Tolerance and breakdown of topological protection in a disordered waveguide

Topologically protected state is generally known to be stable and robust against perturbations. This topological protection (TP) holds significant importance in various disciplines, including photonics,^1,2 condensed matter,^3,4 atomic physics,^5,6 plasmonics,^7,8 and acoustics.⁹ In particular, TP provides a solution for certain problems in photonics through the unique transport property of the topological edge mode. For instance, in a nanophotonic device, the transmission of an electromagnetic (EM) wave is often affected by backreflection and scattering losses. The topological waveguide enables the transmission of reflection-free waves, even in the presence of substantial structural disorder.^10,11 In addition, photonic systems with TP have been excellent testbeds for exploring non-Hermitian physics,^12,13 non-linear optics,^14,15 higher-order band topology,¹⁶ and Floquet physics.^17,18

This TP, however, can be influenced or even broken by specific environmental conditions. In condensed matter physics, extensive studies were performed regarding the breakdown of TP. For example, in graphene, dissipationless transport was undermined due to crosstalk between counterpropagating pairs of downstream and upstream channels.¹⁹ Also, antidots and upstream modes broke TP of a graphene quantum Hall effect.²⁰ The TP breakdown also occurred in a two-dimensional electron gas by enhanced vacuum fluctuations,²¹ and it was theoretically shown that a smooth edge potential breaks TP via edge reconstruction.²² In contrast, such investigations in photonics are relatively scarce. One study indicated TP breakinga by transitioning from the amorphous to crystalline structural state of the constitutive material Ge₂Sb₂Te₂.²³ Another study revealed a TP breakdown with synchronized rotation of unit cells in a periodic Kekulé medium.²⁴ 

Here, we explore the tolerance and breakdown of TP of an edge mode in a disordered waveguide. Our topological waveguide (TW) consists of a trivial lattice interfaced with a non-trivial lattice. When both the position and radius disorders are applied to the trivial lattice, the unidirectionality of the edge mode sustains, exhibiting the TP nature of the mode. The unidirectionality also maintains under the position disorder of the non-trivial lattice. However, when the radius disorder is applied to the non-trivial lattice, all differently localized defect modes prevent the formation of the topological bandgap, which breaks the edge mode. Moreover, as the radius disorder increases, a frequency domain of bulk modes becomes a non-trivial bandgap, accompanying Anderson localization (AL) of EM waves in the topological regime. Such phenomenon is characterized with the Bott index (BI), a topological invariant obtained from the real-space wavefunctions.

Figure 1(c) shows the dispersion diagram of the trivial mirror with the three trivial transverse magnetic (TM) bandgaps of BG_t1, BG_t2, and BG_t3. The first two gaps of BG_t1 and BG_t2 are Mie bandgaps with TM₀₁ and TM₁₁ modes, respectively. The TM₀₁ and TM₁₁ modes constitute pure Mie bandgaps that exhibit high tolerance to the position and radius disorderings, and, thus, the penetration of EM waves into the trivial mirror is well suppressed. Differently from BG_t1 and BG_t2, both Mie and Bragg scatterings generate BG_t3 with the TM₂₁ mode, less tolerant to the disorderings, causing that the EM waves would penetrate gently more into the trivial lattice than BG_t1 and BG_t2.²⁵ 

In the non-trivial mirror, we find three bandgaps of BG_n1, BG_n2, and BG_n3 in the dispersion diagram of Fig. 1(d). The lowest bandgap of BG_n1 is a trivial Mie bandgap with the TM₀₁ mode. The second and third bandgaps are non-trivial topological gaps. Due to the broken time-reversal symmetry in the non-trivial lattice, the high-symmetry points of M and Γ in the irreducible Brillouin zone break, resulting in the creation of BG_n2 and BG_n3, respectively.^10,26 Breaking the time-reversal symmetry originates from the fact that the anisotropic material of YIG has imaginary off diagonal elements in the permeability tensor with $μ ≠ μ T$⁠, where T is the transpose operation.²⁷ The calculations of the dispersion diagram are elaborated in the supplementary material. The calculations are performed using the finite element method in the COMSOL Multiphysics software.

We proceed to the calculation of Chern numbers in both the trivial and non-trivial lattices. The Chern numbers over the first Brillouin zone are zero for all bands of the trivial lattice. The zero value of the Chern numbers is based on preserving the time-reversal symmetries.^10,26,28 On the contrary, we obtain the Chern numbers of $0 , 1 , − 2 ,$ and –1 for the first four bands, from low to high frequency, in the non-trivial lattice (supplementary material).

Each band's Chern number constitutes the value of the Chern number for each bandgap, i.e., the bandgap's Chern number (C_g) is defined as the sum of bands' Chern number below the bandgap.^10,26,28 As a consequence, we obtain $C g = 0 , 1 ,$ and −1 for BG_n1, BG_n2, and BG_n3, respectively. The overlap between BG_t2 and BG_n1 gives rise to the creation of the bandgap of BG_w at normalized frequencies $0.353 < a / λ < 0.451$ in the dispersion diagram of the TW: This results in no edge mode because the difference between C_g of BG_n1 and that of BG_t2 is zero. Differently from BG_w, the overlap between BG_t3 and the two bandgaps of BG_n2 and BG_n3 creates two unidirectional edge modes at $0.556 < a / λ < 0.578$ and $0.613 < a / λ < 0.637$ with positive and negative group velocities, respectively, as shown in Fig. 1(e). The edge modes are visualized in Figs. 1(a) and 1(b). The out-of-plane electric fields, $E z ( x , y )$⁠, at $a / λ = 0.570$ and 0.630 represent the propagation of EM waves to the right and left directions, respectively.

Next, we explore the impact of disorders to the unidirectionality of the edge modes. We consider two types of disorderings, in the position and radius of the rods. The disordered position of the rod i is defined as $( x i , y i )$⁠, where $x i = x 0 i + σ P F x i$ and $y i = y 0 i + σ P F y i$ with the original position $( x 0 i , y 0 i )$⁠, the strength of the position disorder is $σ P$⁠, and $F x i$ and $F y i$ are uniformly distributed random variables between −1 and 1 for the ith rod along the x and y directions, respectively. The position disordering parameter is defined as $η P = σ P / a$⁠. In a similar way, the disordered radius of the rod i in the trivial (non-trivial) lattice is $R t ( n ) i = R t ( n ) , 0 + σ R F R i$⁠, where $R t ( n ) , 0 , σ R$⁠, and $F R i$ stand for the original radius of the rod in the trivial (non-trivial) lattice, strength of the radius disordering, and the uniformly distributed random parameter over the interval of $[ − 1 , 1 ]$⁠, respectively. We define the parameter of the radius disordering as $η R = σ R / R t ( n ) , 0$⁠.

We first obtain $E z ( x , y )$ under the influence of the position disordering. The parameters of the trivial mirror are $R t , 0 = 0.30 a , ɛ t = 18$⁠, and $μ t = 1$⁠, and those of the non-trivial mirror are $R n , 0 = 0.11 a , ɛ n = 15$⁠, and $μ n = μ$⁠. The normalized frequency of the TM polarized dipole source is $a / λ = 0.570$⁠, where the edge mode of a positive group velocity appears. The calculation results are shown when the disorder is $η P = 40$% in either the trivial or non-trivial mirror [Figs. 2(a)–2(f)]: The unidirectionality of the EM waves is preserved at this substantial strength of disorder.

We explore the unidirectionality more quantitatively by calculating the transmittance $T LR$ of the edge mode. $T LR$ is calculated as the EM wave propagates from −x to $+ x$⁠. We judge that the EM wave has the unidirectionality transmission when $T LR > 0.5$⁠. Figures 2(i) and 2(j) show $T LR$ at several disorderings and $ɛ t$⁠. Each transmittance is obtained by averaging the results of 100 numerical simulations. Under applying the position disorders to both the trivial and non-trivial mirrors, the unidirectionality is preserved, and $T LR$ moderately decreases as the disorder increases from $η P = 0$ to 50%.

Similar behavior is observed when the radius disorder is applied to the trivial lattice [Figs. 2(c), 2(g), and 2(k)]. However, the result is entirely different when the radius of non-trivial lattice is disordered. Figures 2(d) and 2(h) show that the EM waves propagate to $+ x , − x$⁠, and −y directions, and the penetration to the trivial lattice (⁠ $+ y$ direction) is suppressed. $T LR$ abruptly decreases from 1 to $∼ 0.4$ at $η R = 10 %$ and to $∼ 0$ at $η R = 20 %$—the unidirectionality is broken completely [Fig. 2(l)]. The simulations are conducted using the finite element method in the COMSOL Multiphysics software.

We describe the origin of the tolerance and breakdown of the edge mode. Figure 3 shows the dispersion diagrams of the TW, containing a supercell with 5 × 8 of dielectric rods with $ɛ t = 18 , ɛ n = 15$⁠, and 5 × 8 YIG rods of $μ$ (supplementary material). When the position disorder $η P = 40 %$ is applied to either the trivial or non-trivial lattice [Figs. 3(a) and 3(b)], the edge mode with a positive group velocity still exists, and, thus, the unidirectionality of the EM wave maintains. This feature is associated with the mode profile in the rods and the interference of the EM waves. In Figs. 2(e) and 2(f), we identify that the localized modes inside each dielectric and YIG rod under the position disordering are TM₂₁ and TM₁₁, respectively. The unidirectionality to the $+ x$ direction is a result of preserving the localized mode inside each rod in the non-trivial lattice. The gentle penetration of EM waves to the $+ y$ [Figs. 2(a) and 2(e)] and −y [Figs. 2(b) and 2(f)] directions is owing to the random increase in the distances between some rods. The increased distances result in decreasing the coupling between quasi-bound states, which makes the EM waves penetrate between the rods.²⁹ However, all identical TM₁₁ modes in YIG rods sustain the slope of the edge mode.

In the case of radius disorders, defect modes affect the directionality of the edge mode. In this configuration, each disordered rod corresponds to a point defect of the waveguide. The point defects with larger (smaller) radii increase (decrease) the effective refractive index, resulting in the appearance of donor (acceptor) modes. The donor (acceptor) modes, defect modes, fall into the adjacent bandgap from the upper edge (lower edge) of the gap, resulting in filling the bandgap with the defect modes. In other words, donor and acceptor modes experience red and blue shifts, respectively; the acceptor modes between BG_t2 and BG_t3 (radius disorder to trivial lattice), acceptor modes between BG_n1 and BG_n2 (radius disorder to non-trivial lattice), and donor modes between BG_n2 and BG_n3 (radius disorder to non-trivial lattice) generate bulk modes that occupy the gap for the edge mode of a positive group velocity.

Note that this description holds for the cases of weak or moderate radius disorders. As the disorder increases, the impact of donor mode becomes more dominant than that of accepter mode because more modes can exist in a rod of larger radius [see the explanation for Fig. 4(b)]; as the radius disorder increases strongly, the bandgap shifts to red.

The impact of the defect modes is different between when occurred in the trivial or non-trivial lattice. Given radius disorder in the trivial lattice, the acceptor modes between BG_t2 and BG_t3 would fill up the gap at $0.556 < a / λ < 0.578$ for the edge mode of a positive group velocity. However, the localized eigenmodes in YIG rods do not change, which persists the directionality of the edge mode. All identical TM₁₁ mode profiles in YIG rods are identified in Figs. 2(c) and 2(g), and accordingly, the positive sign of the group velocity is preserved, as shown in Fig. 3(c).

In contrast, when the radius disorder is applied to the non-trivial lattice, the topological nature is broken totally. As shown in Fig. 3(d), both the acceptor modes from the frequencies between BG_n1 and BG_n2 and the donor modes between BG_n2 and BG_n3 occupy the gap, and the resulting bulk modes exhibit no directionality. The underlying reason for this breakdown is that all the eigenmodes in YIG rods become different under the radius disorder. All distinct these modes disturb the formation of the topological bandgap of the non-trivial lattice, which herein causes the TP breakdown. This feature is also found in Figs. 2(d) and 2(h), showing that different modes are localized in every YIG rod.

We further investigate this breakdown of TP with another topological variable, the BI.^30–33 The BI, obtained from the real-space electric-field distribution, is particularly useful for characterizing disordered structures where the Chern number cannot be defined. The BI and Chern numbers manifest the topological nature of a band with nonzero values, having the same absolute value with the opposite sign. More details of the BI calculation are provided in the supplementary material. We use the COMSOL Multiphysics software to perform the simulations.

The calculation results of the BI are presented in Fig. 4. We consider a supercell of 6 × 6 YIG rods in air under position and radius disorderings over the frequency interval of $0.430 ≤ a / λ ≤ 0.645$⁠. The BI at each frequency and disordering is obtained by averaging the results of 50 simulations. In the case of position disorder, we find two topological bandgaps of BG_n2 and BG_n3 with the BI of −1 and +1, respectively, which do not change as the position disorder increases [Fig. 4(a)]. This shows that the edge mode is not influenced by the position disorder of the trivial and non-trivial lattices, agreeing with the results in Figs. 2(a) and 2(b), and 3(a) and 3(b). The localization of TM₁₁ mode in all YIG rods gives the same BI, which is independent of position disordering: This supports the unidirectionality of the edge mode.

As shown in Fig. 4(b), the behavior of the topological bandgap, under the radius disorder of the non-trivial lattice, is very different from that under the position disorder: Both the bandgaps of BG_n2 and BG_n3 undergo red shift as $η R$ increases. This frequency shift is attributed to the donor modes in the point defects of larger radii. Under the radius disorder, defects with both larger and smaller radii are present. While many bulk modes can appear in the rods of larger radii, the number of modes that can exist in smaller defects would be much less; for instance, certain modes cannot survive if the size of a rod is smaller than a critical radius. Dominated by the impact of defects with larger radii, the effective index of refraction of the lattice increases, causing the red shift of overall defect modes—the bandgaps shift to red as well.

This red shift brings about two phenomena: The breakdown of the TP and the emergence of AL.^34,35 First, we consider an edge mode at $a / λ = 0.570$⁠. As $η R$ increases, the topological bandgap is occupied with defect modes, resulting in the disappearance of the edge mode (BI changes from –1 to 0): This is the reason for the breakdown of the unidirectionality as discussed above. Second, we concentrate on a region of $0.430 < a / λ < 0.528$ in Fig. 4(b). While this domain is occupied with bulk modes at $η R = 0$⁠, these modes gradually disappear, and the topological bandgap emerges as $η R$ grows. Accordingly, the edge mode appears in this frequency region, accompanying not only the unidirectionality of the mode but also the localization of the EM waves under the disorder—this corresponds to AL. The behavior of AL is revealed in Fig. 4(e), where $| E z ( a , y ) |$ and $| E z ( 3 a , y ) |$ are fitted with the exponential decay function of $E 0 · exp ( − | y | / ξ )$⁠. The localization length is given by ξ. From the fitting, we obtain $ξ ≃ 3.3 a$ for $| E z ( a , y ) |$ and $ξ ≃ 3.8 a$ for $| E z ( 3 a , y ) |$⁠, and this exponentially decaying feature proves the genuine nature of AL. In summary, the phenomena here, as the disorder increases, transition from the metal conducting gap to non-trivial bandgap happens, in tandem with a disorder-induced emergence of AL in this topological regime.

We show the evolution of bandgap under position and radius disorder by calculating the normalized localization length (⁠ $L loc$⁠) as depicted in Fig. 5. To compute $L loc$⁠, we utilize a supercell of $L x × L y = 10 × 10$ YIG rods under position and radius disorder. For each configuration of position and radius disorder, we define $L loc = − L x Ln ( S )$⁠, where S is the transmittance. The transmittance is determined by placing two ports on the right and left sides of the supercell and employing scattering boundaries on the top and bottom for EM wave absorption. Notably, the topological bandgaps of BG_n2 and BG_n3 remain largely unchanged under position disorder, whereas both bandgaps exhibit red shifts.

We finally remark two points regarding our work. First, as far as we are aware, our work explores the impact of radius disorder in a topological waveguide. While previous studies were done with position disorders,^36–38 we investigate how the radius disorders affect the edge mode, making it possible to show the TP breakdown. Second, we herein envision the possibility of a similar study in the optical and infrared (IR) frequency domains. The deployed YIG material exhibits the nature of broken time-reversal frequency near 4.5 GHz.^10,11 In order to investigate such topological behavior in optical frequencies, one can utilize a honeycomb lattice including a dielectric-helical waveguide, in an ambient medium with a refractive index of 1.45.¹⁷ The symmetry breaks in the helical direction, which acts like time-reversal symmetry breaking in YIG, leading to the unidirectional EM wave around the lattice at optical frequencies. In the IR region, one can make use of dielectric rods with a refractive index of 3.42 in a honeycomb pattern, revealing pseudo-time-reversal symmetry preservation. This would result in the generation of unidirectional chiral edge states in the IR domain.³⁹ 

In conclusion, we have studied the tolerance and breakdown of TP in a disordered waveguide. Both the position and radius disorders are applied to either the trivial or non-trivial lattice. The edge mode disappears under the radius disorder of non-trivial lattice because all different donor and acceptor modes prevent the creation of the topological gap. Moreover, through the calculation of a topological variable of the BI, we show that EM waves are localized in a certain frequency region, which is an AL effect associated with the topological gap. Our work offers understanding of the one-way light propagation in nanophotonics and also gives an insight into the development of topological optical circuits and integrated photonic devices.