first type is a position disorder, which we represent in Figure 3A for the bridge valley interface and corresponds to random shifts in the position of the resonators in any direction and in the range [0, R_def], resulting in a local rupture of the lattice symmetry. In the second type of disorder (Figure 3A), the resonators are not moved, but their resonance frequencies are shifted by a random value in the range [−δf, δf], also affecting the local periodicity of the interface. Both types of defects can lead to backscattering, since they either increase the breaking of C6-symmetry or potentially induce intervalley-coupling. Finally, we also considered the typical sharp turn (Figure 3C) often seen in papers about VH insulators, in which an edge mode propagating along ΓK₁ is routed towards ΓK₂.

Figure 3:

Different types of disorders and obstacles studied in this work.

(A) Position and (B) frequency disorder (top panels) are introduced by displacing the resonators or changing their resonance frequency. The disorders are contained in a box of size 2W×L along the interface, where W is the transverse mode width and L is the varying length of the box. Each resonator within this box is affected by a random disorder in the range [0, R_def] for position and [−δf, δf] for frequency disorders. (C) Another type of obstacle is a sharp turn (top panel) and corresponding electric field distributions for VH metamaterial TI with bridge interface (middle panel) and C6 metamaterial TI with zigzag interface (bottom panel), respectively. The electric field color maps in (A) and (B) are generated for a box length L=W and disorder strengths in the range of R_def=[0, 0.2W], δf =[−BW, BW], for position and frequency disorders, respectively (BW is the bandwidth of the transmission band for each waveguide).

Let us first discuss the cases of position and frequency disorders, whose effect on the edge mode propagation requires a statistical approach, in order to average their quantitative behavior out of an ensemble of realizations of disorder. In order to carry on statistical studies, we resorted to a semi-analytical model based on coupled two-dimensional dipoles [38], with the advantage of faster computing times with respect to three-dimensional FEM simulations [22]. In such model, the resonant inclusions are modeled as dipoles with electrical polarizabilities and coupled to each other through the 2D free-space Green’s functions (see Supplementary material for more details). This semi-analytical code was checked to be consistent with the full three-dimensional case in terms of mode profiles and dispersion (examples of semi-analytical mode profiles for particular interfaces are shown in the bottom panels of Figure 3). Similar to our previous work [30], this technique allowed us to extract the transmission coefficient S₂₁ of an edge mode through a disorder box of length L along the interface (the width of the box is set to twice the mode width at half intensity maximum W). For a given length we evaluated the transmission coefficient (S₂₁) as a function of disorder strength in case of position [frequency] disorder S₂₁(R_def)[S₂₁(δf)], and averaged it over many realizations of disorder, keeping the strength of the disorder constant. In order to treat edge modes with different localization and group velocity on an equal footing, the ratio R˜def=Rdef/WC6,VH for position defects and δf˜=δf/BWC6,VH for frequency defects is fixed (where W_c6,VH is the mode width at half intensity maximum and BW_C6,VH is the bandwidth of the transmission band for C6 and VH waveguides, respectively). It is important to stress that, unlike the technique used in our previous work on a chiral metamaterial waveguide [30], this technique is applied to different possible interfaces between the TIs while ensuring a fair comparison between these different systems. We also stress that the frequency at which this study is performed on the edge mode band is carefully chosen to be the one where the transmission is the most robust, which is near Γ for the C6 design, and near K for the VH design. In case of the sharp turn, we do not need to perform statistical averages and simply analyze the transmission coefficient (S₂₁) as a function of operating frequency S₂₁(f) in case of the different interfaces. Examples of such sharp curved waveguides for valley (bridge interface) and C6-symmetry (zigzag interface) with semi-analytical color maps of the electric field distribution are shown in Figure 3C (middle and bottom panels), and other cases in the Supplementary material.

3 Results

The transmission efficiency (S₂₁) of each waveguide geometry averaged over 150 different realizations of disorders for all possible interfaces is represented in Figure 4A and B, assuming a fixed disorder strength R˜def=0.055 and δf˜=1, while varying the defect box length L. In these figures, we do not represent directly the average of the magnitude of S₂₁, but instead the average of its logarithm as a function of L. They are denoted as ⟨ln(T_p)⟩ and ⟨ln(T_f)⟩ for disorders in the position and frequency, respectively. From the figure it is clear that the different waveguides are not equivalent, with some designs handling much better presence of disorder as the length L gets larger. One can also clearly observe that the average logarithms ⟨ln(T_p)⟩, ⟨ln(T_f)⟩ computed for fixed strengths of disorder (R_def, δf=const) drop linearly with the increasing length of the disorder box L, which is a clear sign of 1D Anderson localization [26], [39].

Figure 4:

Statistical analysis of the robustness of topological subwavelength-guiding systems in the presence of position and frequency disorder: the C6-based waveguides with armchair (blue circles) and zigzag (red triangles) interfaces, VH-based waveguides with bridge (green squares), armchair (gray stars) and zigzag (purple rhombs) interfaces.

Ensemble average logarithm of the numerically obtained transmission 〈ln(T_p,f)〉 as a function of the disorder box length L for (A) position (with normalized strength R˜def=0.055) and (B) frequency (with normalized strength δf˜=1) disorders. Normalized Anderson localization length (L˜loc=−(L/W)/〈ln(Tp,f)〉) as a function of the normalized amplitude of (C) position (R˜def) and (D) frequency (δf˜) disorders. Each point represents an inverse of the slope 〈ln(T_p,f)〉 as a function of the disorder box length L shown in (A) and (B). The vertical error bars represent the standard deviation of L˜loc for each disorder amplitude (they are barely visible).

Thus, as a quantitative metric, we can calculate the normalized Anderson localization length L˜loc=−(L/W)/〈ln(Tp,f)〉 by extracting the inverse slopes of the curves of Figure 4A and B, and repeat this operation for other values of R_def and δf. The obtained normalized Anderson localization length L˜loc for each guiding system and each type of disorder are shown in Figure 4C–D as a function of the disorder strength. From these figures, it can be observed that the robustness of the propagating edge waves strongly depends on the geometry and on the type of interface. For instance, consistent with results previously obtained in Ref. [30], the robustness of the C6-based armchair interface is poor when it comes to position disorder, resulting in a quick drop of L˜loc below 10W for disorder strength as low as 5% of W. The VH-based armchair interface, which is aligned in the “wrong” direction (namely ΓM), is very weak to both position and frequency disorders. The VH-based zigzag interface mode is instead quite robust to both disorder types. C6-based interfaces are not very robust to position disorder, since they inherently break C6 symmetry; however their performance in terms of frequency robustness is equivalent to most other designs.

Finally, we move to the study of the turns. We obtained the transmission coefficient as a function of operating frequency S₂₁(f) in the presence of the sharp turns along the propagation path for all waveguides. Due to C_3v- and C_6v-symmetries of the unit cells, the propagation path can be curved with a limited number of angles (namely 60° steps). Here, for the sake of simplicity the transmission is analyzed only for a 120° turn, which respects the valley polarization and is compatible with all the C6-based interfaces. We discard the other possibilities which are known to transmit very poorly in the VH case. The extracted semi-analytical S parameters for all interfaces of C6- and VH-based waveguides are shown in Figure 5A–E.

For the armchair interface of the C6-based waveguide (Figure 5A), the higher frequency band has two transparency windows of transmission, while the transmission drops to zero around Γ point (this corresponds to the point close to the minigap in the band structure of Figure 2A). In the zigzag interface of the C6-based waveguide (Figure 5B), the transparency window is now located next to the minigap or around Γ point, which is consistent with previous observations [37]. In the case of the VH-based waveguides, none of the interfaces provide a total transmission within the analyzed frequency bands. Moreover, as expected of such designs, the transmission drops to zero around Γ point and increases near K points (see Figure 5C–E).

Figure 5:

The extracted S parameters for the 60 degrees sharp turn.

Spectrum for (A) the armchair and (B) zigzag interfaces of the C6-symmetry-based waveguide, (C) bridge, (D) armchair and (E) zigzag interface of the VH-based PTI waveguide. The extracted S₁₁ parameters are represented with blue dashed lines and S₂₁ with red solid lines.

4 Discussion

The quantitative study presented in this work clearly demonstrates that time-reversal invariant metamaterial topological insulators can be used to induce an interesting class of subwavelength topological edge modes, possess a degree of robustness that in the best case, only slightly outperforms that of other subwavelength slow-waveguides based on other means [33]. Under no circumstances, they can provide total immunity against backscattering, as it is the case of Chern topological insulators, as one can sometimes read in prior arts. The most robust of all the designs explored here is the VH-based zigzag interface, which possesses the unique property of being significantly robust to both position and frequency disorder. Similar performance has already been reported with a different subwavelength waveguide design, based on chirality [30], [40]. The other waveguiding schemes compared in this work demonstrate a certain level of protection, stronger for either position or frequency disorder (see Figure 4C and D), which can be explained by the physical symmetries underlying the topological protection. In the C6-based waveguides, any random displacements of the resonators will locally break the symmetry of the hexagonal clusters, mixing the two pseudospin states and reducing the Anderson localization length for both armchair and zigzag interfaces (blue spheres and red triangles, respectively in Figure 4A). However, by introducing a zigzag interface, which reduces the minigap between the edge mode bands [37], these modes are less affected by the lattice disorders. In case of frequency disorders, the C6-based design is less affected by the local changes in the resonance frequency of resonators, which results in a more robust behavior (blue spheres and red triangles, respectively in Figure 4B). In the VH-based waveguides, the direction of the interface is very important, and the edge mode is poorly robust in any direction other than ΓK. Finally, the study of a turn demonstrates the importance of quantitative metrics for the evaluation of topological robustness, highlighting a largely incomplete degree of robustness that depends on frequency, and on the specific design, despite the fact that all these designs are often reported as “topologically protected”.

To summarize, we have performed a statistical study that quantifies with a clear universal metric the robustness to disorder of two different types of waveguiding systems based on time-reversal invariant metamaterial topological insulators, namely systems based on six-fold crystalline symmetry and valley conservation. The obtained results have demonstrated the importance of quantitative metrics for the study and discussion of topological protection and related properties, and highlighted the good performance of the zigzag interface of VH metamaterial insulators when compared to the existing literature. We want to stress here that the study focused on locally resonant periodic metamaterials that are based on periodic arrangements of subwavelength resonators, interacting via multiple scattering, and Fano interferences [41]. A possible extension of this work is to consider whether our conclusions still hold in the case of non-resonant photonic crystals which are based on different physics, namely Bragg interferences. We believe that such systematic quantitative robustness study is an indispensable tool for the practical realization of disruptive subwavelength-guiding systems that are robust to fabrication tolerances and external factors, not only at microwaves, but also for plasmonics or in the area of dielectric-based nanophotonics.