1. Introduction

The interface between two electronic materials with topologically distinct band structures necessarily supports a topologically protected mode [1–3]. The robustness of this mode arises from global symmetries like parity, space inversion, and time reversal [4,5]. In electronic systems, the topology of the electronic band structures can be controlled relatively easily by either using materials properties (e.g. picking a suitable semiconductor [6]) or by materials engineering (e.g, by making hybrid structures like quantum wells out of suitable materials [7]). The combination of robustness of topological edge modes and the relative ease of fabrication has resulted in an explosion of interest in these systems with applications ranging from decoherence-free quantum state manipulation [8] to electronic device design [9–11].

Following the seminal work of [12], it was realized that topological band structures can also exist in photonic systems. These ideas have been explored in a number of theoretical proposals [13–19]. They have also been realized experimentally in the following photonic crystal systems: gyromagnetic photonic crystal at microwave frequencies [20], coupled whispering gallery mode optical resonators [21], optical waveguides [22,23], and plasmonic arrays [24–26].

The analysis of Refs. [22], which provides a mapping between electronic systems and paraxial light propagation in photonic and plasmonic systems, inspired a number of works. Here, we are concerned with the one-dimensional topological model of Su-Schrieffer-Heeger (SSH) [27,28], which was originally used to describe electron motion in polyacetylene chains. The SSH model hosts topologically protected modes on domain boundaries, the protection being provided by the sub-lattice (or chiral) symmetry which belongs to the AIII-class of [4]. The mapping inspired by Rechtsman et al’s work was adopted to the SSH model and explored experimentally in plasmonic nanowire arrays [24, 29] as well as photonic crystals [23]. The propagation of plasmonic topological modes in graphene nanowires was also explored theoretically in [30]. These works serve to establish the existence of optical topological defect modes.

In this work, we explore how to use SSH model plasmonic topological modes to manipulate light. We provide an in-depth analysis of the topological defect modes, similar to the ones experimentally observed in [23, 24], and propose applications of these modes in plasmonic systems. We show that the proposed setup allows us not only to guide light but also to robustly manipulate it by shifting the topological defect modes, as a function of axial position, inside the structure. Using full 3D finite-difference-time-domain (FDTD) solutions of the Maxwell equations, we demonstrate a beam splitter and a spatial mode filter (that couples light from a pair of bulk modes to a pair of topological defect modes). Further, we analyze tolerance to manufacturing defects. While topological robustness to disorder that is invariant in the axial direction (i.e. time-independent disorder in electronic systems) has been extensively studied before, here we analyze tolerance to disorder that varies in the axial direction (i.e. time-dependent disorder in electronic systems). We investigate two types of defects: (a) defects in nanowire positions, that violate sublattice symmetry exponentially weakly and (b) defects in nano-wire diameter, that violate sublattice symmetry strongly. Using the 3D FDTD calculations we show that our structures are very tolerant to type (a) defects and reasonably tolerant to type (b) defects with transverse length-scales comparable to inter-nanowire spacing, as long as the axial length-scales are larger than the inverse topological band gap.

2. Guiding light using topological defect modes

Consider a system that is almost translationally invariant along the axial, i.e. z-direction. Our goal is to describe paraxial modes, time-harmonic electromagnetic waves that propagate at small angles to the z-axis. For paraxial TM-modes it is natural to focus on the transverse components of the electric field [31,32] and to separate out the fast oscillating part, E_x = ψ_x(x, y, z)e^iβ₀z and E_y = ψ_y(x, y, z)e^iβ₀z, where β₀ = ω/c. The propagation of paraxial TM-modes is governed by the paraxial Schrödinger equation (see appendix for details)

Let us now consider the SSH model, which was originally used for studying electrons in polyacetylene. The backbone of polyacetylene is a chain of carbon atoms with staggered single and double bonds, schematically depicted in Fig. 1(a). (Here, we assume that the pattern of single and double bonds does not have intrinsic dynamics, but is instead a prescribed function of time.) The discretized version of the Hamiltonian describing the hopping of spinless electrons along the backbone of polyacetylene is

 

Fig. 1 (a) Single-double bond pattern in a polyacetylene chain with a kink defect. The red disks show the carbon atoms. (b) Analogous nanowire array with staggered spacing and a kink defect. (c) Staggered nanowire spacing leads to a band gap for the bulk modes (red), the topological defect mode appears in the middle of the band gap (blue). (d,f) The positions of nanowires in the array without (d) and with (f) a domain wall. (e,g) Spreading of light in the nanowire array without (e) and with (g) a domain wall. The topological defect mode bound to the domain wall is observed guiding light in (g). For this figure we used ∊ = −45.83 + 2.49 × 10⁻⁹i (the real part of the permittivity is identical to that of silver and the imaginary part was chosen to be very small to highlight the guidance by the topological defect mode.) Geometry parameters used: diameter 200 nm, center-to center spacing 300 and 500 nm, wavelength λ = 1 μm.

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A domain wall in the single-bond double-bond pattern of the chain, as depicted in Fig. 1(a), is a topological defect: locally changing the bond strength around the kink cannot eliminate it as the single- double-bond pattern is a non-local property. Domain walls can, however, be eliminated in pairs by moving them towards each other, as the single-double-bond pattern far away from where the domain walls are being merged is not effected by the merger. Moreover, each domain wall must host a mid-gap state, a topological defect mode, that is localized in the vicinity of the domain wall.

We now investigate the optical equivalent of the SSH topological defect modes in plasmonic crystals. Consider a plasmonic crystal that consists of an array of parallel nanowires with staggered spacing as depicted in Fig. 1(b). The Helmholtz-Schrödinger analogy tells us that for each eigenmode of the SSH Hamiltonian Eq. (2), there is an equivalent electromagnetic mode in the plasmonic crystal. We note that the fermionic commutation relations do not play a role here as we are considering the non-interacting case; hence, we can replace the operator $c_i^{†}$ that creates an electron in a carbon atom atomic orbital by the operator $b_i^{†}$ that create a surface plasmon on the i-th nanowire. (We present the details of the connection between the continuous and discrete Hamiltonians in the appendix.) Thus the band gap in the electronic system maps onto a β_z gap in the plasmonic system. Moreover, electronic states that are localized on domain walls (and appear inside the band gap) map directly onto guided plasmonic states that propagate along the domain walls in the z-direction [and appear in the β_z gap, see Fig. 1(c)].

Now consider injecting a spatially truncated plane wave into a plasmonic crystal with staggered spacing but no domain walls [Fig. 1(d)]. This is equivalent to injecting an electron into polyacetylene using a local probe like an STM tip. Because the electron is being injected locally, it overlaps many k-modes, and hence the electron wave-packet will spatially spread out as time advances. Similarly, the plasmonic wave packet will expand in the transverse direction as it advances along the z-direction. To illustrate this expansion we introduce the normalized Poynting vector P_z in the x–z plane that cuts through the middle of the nanowire array

The introduction of a topological defect into the plasmonic crystal [see Fig. 1(b)] gives rise to a localized mode. We expect that light injected in the vicinity of the topological defect couples both to the localized defect mode as well as to the bulk modes. We plot the results of injecting a truncated plane wave into a plasmonic crystal with a topological defect [Fig. 1(f)] in Fig. 1(g). In accord with our expectations, we observe that light coupled into the bulk modes forms a diffracting fan, while light coupled into the topological mode propagates without spreading transversely.

We comment that light guided by a topological defect mode is spatially concentrated. As a figure of merit, we consider the quantity λ²P_z, where λ is the free space wavelength, which measures how much the light is squeezed spatially as compared with diffraction limited optics (λ²P_z ≈ 1 at diffraction limit). Plasmonic confinement of light in the topologically guided mode of the structure depicted in Fig. 1(g) results in λ²P_z ≈ 16.6.

3. Manipulation of light using topological defect modes

In the electronic system, there are two well-established operations for manipulating topological defect modes: (i) shifting the position of a topological domain wall causes the associated topological defect mode to be carried along with the domain wall, and (ii) pairs of domain walls can be nucleated and pulled apart, causing two of the bulk modes to be turned into topological defect modes. In this section, we explicitly apply these operations to achieve topological manipulation of light in the plasmonic crystal of metal nanowires.

In the Helmholtz-Schrödinger correspondence, ∂_t is mapped to c∂_z, so that time dependent manipulation in the electronic system is mapped onto axial dependent manipulation in the plasmonic crystal. For instance, the time-dependent shifting of domain walls in the electronic picture is mapped to the shifting of the domain walls as a function of the axial position z in the plasmonic crystal, which we achieve through the variation of the nanowire spacing as a function of z.

3.1. Topological spatial mode filter

Nucleating two domain walls in the middle of the SSH chain and adiabatically moving them apart results in a spectral flow in which a pair of delocalized electronic states, one from the upper bulk band and one from the lower, are adiabatically transformed into the two mid-gap states spatially localized on the domain walls. We take advantage of this spectral flow to perform spatial mode filtering of light using a plasmonic crystal.

Specifically, we design a plasmonic crystal with a pair of domain walls that are created inside the array and shifted apart as a function of z, see Fig. 2(a). Two of the bulk modes at the input side (z = 0) of the array are mapped into the two topological defect modes on the output side (z = 200 μm) by the spectral flow, see Fig. 2(b). If one of these two bulk modes is injected into the array, after propagation, the maximum of the light intensity on the output side will be strongly localized around the kinks. To verify this behavior, we perform full 3D FDTD simulations of the structure depicted in Fig. 2(a), and plot the results in Fig. 2(c). We observe that the majority of the light flux is indeed guided into the topological defect modes. On the other hand, if we inject any other bulk mode on the input side it will be rejected by the mode filter and the output light flux in the vicinity of the domain walls will be small (see appendix). We note that the topological defect modes are not always in the center of the band-gap [see Fig. 2(e)] as we are locally slightly breaking the sub-lattice symmetry. Indeed, topological protection endows our device with robustness against exactly these types of perturbations.

 

Fig. 2 (a,d) Nanowire array geometry for a spatial mode filter (a) and a beam splitter (d). Blue lines indicate the positions of the nanowires and red dashed lines the position of the kinks. (b,e) Spectral flow of bulk modes (blue) and the topological defect modes (red) as a function of z for the spatial mode filter (b) and the beam splitter (e). (c,f) Light propagation through the spatial mode filter (c) and the beam splitter (f), red dashed lines indicate the position of the kinks (see text). Parameters used are same as Fig. 1.

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3.2. Beam splitter with topological defect modes

Consider the geometry of a nanowire array depicted in Fig. 2(d): two domain walls that are well separated spatially on the input and output sides of the structure are brought close together in the middle of the structure. The propagation of the two topological defect modes can be described by the Landau-Zener Hamiltonian, in which the coupling is controlled by the spatial separation of the domain walls. The spectral flow of the array along the z direction is shown in Fig. 2(e). When the domain walls are well separated, the topological defect modes are non-interacting, and hence both are in the middle of the gap. As the domain walls are moved closer, the topological defect modes begin to interact and the degeneracy is broken. By controlling the interaction strength and the length of the interaction region it is possible to construct a 50–50 beam splitter. We perform 3D FDTD simulation of the beam splitter in which we inject light into one of the defect modes on the input side and observe an equal superposition of light in the two defect modes on the output side [see Fig. 2(f)].

4. Tolerance to perturbations in wire placement and diameter

Consider a structure in which a topological defect mode is guided by a kink that is being shifted as a function of z [see Fig. 3]. To test the tolerance of the light manipulation to perturbations (e.g. manufacturing defects), we measure the amount of light that leaks from the topological defect mode into the bulk modes due to perturbations in the plasmonic nanowire structure. This is a particularly stringent test of mode transport fidelity, as perturbations near the domain wall can affect the structure of the defect mode directly as well as how the defect mode hops from nanowire-to-nanowire as the domain wall is shifted. As it is not possible to test the tolerance to all kinds of perturbations, we focus on two specific types that are likely to occur in experimental situations: meander of a single nanowire and distortions in the diameter of a single nanowire. We note that topological robustness only protect defect modes when they are moved infinitely slowly. In a realistic plasmonic system we would rather move the defect modes as fast as possible to avoid absorption losses which has the potential to break the adiabatic approximation. It is the tolerance of light manipulation rather than topological mode robustness protected by the chiral symmetry that is examined in this section.

 

Fig. 3 (a,b) Topological defect mode propagation in a nanowire array with a perturbation. The red nanowire is shifted away from its original position indicated by the black dashed line. The smoothness parameter is w = 25 μm in (a) and w = 6.25 μm in (b). (see text for details) (c) Fraction of flux retained in the topological defect mode η in a structure with a shifted nanowire perturbation. We vary the smoothness parameter w and fix the displacement parameter δx = 150 nm. (d) η in a structure with a nanowire diameter perturbation, the nanowire diameter parameter δd is varied and w = 25 μm is fixed. Materials and nanowire geometry parameters are same as Fig. 1.

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In Figs. 3(a) and 3(b), we compare the results of 3D FDTD simulation for two test structures in which we displace one of the nanowires by the addition of a meander M(z) = δx e^{−(z−z₀)2}/w². In Fig. 3(a) the meander is smooth with w = 25 μm, while in Fig. 3(b) the meander is abrupt with w = 6.25 μm. The positions of unperturbed wires are indicated with black lines, the original position of the perturbed wire is indicated with black dashed line and its new position with the red line. The color scale shows P_z for light that is injected into the topological defect mode on the left side of the structure. We observe that if the meander is sufficiently smooth the topological mode remains guided along the domain wall [Fig. 3(a)]. However, if the meander is too abrupt there is significant leakage of light into the bulk modes [Fig. 3(b)].

What role does topological protection play here? The spectral flow calculation (see Appendix D) shows that the topological defect mode β_z shifts as M(z) becomes large. This feature is due to the existence of next-nearest-neighbor (NNN) coupling between the nanowires which breaks the sublattice symmetry. As the NNN coupling is exponentially small, the topological defect mode is well protected from nanowire meander disorder (see Appendix D for details).

Despite the topological protection from perturbations that respects sub-lattice symmetry, these perturbations can still break adiabaticity (i.e. an abrupt meander) to give heavy leakage. This is a consequence of the fact that while topological protection ensures that the defect mode is pinned to the middle of the topological band-gap, the wave function of the topological mode still depends on the perturbation. Thus if the perturbation is turned on sufficiently abruptly, on a length scale ≈ Δβ_z, adiabaticity will break down. To quantify the amount of leakage caused by a perturbation, we introduce the quantity η,

Perturbations of the nanowire diameter, as opposed to position, are potentially more problematic as these explicitly break the sublattice symmetry [by affecting the plasmon self-energy]. We investigate a series of structures similar to the one depicted in Fig. 3(a), but instead of shifting the red wire, we modify its diameter D(z) = D₀ − δd e^{−(z−z₀)2}/w², where D₀ = 200 nm is the unperturbed diameter. We plot η as a function of δd in Fig. 3(d). We observe robustness to small perturbations followed by a sharp drop in η for large perturbations. The loss of transport fidelity is again associated with the loss of adiabaticity. However, due to the sublattice symmetry breaking nature of the perturbation, the loss of adiabaticity occurs prematurely as the defect mode is pushed close to the edge of the topological band-gap. That is the defect mode is indeed protected from small perturbations in the nanowire diameter (and other non-avoidable symmetry-breaking perturbations) by the topological band gap.

In summary, the device is quite tolerant to perturbations in both the nanowire position and diameter. Indeed, the device is very tolerant to nanowire position errors, which preserves the chiral symmetry if the NNN interaction is ignored. In order to see strong leakage from the topological defect mode into the bulk modes we need to displace a nanowire until it is almost touching its neighbor. The device is more sensitive to nanowire diameter which breaks the chiral symmetry, however it is still tolerant to diameter errors of ∼ 30%.

5. Outlook and summary

One of the key issue of plasmonic devices is the absorption of light due to finite optical conductivity in metals and hence a small but finite imaginary part of the dielectric constant. A useful figure of merit is FOM = Δβ_z L_d/(2π) the product of the decay length and the topological band gap. The FOM counts how many adiabatic operations we can do on a topologically guided mode before it decays by 1/e. The proposed topological mode filter and beam splitter both require ∼ 50 operations. For silver nanostructures with λ ∼ 900 nm (in vacuum), we find that the FOM ∼ 5 − 20 depending on the details of the structure (see appendix). From the experimental perspective, these are appealing length scales due to the availability of lasers and the ease of nano-fabrication. Moreover, for demonstration purposes the attenuation is quite reasonable. Applications would require further optimization of the structure to limit attenuation.

In summary, we have provided an in-depth analysis of the plasmonic analogue of the topological protected defect modes in the SSH model theoretically. We have explored using topological defect modes to manipulate light. Specifically, we numerically demonstrated the functionality of two devices: a mode filter and a beam splitter. Moreover, we showed that light manipulation is tolerant to fabrication errors in nanowire diameter and very tolerant to fabrication errors in nanowire displacement.

Appendix

Appendix A. The paraxial Schrödinger equations

The propagation of electromagnetic waves is governed by the Helmholtz equation

(6)$$\left({\nabla }^2+\frac{{\omega }^2}{c^2}∊\right)\left\{\begin{array}{c}\overrightarrow{E}\\ \overrightarrow{B}\end{array}\right\}=-\left\{\begin{array}{c}\nabla \left({∊}^{-1}\nabla ∊\cdot \overrightarrow{E}\right)\\ {∊}^{-1}\nabla ∊\times \left(\nabla \times \overrightarrow{B}\right)\end{array}\right\}$$

where ∊(x, y, z) is the position-dependent relative permittivity, E⃗(x, y, z), B⃗(x, y, z) are the electric and magnetic field components, and the right hand side encodes the boundary conditions at the metal-air interface. For the case of dielectric waveguides, the dielectric constant tends to vary gently, and hence the terms on the right hand side of Eq. (6) can be neglected. Therefore it is natural to obtain the paraxial Schödinger equation from the Helmholtz equation for E_z. For metallic structures variations of ∊ cannot be neglected. However, for structures that are translationally invariant in z-direction, ∇_∊ only has transverse components. Hence, it is natural to focus on the transverse components E_x and E_y, as the Helmholtz equations for those components have closed form. For TM-modes propagating at small angles to the z-axis it is natural to implement the paraxial approximation, which leads us to the paraxial Schrödinger equation, Eq. (1). The potential energy operator, that appears in Eq. (1), is given by

$$V(x,y,z)=\frac{\omega }{2}(1-∊)+c\left(\begin{array}{cc}{\nabla }_x\left(\frac{{\nabla }_x∊}{∊}\right)& {\nabla }_x\left(\frac{{\nabla }_y∊}{∊}\right)\\ {\nabla }_y\left(\frac{{\nabla }_x∊}{∊}\right)& {\nabla }_y\left(\frac{{\nabla }_y∊}{∊}\right)\end{array}\right),$$

where the second term accounts for the boundary conditions at the metal-air interface.

Appendix B. Connecting the continuous and discrete Helmholtz equations

In this appendix, we connect the continuous description of electromagnetic waves Eq. (1) with the discrete description of the SSH model given by the Hamiltonian

(7)$$H_{\text{SSH}}=-\sum _i{t}_{i,i+1}\left(c_i^{†}{c}_{i+1}+c_{i+1}^{†}{c}_i\right)+v_i{c}_i^{†}{c}_i.$$

This Hamiltonian is an extension of Eq. (2), where we have added the on-site energy v_i which will be used to describe the plasmon self-energy. We note that this connection is only precise in the limit of weak coupling (i.e. when the distance between nanowires is sufficiently large compared to the wavelength of light). In the strong coupling limit the topological properties of the Helmholtz equation remain intact but the tight-binding model can no longer be used to accurately describe light propagation. Consequently we use full 3D FDTD solutions of the Helmholtz equation throughout the main text.

Our strategy to make the connection is to (1) describe the plasmon “self-energy” by modeling a single nanowire, and (2) describe the plasmon hopping by modeling two nanowires. We begin by considering a single nanowire of the type that makes up the plasmonic crystal. The single nanowire has a well defined plasmon mode, i.e. a radially symmetric solution of Eq. (1) of the form ψ(x, y, z) = ψ₁(x, y)e^iβ₁z. We can capture the plasmon self-energy β₁, by setting v_i = c(β₀ − β₁) in Eq. (7).

Next, we consider the plasmon modes of a system of two parallel nanowires separated by distance s. The plasmon spectrum is now composed of two modes ψ_± with eigenvalues β_±. These are approximately the symmetric ${\psi }_{+}\approx \left({\psi }_1(x+s/2,y)+{\psi }_1(x-s/2,y)\right)/\sqrt{2}$ and the antisymmetric ${\psi }_{-}\approx \left({\psi }_1(x+s/2,y)-{\psi }_1(x-s/2,y)\right)/\sqrt{2}$ combinations of the single nanowire modes. Comparing this spectrum with the spectrum of the two site discrete model, we identify t = (β₊ − β₋)/2. In Fig. 4(a) we plot β_z of the symmetric and antisymmetric modes obtained using the Helmholtz equations as a function of s. We observe that the splitting of the symmetric and antisymmetric modes with respect to the single-nanowire β₁ (black dash line) is even for s > 0.5 μm and hence we can extract the tight binding parameter t. For s < 0.5 μm the splitting becomes uneven signaling the breakdown of the tight binding model. We plot the extracted tight-binding parameter t = (β₊ − β₋)/2, which is applicable for s > 0.5 μm, in Fig. 4(b).

 

Fig. 4 (a) The splitting of the symmetric and antisymmetric plasmon modes (for λ = 1 μm in vacuum) in a system of two parallel silver nanowires with 100 nm radius, as a function of the nanowire center-to-center separation s. The black dashed line indicates the single nanowire eigenvalue β₁. (b) Extracted tight-binding parameter t as a function of spacing between two nanowires. The tight-binding model starts to break down when the spacing between the two nanowire goes below 0.5 μm (indicated by the dashed red line).

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Appendix C. Mode filtering

In Fig. 5 we demonstrate an example of mode rejection by the mode filter. We inject a mode that is not adiabatically connected to the two topological defect modes in the structure. The light flux spreads out over the whole structure except for the area in the vicinity of the two topological defects.

 

Fig. 5 Mode rejection by the mode filtering nanowire array. On the input (left) side of the nanowire array, a bulk mode that is orthogonal to the two select modes, is injected into the nanowire array. On the output (right) side there is essentially no light in the vicinity of the two topological defect modes.

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Appendix D. Tolerance to fabrication error

In this set of figures we provide additional data for the same set of structures that were used to construct Fig. 3 of the main text. The data demonstrate the tolerance of the topological defect mode manipulation against perturbation in nanowire position and diameter. The top panel of Fig 6(a) shows the light flux computed using 3D FDTD in a structure with a single topological defect shifting across the nanowire array. Light is injected into the topological defect mode on the left side of the array. The light is guided in the middle of the nanowire array and remains confined to the topological defect until it exits the array on the right side. The bottom panel of Fig 6(a) shows the spectral flow of the β_z spectrum as a function of position along the wire (similar to Figs. 2(b) and 2(e) of the main text). The spectrum plot shows that the defect mode is well separated from the bulk modes throughout the structure.

 

Fig. 6 Light propagation along structures with a perturbation on the position of one of the nanowires. (a) Light flux (top panel) and the spectral flow (bottom panel) in a structure with no perturbation. (b)–(f) Light flux (top panel), meander M(z) (middle panel), and spectral flow (bottom panel) in structures with one of the nanowires displaced from its original position (dashed line in top panel) to a new position (red line in top panel). The nanowire is displaced by δx = 150 nm over a width of w = 25.00 μm (b), 18.75 μm (c), 12.50 μm (d), 10.00 μm (e), 6.25 μm (f).

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Next, we test tolerance to perturbations by displacing the red nanowire by a Gaussian with maximum displacement of δx and a width w. In Figs. 6(b)–6(f) we plot the light flux, wire displacement, and the spectral flow of the β_z spectrum as a function of z for δx = 150 nm and w ranging from 25 μm to 6.25 μm. From the light flux plots (top panels), we observe that the light intensity is well guided when w ≳ 12.5 μm. For w ≲ 10.00 μm, light flux heavily leaks into the bulk modes. The reason for this, is the breaking of adiabaticity as w ≲ 1/Δβ_z.

Figure 7 is similar to Fig. 6, except we perturb the diameter D of the red nanowire. The unperturbed nanowire has a diameter of 200 nm, and we shrink it by 25.0nm to 100.0 nm in Figs. 7(a)–7(f). The perturbation profiles have a fixed width of 25 μm. Topological mode guidance starts to break down when the diameter shrinks by 87.5 nm as the β_z of the guided mode is approaching the bulk spectrum.

 

Fig. 7 Same as Fig. 6, except we perturb the diameter D(z) of the red nanowire. We fix the perturbation width at w = 25μm, and shrink the nanowire diameter by δD = 25.0 nm (a), δD = 50.0 nm (b), δD = 62.5 nm (c), δD = 75.0 nm (d), δD = 87.5 nm (e), δD = 100 nm (f).

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We note that the topological defect modes in Fig. 6 are shifted slightly from the middle of the band gap when we introduce disorder on the nanowire position. This shift is caused by the exponentially weak next-nearest-neighbor (NNN) hopping of the plasmons, which breaks the sub-lattice symmetry. Indeed, the effect of the NNN interaction on the energy of the topological defect mode (in polyacetylene) has been studied before [36–39].

Appendix E. Decay of the topological defect mode

In this section, we compute the decay length of the topological defect modes using complex susceptibility for silver nanowire structures. We use Lumerical Mode Solution numerical eigenmode solver to obtain complex β_z and extract the decay length for two types of topological defects (see Fig. 8 inset). Specifically, we simulate a nanowire array consisting of 7 nanowires with periodic boundary conditions. The nanowires have a diameter of either 400, 800, or 1200 nm and staggered surface-to-surface spacing of 50 nm and 350 nm. The key figure of merit is FOM = Δβ_z L_d/(2π). Here, Δβ_z is the topological band gap, which sets the length scale for adiabatic manipulation of topological defects; L_d = 1/Im[β_z] is the decay length for the topologically guided mode.

 

Fig. 8 The Figure of Merit (ΔβL_d/(2π)) as a function of the free space wavelength (λ₀) for a silver nanowire array with nanowire diameter 400 nm, 800 nm and 1200 nm. The major and minor surface-to-surface spacing is 350 nm and 50 nm respectively. The dashed lines show the topological defect with minor spacing around the defect while the solid lines show the topological defect with major spacing, as indicated in the insets.

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In Fig. 8 we plot the FOM as a function of free space wavelength (λ₀) for two flavors of topological defects: domain wall with major spacing (solid lines) and domain wall with minor spacing (dashed lines). We observe that FOM for topological defects with major spacing is larger than the FOM for defects with minor spacing. The reason for this is the large penetration of the electric field into the nanowires for the kink with minor spacing [see plots of the mode structure, Fig. 9]. We also observe that the FOM has a broad maximum around 900 nm, which defines an optimal wavelength for silver structures. The maximum is a result of the competition between the topological band gap (which decreases with increasing wavelength) and the decay length (which increases with increasing wavelength). Finally, we observe that the FOM for domain wall with major spacing is larger for larger diameter nanowires, which is a consequence of the suppression of L_d as the wire diameter becomes smaller than wavelength.

 

Fig. 9 The electric field z component (E_z) of the topological defect mode around two kinds of defects, (a) a topological defect with major spacing, and (b) a defect with minor spacing. The E_z field is calculated based on a 400 nm diameter silver nanowire array with a staggered surface-to-surface spacing 50 nm and 350 nm. The dashed circles show the position of the nanowires.

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Appendix F. Phase matching to improve coupling to the topological defect mode

Finally, we make a remark regarding [24], which claims to experimentally detect topologically guided modes in the type of structures that we propose. We found that the analysis carried out in [24] is flawed. Specifically, (i) there is breakdown of the coupled-mode theory, which predicts velocities of modes that are faster than the speed of light (ii) there is zero overlap of the injected mode with the topological mode in minor-spacing defect array due to symmetry reasons.

The the self-energy in the coupled mode theory results in the real part of mode index β = 1.047k₀, where k₀ is the free space wavenumber. According to their coupled mode theory calculation, the lower band edge reaches β_z ≈ β − 0.225 k₀, which corresponds to β_z ≈ 0.822k₀. This value of the mode index indicates that the mode is propagating faster than the speed of light, which signals the breakdown of the coupled mode theory.

The topological protected defect modes in major-spacing case is symmetric while minor-spacing case tends to have an anti-symmetric nature [see Fig. 9]. In real experiment, if the plasmonic mode is injected directly into the beginning of the nanowire array without any modification of the plasmonic field, the mode is likely to be symmetric and fails to couple to the topological defect mode in minor-spacing case. Here we proposed a way of modifying the length of the nanowires at the beginning of the nanowire array to induce phase difference between the nanowires [see Fig. 10], in order to improve topological mode coupling efficiency. If we assume a truncated plane wave injected into the plasmonic nanowire arrays shown in Fig. 10 from above and the plasmon is excited at the edges of each nanowire, in order the achieve the π-phase shift between the defect nanowire and the nearby nanowires [see Fig. 9], the nanowire length difference should be Δl = π/β, where β is the wave-number of the plasmonic mode. The different staggered pattern in minor-spacing defect array [Fig. 10(a)] and major-spacing defect array [Fig. 10(b)] is because of the distinct phase profile of the topological defect modes in these two nanowire arrays.

 

Fig. 10 The schematic nanowire array geometry at the beginning of an array with minor-spacing defect (a), and major-spacing defect (b). The beginning of the nanowire array is staggered to match the phase different between the nearby nanowires in topological defect mode. The length difference between the nanowire is determined by the exaction method and plasmonic mode wavelength in the nanowire array.

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Funding

Charles E. Kaufman Foundation (KA2016-85221 and KA2014-73919).

Acknowledgments

C. L. was partially supported by a Pittsburgh Quantum Institute graduate student fellowship. We acknowledge the support from the Charles E. Kaufman Foundation.

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