## Abstract

Plasmonic cluster arrays have demonstrated rich physics in topological photonics, but they are seriously affected by the material loss and limited by the requirement of high-precision machining. Here, we propose a kind of ultra-thin metaparticle arrays which can mimic the coupled localized plasmonic resonances at lower frequency ranges and so that can overcome the loss and fabrication problems in real metal plasmonic systems. The metaparticle is a metallic disk with circuitous grooves that can support both spoof electric and magnetic localized resonances, and these resonances can be pushed to a subwavelength region through tuning the geometric parameters. In virtue of the highly field confinement of these localized resonances, it is thought to be an ideal experimental platform to be an analogy with various near-field interactions in topological materials. As a first proof-of-concept study to show this feasibility, the subwavelength topological edge states at the zigzag metaparticle chain boundaries are numerically and experimentally demonstrated at microwave ranges. Moreover, the subwavelength topological edge states in this zigzag chain can be excited simply by the plane wave incidence, and the edge modes at two ends can be selectively excited by controlling the polarization direction. Therefore, this kind of metaparticle array not only provides an ideal platform to experimentally study various near-filed interaction dominated topological systems but may also find massive potential applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Massive researches have shown the great potential of surface plasmons (SPs) as a new route for various functional photonic devices, e.g., data storage, light-emitting transistor, surface-enhanced Raman scattering, photobiology and photomedicine [1,2]. One of the most striking properties of SPs is the effect of subwavelength field confinement and the associated energy concentration in nanostructures. However, this phenomenon is inevitably limited in visible or near-infrared frequency ranges with regard to noble metals according to the material response. At lower frequencies, electromagnetic (EM) modes are weakly bounded near the metal surface, and in the extreme case of the perfect electric conductor (PEC), SPs can not be supported. Fortunately, the concept of SPs was transferred to geometry-induced bound EM modes in lower frequency ranges by analogy with their similar features, and these plasmon-like modes are called spoof surface plasmon polaritons (spoof SPP) [3,4]. In contrast to the real SPs, the spoof (or designer) SPP can exist on the grooved metallic surface in the PEC limit with negligible losses, and its dispersion can be tuned by simply changing the geometric parameters. The transverse confinement of spoof SPP has been reported in various grooved systems, including the perforated 2D planes [3–5] and 1D waveguide chains [6, 7]. Moreover, it was also demonstrated that the textured closed surfaces can support localized spoof SPP both in 2D and 3D systems [8, 9]. For spoof SPP in three-dimensional spaces, the thickness of the metallic groove structures can be reduced to being infinitely thin due to the intrinsically field confinement away from the surface. By virtue of this property, the spoof SPP modes can be supported by metallic metasurface structures which are experimentally accessible using standard printed circuit board (PCB) technology. And the flatness of the metasurface facilitates the local field measurement in experiments [10, 11]. In addition, the spoof SPP can also be found in their complementary structures according to Babinet’s principle [12–14], and the high-order resonances of both electric and magnetic modes are easy to be implemented through tuning the geometric parameters while high-order modes of magnetic resonances are rarely observed in real plasmonic systems [11,14–18]. The spoof SPP modes of these artificial subwavelength structures with high degrees of freedom to design can be regarded as the low frequency analogues of the localized surface plasmons (LSPs) on nanoparticles and should share the same features as its optical counterpart.

In real plasmonic optics, the 1D chains or 2D lattices consisting of subwavelength nanoparticles exhibit fantastic optical guiding and energy localization potentials [19–21]. At a subwavelength scale, the inter-couplings of low-order LSPs modes approximate to tight-binding interactions and can be solved through self-consistent coupled equations [19]. This also makes the plasmonic lattice be a proper platform to mimic the electron hoppings in condensed matter. Particularly, similar to the split-ring resonator chains [22], the topologically protected edge modes firstly proposed in electronic systems were also found to exist in 1D and 2D plasmonic systems [23,24]. The topological edge states in zigzag chains of plasmonic nanodisks has also been found in simulations [25] and experimentally observed through near-field scanning optical microscopy [26]. The plasmonic disk chains or lattices greatly promote to a variety of compact nanodevices in a subwavelength scale such as polarization-controllable nanoantennas, environmental sensors and optical circuits [27–29]. However, the plasmonic chains or lattices require a rather high manufacturing accuracy which is difficult to the current e-beam lithography technology and the field-mapping measurement is greatly disturbed by probe-sample interactions. Moreover, the real plasmonic resonances limit the wavelength to visible and near-infrared ranges only, and the nanodevices based on plasmonic resonances suffer heavily from the intrinsic material losses.

In this letter, we first theoretically analyzed and numerically studied both the electric and magnetic modes of an ultra-thin metallic metaparticle and scale them down to a subwavelength region through tuning the geometric parameters. By comparing its spoof localized resonances with real SPs in metallic materials, we declare this metaparticle can act as a low frequency analogy with the LSP in nanostructures. Therefore, the interaction of localized resonance modes in this metaparticle arrays can perfectly mimic the coupled LSP resonance in nanoparticle arrays, and thus to be an ideal platform to study the topological physics for those are based on the tight-binding interactions. Under the approximation of dipole-dipole interactions, we numerically simulated and experimentally demonstrated the existence of the topological edge states in a zigzag chain consisting of this spoof LSPs planar metaparticles, and the edge states at two ends can be selectively excited by controlling the polarization of the normally incident plane wave. More interestingly, the frequency sequence of these electric resonance modes is reversed with respect to their azimuthal quantum numbers. Such spoof SPP structures can be fabricated by the standard PCB technology and directly measured by field scanning process while it has only a little loss.

## 2. Localized spoof SPP in deep subwavelength

Arrays of noble-metal nanoparticles can support coupled localized surface plasmon resonances in optical ranges, and the subwavelength field confinement of these localized modes lead to nearest neighbor coupling dominating. Therefore, the optical properties of the nanoparticle arrays can be conveniently manipulated by tuning the dimensions of the particles and the periods, which has triggered many thrilling researches and applications [28,30,31]. However, these plasmonic lattices and related applications are severely hampered by the material loss and the fabrication limitation. The all-dielectric materials can overcome the disadvantages of material losses but the large-refractive-index materials are required [32,33]. Moreover, the all-dielectric particle of subwavelength size is dominated by the Mie scattering resonance modes in contrast to LSP, so that the energy of these resonance modes can not be totally localized [33]. In consideration of these problems, scaling down the frequency to microwave ranges seems to be a good solution since the metal can be regarded as PEC in the radio-frequency range. However, natural metallic materials can hardly support the SPs or LSPs in RF ranges as the negative dielectric constants are far away from zero at microwave ranges. Therefore, the metamaterial which supports spoof SPs and LSPs at microwave frequencies can be a good pathway to substitute for conventional metallic materials.

Here, we propose an ultra-thin subwavelength metallic disk with circuitous grooves that can support spoof LSPs in microwave ranges, and the metaparticle does not demand any high-dielectric-constant materials. It has also to be noted that the subwavelength size of the metaparticle is required for low-order dipole-dominated response which is similar to LSP in real metals. Started from the textured PEC cylinders and we will show how the resonances are scaled down to a subwavelength region. According to the previous results [8, 9], the subwavelength regime can be achieved by increasing the groove length or the refractive index inside the grooves. After massive calculations and simulations, we found the circuitous groove is one of the most effective way to lower the resonance frequencies of the corrugated PEC cylinder with a fixed outer radius *R*. Here, we try to avoid using large index materials in the grooves due to the difficulty of experimental realization. In principle, the corrugated cylinder can be strictly approximated to a spoof plasmonic cylinder only if the number of grooves *N* approaches to infinite large. However, all of the phenomena are almost same to the ideal one when *N* is large enough. In fact, the final design of the unit cell is a trade-off between all of the requirement (see details in Appendix A). As shown in Fig. 1(a), the structure consists of an inner cylinder of radius *r* overlaid with 6-fold circuitous grooves, and the period along circumferential surface is *d* = 2*πR*/*N* where *N* = 6 in this case.

In order to reach the subwavelength regime and ensure the minimum geometric element is experimentally accessible, a set of parameters of this corrugated cylinder are assigned (*R* = 10 mm, *r* = 1.2 mm, *w _{g}* =

*w*= 0.4 mm). It should be mentioned that this meta-cylinder is reminiscent of the ultra-sparse metasurface for low-frequency airborne sound based on Mie resonances which exhibits a negative bulk modulus supported by the monopolar Mie resonance [34]. As an equivalent function, the spoof plasmonic metastructure we present here exhibits a negative permittivity supported by the bound EM modes. However, they are essentially different in regard to field localization since the resonance modes supported by acoustic Mie resonances is inevitably radiative while the spoof plasmonic resonances can be localized on the metastructure.

_{m}Without considering the Ohmic losses of the metal, the fundamental resonance modes of the corrugated cylinder surrounded by air can be solved directly by the eigenmode analysis. Furthermore, the spoof plasmonic resonances can also be identified from the peaks of the scattering cross section (SCS) of the cylinder for a TM-polarized incident plane wave (*H⃗* pointing along *z* axis). In Fig. 2(a) (blue solid line), we show the normalized SCS spectrum as a function of frequency, which is numerically calculated by the commercial software Comsol Multiphysics. Clearly, the resonance frequencies are very close to the asymptotic spoof SPP frequency *f _{a}* =

*ω*/2

_{a}*π*≈

*c*/4

*h*= 1.2167 GHz of a corresponding 1D array of grooves with the same period

_{e}*d*and effective groove depth

*h*[4], where

_{e}*h*=61.5882 mm can be directly obtained from the geometric relation. Actually, the plasmonic resonances fall into subwavelength regime when the frequency approximates to the asymptotic frequency (orange dashed line) where the dispersion curve (black solid line) moves away from the light line (red solid line) as shown in the inset of Fig. 2(a). Through the eigenmode calculations, as shown in Fig. 2(b), we found that the resonance frequencies of the magnetic dipole (MD) mode and electric dipole (ED) mode are very close in this 2D system. Indeed, these two nearly-degenerate modes can not be completely separated by tuning the geometric parameters or filling the large index materials into the grooves, as a result of the intrinsically broad band width of the MD resonance and relatively narrow band width of the ED resonance. Therefore, the first peak of the SCS in Fig. 2(c) is a consequence of the superposition of magnetic and electric dipole resonances. Since the magnetic dipole resonance has a rather large spectral width (see details in Appendix A), it still contributes to the SCS around the frequencies of higher-order electric modes. Thus, the frequencies of the SCS peaks have a little shift in comparison with the eigen frequencies. The effect of nearly degenerate magnetic and electric dipoles is interesting which is hard to realize in LSP resonances of nanostructures by natural materials. However, this effect also causes the complex when one of these two overlapping modes is of interest. Nevertheless, such two quasi-degenerated dipole modes split away spontaneously in 3D systems as we will show in the following part.

_{e}The ultra-thin 3D structure with the same corrugation is located on the top of a substrate and possesses a thickness of *t _{m}* along

*z*direction as shown in Fig. 1(b). To achieve a low loss level in the substrate and a good flatness of the sample, the F4B material with 2 mm thickness is utilized as the substrate. Then, we show the SCS spectra in Fig. 3(a) for PEC disks with the same corrugation but various thicknesses, where the incident plane wave propagating along the

*x*axis is polarized with

*E*. As we can see, the resonance frequencies shift with the thickness of the disk, and the magnetic dipole resonance moves to a much higher frequency than the electric dipole, so that the first peak of SCS for each case in Fig. 3(a) is a net electric dipole response. This separated electric and magnetic dipole modes are further proved by the field distribution of

_{y}*E*at

_{z}*z*= 0.1 mm above the top surface of the disk, as shown in the inset of Fig. 3(b) and (d). In contrast with the 2D case, the spoof resonances of both ED and MD modes also decay evanescently away from the metal surface. This localized subwavelength electric dipole mode (in 3D space) is very similar to the optical LSPs on a nanocylinder of noble metals. Therefore, we can expect that the collective resonant behaviors of a spoof localized surface plasmonic polaritons (LSPP) metaparticle array must be consistent with the real LSPP resonances of a metal nanoparticle lattice with the same lattice structures. Accordingly, this kind of spoof SPP metaparticle array provides an ideal platform to mimic various fascinating phenomena that was believed only exist in real plasmonic lattice previously. And we will show an explicit system based on this metaparticle in the next section. More interestingly, we found that the electric mode sequence reverses in 3D systems, i.e. the higher-order electric modes possess lower frequencies, and the electric modes have lower frequencies than magnetic modes with the same order (see details in Appendix B). This frequency-ordering reverse is remarkable, which has been rarely reported in conventional plasmonic systems. Complex systems involved multipolar resonances could benefit from this unusual property in some cases, e.g. the mode hybridization of resonances with different orders.

## 3. Plasmonic zigzag chain

Plasmonic particle arrays have been widely investigated due to the near-field coupling property, and this virtue indicate they can be a good experimental carrier to study many theoretical models determined by near-field interactions. Indeed, there are many works to study these mode interactions and topological effects based on these systems both in 1D chains and 2D lattices [20,23,24,26,35]. However, such systems based on nanostructures are difficult to realize and measure in current nano technology and the material loss in nanostructures further hinders the development of LSP based systems. Here, we propose a kind of spoof surface plasmonic lattices, which are composed of periodic metaparticles previously studied. As a proof-of-concept study, we use the spoof surface plasmonic zigzag chain (as shown in Fig. 4) as an example to theoretically and experimentally show this kind of meta-lattice is almost lossless, mature in fabrication and easy to measure in microwave ranges. Finally, we use this zigzag chain as an example to show that this kind of spoof LSPP metaparticle arrays are a good platform to experimentally study topological physics for those near-field interaction dominated systems.

This semi-infinite zigzag chain with bond angle *θ* has a period *p* = 2*d _{s}* cos

*α*along

*x*direction, where

*α*= (

*π*−

*θ*) /2 is the angle between the bond connecting two adjacent particles and

*x*-axis, so that the displacement vector of particle

*n*and

_{A}*n*can be expressed as:

_{B}*R*/

*d*≪ 1/3, the dipole interactions dominate the coupling in this chain. Since the zigzag chain is a compound lattice with two inequivalent sublattices, as shown in Fig. 4, we can combine the dipole moments of two sublattices within one unit cell (${\mathit{P}}_{n}^{A}$ and ${\mathit{P}}_{n}^{B}$) into a single polarization vector ${\mathit{P}}_{n}={\left({\mathit{P}}_{n}^{A},{\mathit{P}}_{n}^{B}\right)}^{T}$, which are located in the

_{s}*xy*plane since the ultrathin metallic disk can only support the in-plane polarized modes. Under these conditions, the eigenmodes of this system can be solved directly by a self-consistent coupled-dipole equation [19],

*A*/

*B*in the

*n*

^{th}unit cell generated by all of the dipoles on other lattice points,

*α⃡*is the in-plane polarizability tensor with nonzero diagonal components with

*α*=

_{xx}*α*, and

_{yy}**(**

*G***) denoting the Green tensor of a quasi-static dipole source:**

*r**α*(

*ω*)

^{−1}, On the basis of Bloch states

*ψ*

_{k}, the eigen equation of the Hamiltonian becomes

*H*

_{k}

*ψ*

_{k}=

*α*(

*ω*)

^{−1}

*ψ*

_{k}with the Hamiltonian in Bloch representation ${H}_{\mathit{k}}=\sum _{m}{H}_{m,n}{e}^{-i(m-n)kp}$. Since dipole field decays rapidly, only the interactions between close particles are important, if we only consider the nearest neighbor coupling [33], the bulk 4 × 4 Hamiltonian reads where $\mathit{Q}={h}_{0}+\mathit{h}\cdot \mathit{\sigma}=\sum _{i=1}^{2}\overleftrightarrow{\mathit{G}}({\mathit{r}}_{i}){e}^{-i\mathit{k}\cdot ({\mathit{r}}_{i}-{\mathit{r}}_{2})}$ with

*r**(*

_{i}*i*= 1, 2) denote two nearest interactions respectively and

**are the Pauli matrices. On the other hand, the coupling can be decomposed into copolarized (**

*σ**t*

_{‖}) and cross-polarized (

*t*

_{⊥}) mode interactions with respect to the link vector Δ

**. Therefore,**

*r**h*

_{0}=

*t̄*(1 +

*e*

^{−ikP}),

*h*=

_{x}*t̃e*

^{−ikP}sin 2

*θ*,

*h*= 0 and

_{y}*h*=

_{z}*t̃*(1 +

*e*

^{−ikP}cos 2

*θ*) where

*t̄*= (

*t*

_{‖}+

*t*

_{⊥})/2,

*t̃*= (

*t*

_{‖}−

*t*

_{⊥})/2. It is easy to know from the bulk Hamiltonian that the band gap opens when |

*t̃*cos

*θ*| > |

*t̄*|, i.e. 70.5° <

*θ*< 109.5°.

## 4. Topological edge states

The Hamiltonian in Eq. (6) possesses chiral symmetry: *CH _{k}C* = −

*H*, where

_{k}*C*= diag (

**, −**

*I⃡***). This indicates that topologically protected zero-eigenvalue edge states may exist on a finite zigzag chain with open boundaries [36]. According to our previous results [37], the number of pairs of zero-eigenvalue edge states (NPE) is characterized by the generalized winding number**

*I⃡**W*(det

*Q*

^{†}) and the summation

*Z*of the Zak phases of the bands below the mid band gap divided by

_{s}*π*,

*c*can be any path including the coordinate origin and

*j*= 1, 2 represents the band number below the gap. As the spoof LSPs has similar properties with LSPs real plasmonics, so the Drude model with plasmon frequency

*f*that possesses same dipole resonance frequency

_{p}*f*as simulations is adopted for simplicity. Figure 5(a) shows the numerically calculated band structure with tight-banding method of a finite zigzag chain with

_{d}*d*= 3

_{s}*R*, and it is clear that the gap opens in the range of 70.5° <

*θ*< 109.5°. The flat edge states with calculated frequency

*f*= 1.8375

_{e}*GHz*exist in the gap with a bond angle that possesses nontrivial topological number as the theory predicted. It has to be stressed that the tight-binding model is not suitable when the bond angle

*θ*is too small which we are not interested here. Actually, the flatness of edge states dispersion curves are bent when long range couplings are included [37]. The edge states for

*θ*= 90° is also confirmed by eigen mode simulations, and the edge modes at two ends have different polarizations (see details in Appendix C). The polarization directions of these two edge modes are perpendicular when

*θ*= 90°, so it can be selectively excited by linearly polarized plane wave with different polarizations. On the contrary, the dipole moment orthogonality of the topological edge states at two ends is not satisfied when

*θ*≠ 90°.

## 5. Experimental verification at microwaves

In order to observe the topological edge states on the proposed spoof LSP zigzag chain with open boundaries, we performed a proof-of-principle microwave experiment. The sample of 5 “atoms” with bond angle *θ* = 90° was fabricated by the standard PCB printing technology, and the geometric parameters are the same as previous calculations with a 2 mm thick substrate and 70 *μ*m thick copper patterns. Here, we select the F4B material as the substrate in virtue of its small loss tangent *∊*_{F4B} = 2.65(1 + 0.001*i*), and the metal patterns with *d _{s}* = 3

*R*(

*R*= 10 mm) are printed with copper which has a good conductivity at low frequencies. As shown by the experiment setup in Fig. 6(a), the sample is illuminated by a plane wave linearly polarized in

*xy*plane entering from −

*z*direction. The polarization is characterized by the angle

*β*between

**field and**

*E**x*-axis. And the field distributions of |

*E*| are measured at

_{z}*z*= 0.5 mm above the top surface. The plane wave is generated by a standard microwave horn placed 1 m away below the sample. In Fig. 6, we show the field distributions of |

*E*| obtained by both numerical simulations and experiments for 3 different polarizations of the incident wave:

_{z}*β*= 0°, −45°, 45°. The experimental images are reconstructed by a simple point-to-point consecutive scanning process (with step length 0.1 mm in

*x*and

*y*directions) without any deconvolution or post-processing. Obviously, the subwavelength topological edge states at two ends can be selectively excited through controlling the polarization angle

*β*, which is determined by the field orthogonality of topological edge modes in this zigzag chain. In general, the experiment results are in good agreement with numerical simulations with only a small frequency shift. This frequency shift is believed to be a result of parameter differences between simulations and experiments, which does not affect the phenomenon of polarization-dependent topological edge states. The noise-like signals along the horizontal direction in experiment images result from mechanical drifts during the scanning process since the scanning probe has been moving in this direction. It must be pointed out that the polarization-dependent topological edge states are not limited to number of particles

*N*= 5. In fact, this effect with

*N*> 5 when the whole size is still in a subwavelength range has no essential difference since this effect is determined by neighbor couplings. Here,

*N*= 5 is utilized in consideration of the calculation memory requirement and field scanning reasons. Indeed, a sample with

*θ*inside the topological nontrivial range 70.5° <

*θ*< 109.5° should possess mid-gap edge states at the two ends, but they cannot be selectively excited merely by controlling the polarization of the incident wave, because the dipole moments of these two edge states are not perpendicular anymore. Another feature is the experimental field distribution does not look like a typical spoof LSP dipole mode. There are four main reasons contributing to this result. Firstly, the spoof resonance mode’s intensity profile does not take exactly after a typical dipole unless the number of circumferential periods of grooves is sufficient large. But the number of periods N is set to be 6 in our sample on account of the feasibility in the experiment. Secondly, the fine structure of the resonant unit is too small for a conventional dipole probe at microwave ranges, so that the measured field at a point by the probe is indeed the average electric field of a probe-size area. Thirdly, there exist interactions between the probe and metaparticle when the probe is placed close to the sample under plane wave excitations, which further blurs this dipole profile. Finally, the radiation effect and long range interactions makes the difference between the measured result and an ideal dipole profile, this influence can also be found in simulation results, as shown in Fig. 6.

## 6. Conclusion

In summary, we have demonstrated that an ultra-thin metallic disk with circuitous grooves can support multiple subwavelength electric and magnetic spoof LSP resonances without a metallic ground plane. The spoof LSP multiple resonances has a reverse frequency ordering in comparison with conventional LSP resonances in noble metal nanostructures. Moreover, we show this dipole mode dominated spoof LSP metaparticle arrays resemble the real LSPP coupling in nanoparticle arrays and hence such system can be an ideal experimental platform to mimic various LSPP related effects, including fascinating topological transport, which usually thought only exist in metallic particle arrays in nanoscale. In particular, we studied the subwavelength topological edge states in a zigzag chain composed of this spoof metaparticles, and we showed that these topologically protected edge states can be selectively excited by controlling the polarization of incident plane waves. The microwave experiment results show great agreements with the theoretical predictions and simulation results. Such arrays of ultra-thin subwavelength spoof LSP metaparticles are favorable for a wide variety of potential applications such as index-sensing, signal position controlment and polarization detection.

## Appendix A: The design of the textured cylinders

In 2D structures, the EM response can be solved by the analytical model through the modal expansion method. In the form of cylindrical coordinate, the mode solution can be obtained through matching the boundary conditions at the outer boundary, *ρ* = *R* of the corrugated PEC cylinders surrounding by air with effective groove depth *h _{e}* [9]. The eigen frequency of azimuthal quantum number

*n*is determined by the transcendental equation:

*n*= 0, 1, 2, 3 on the complex plane of

*ω*. The zero point in each panel gives the eigen-frequency for the corresponding multipolar eigenmode, namely

*n*= 0 for magnetic dipole,

*n*= 1 for electric dipole,

*n*= 2 for electric quadrupole,

*n*= 3 for electric hexapole. To be more specific, the real parts of the four eigen frequencies are 1.2062, 1.2088, 1.2131, 1.2144 GHz for

*n*= 1, 2, 3, 4 respectively. We see that the eigen frequencies of the ED and MD are very close. The imaginary parts of eigen-frequencies determine the quality factor of the modes. As a result, the progressively decreasing imaginary parts of the 4 eigen-frequencies indicate the quality factor increases with

*n*, which is consistent with the numerical results shown in Fig. 2. However, these mode frequencies have a small red shift in comparison with the full-wave numerical results. This could be ascribed to the internal reflections at the bends in the grooves. And although the small number of grooves

*N*= 6 lead a little deviation from the assumption

*d*≪

*λ*. The eigen frequencies of the EM modes still can offer a reasonable initial value to search for the exact eigenmodes with numerical simulations.

On the other hand, the SCS of each mode can also be analytically calculated in some
cases by using the radial groove approximations [9]. However, the effective
groove depth here is larger than the outer radius of the PEC cylinder,
so it can not be directly mapped to a radial groove. By checking Eq. (10), we found that the
effective depth of the groove plays an equivalent role as the index of
the filled material inside the groove. Therefore, we conclude that a
disk with radial grooves in the range of (*r*,
*R*) filled with high-index dielectric material share
the same physics. As a disk with circuitous grooves, evidenced by
their extremely similar profile of the SCS peak governed by the
magnetic and electric dipole resonances as shown in Fig. 8(a), where the equivalent disk with radial
grooves has the same *r* and *R*, but
*w _{g}* = 6

*mm*,

*n*= 9.63. In addition, the SCS

_{g}*σ*of a PEC cylinder with radial grooves can be directly calculated through metamaterial approximation [8]:

*n*

*f*=

*J*

_{1}(

*n*

_{g}k_{0}

*R*)

*Y*

_{1}(

*n*

_{g}k_{0}

*r*)−

*J*

_{1}(

*n*

_{g}k_{0}

*r*)

*Y*

_{1}(

*n*

_{g}k_{0}

*R*),

*g*=

*J*

_{0}(

*n*

_{g}k_{0}

*R*)

*Y*

_{1}(

*n*

_{g}k_{0}

*r*)−

*J*

_{1}(

*n*

_{g}k_{0}

*r*)

*Y*

_{0}(

*n*

_{g}k_{0}

*R*), the ${H}_{n}^{(1)}$,

*J*,

_{n}*Y*are the Hankel function of the first kind, Bessel functions of the first and second kinds, for an integer order

_{n}*n*respectively. Fig. 8(b) shows the calculated SCS of

*n*-order, the

*n*= 0, 1, 2, 3 corresponding to magnetic dipole, electric dipole, electric quadrupole, electric hexapole modes respectively. As we can see, the spectral width decreases with the increasing order number, meanwhile, the magnetic and electric dipole resonances are highly overlapped.

## Appendix B: Frequency ordering of the spoof LSPs modes

Generally, the magnetic resonances have lower frequencies than the electric resonances in plasmonic systems, and the higher-order modes have higher frequencies. However, it is remarkable that the frequency sequence of the spoof LSP modes reverses exactly in the present 3D structures. To eliminate the influence of the substrate, we calculated the eigenmodes of the metaparticle without a substrate, as shown in Fig. 9. The reverse of the frequency ordering of ED and MD have been explained by the variation principle according to the numerically calculated field distributions [9], but the fundamental reason of such specific field distribution is still unknown. And as far as we know, and the inverted frequency sequence of high-order electric resonances in such spoof LSP systems has not been reported before. In systems involve hybridization between modes of different orders, an inverted frequency sequence of modes may give rise to new phenomena.

## Appendix C: Simulated eigen modes of the topological edge states

For a finite zigzag chain with five atoms and *θ* = 90°, we calculated the eigen field distributions of the edge states. As shown in Fig. 10, the polarizations of the dipole moment on the end atoms are obviously perpendicular with each other. Clearly, for the two eigen edge states at different ends, the polarizations for both edge states are also perpendicular to the lattice vectors connecting two adjacent metaparticles respectively. This can be understood by the radiation superimpositions, the radiation from others metaparticles cancel with each other so that the left end metaparticle has a dipole moment perpendicular to the adjacent bond vector.

## Funding

The Research Grants Council of the Hong Kong (Project No. AoE/P-02/12); Fundamental Research Funds for the Central Universities (2018CDXYGD0017); “Shenzhen Peacock Plan” (No. KQTD2016022614361432); Major Program of Natural Science Research of Jiangsu Higher Education Institutions (No. 18KJA140003); The Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions; Fundamental Research Funds for the Central Universities (2018CDJDWL0011); Key Technology Innovation Project in Key Industry of Chongqing (cstc2017zdcy-zdyf0338).

## Acknowledgments

The authors would like to thank Prof. C. T. Chan for useful discussions.

## References

**1. **W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

**2. **N. Meinzer, W. L. Barnes, and I. R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nat. Photonics **8**, 889–898 (2014). [CrossRef]

**3. **J. Pendry, L. Martin-Moreno, and F. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847–848 (2004). [CrossRef] [PubMed]

**4. **F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A: Pure Appl. Opt. **7**, S97–S101 (2005). [CrossRef]

**5. **A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science **308**, 670–672 (2005). [CrossRef] [PubMed]

**6. **N. Yu, Q. J. Wang, M. A. Kats, J. A. Fan, S. P. Khanna, L. Li, A. G. Davies, E. H. Linfield, and F. Capasso, “Designer spoof surface plasmon structures collimate terahertz laser beams,” Nat. Mater. **9**, 730–735 (2010). [CrossRef] [PubMed]

**7. **S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. Garcia-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. **97**, 176805 (2006). [CrossRef] [PubMed]

**8. **A. Pors, E. Moreno, L. Martin-Moreno, J. B. Pendry, and F. J. Garcia-Vidal, “Localized spoof plasmons arise while texturing closed surfaces,” Phys. Rev. Lett. **108**, 223905 (2012). [CrossRef] [PubMed]

**9. **P. A. Huidobro, X. Shen, J. Cuerda, E. Moreno, L. Martin-Moreno, F. Garcia-Vidal, T. J. Cui, and J. Pendry, “Magnetic localized surface plasmons,” Phys. Rev. X **4**, 021003 (2014).

**10. **X. Shen and T. J. Cui, “Ultrathin plasmonic metamaterial for spoof localized surface plasmons,” Laser & Photonics Rev. **8**, 137–145 (2014). [CrossRef]

**11. **Z. Liao, B. C. Pan, X. Shen, and T. J. Cui, “Multiple fano resonances in spoof localized surface plasmons,” Opt. Express **22**, 15710–15717 (2014). [CrossRef] [PubMed]

**12. **Z. Gao, F. Gao, Y. Zhang, and B. Zhang, “Complementary structure for designer localized surface plasmons,” Appl. Phys. Lett. **107**, 191103 (2015). [CrossRef]

**13. **Z. Liao, X. Shen, B. C. Pan, J. Zhao, Y. Luo, and T. J. Cui, “Combined system for efficient excitation and capture of lsp resonances and flexible control of spp transmissions,” ACS Photonics **2**, 738–743 (2015). [CrossRef]

**14. **Z. Gao, F. Gao, and B. Zhang, “High-order spoof localized surface plasmons supported on a complementary metallic spiral structure,” Sci. Reports **6**, 24447 (2016). [CrossRef]

**15. **B. Xu, Z. Li, C. Gu, P. Ning, L. Liu, Z. Niu, and Y. Zhao, “Multiband localized spoof plasmons in closed textured cavities,” Appl. optics **53**, 6950–6953 (2014). [CrossRef]

**16. **Z. Li, L. Liu, C. Gu, P. Ning, B. Xu, Z. Niu, and Y. Zhao, “Multi-band localized spoof plasmons with texturing closed surfaces,” Appl. Phys. Lett. **104**, 101603 (2014). [CrossRef]

**17. **Z. Gao, F. Gao, Y. Zhang, X. Shi, Z. Yang, and B. Zhang, “Experimental demonstration of high-order magnetic localized spoof surface plasmons,” Appl. Phys. Lett. **107**, 041118 (2015). [CrossRef]

**18. **F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, and B. Zhang, “Dispersion-tunable designer-plasmonic resonator with enhanced high-order resonances,” Opt. Express **23**, 6896–6902 (2015). [CrossRef] [PubMed]

**19. **W. Weber and G. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004). [CrossRef]

**20. **S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B **69**, 125418 (2004). [CrossRef]

**21. **F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. **104**, 106802 (2010). [CrossRef] [PubMed]

**22. **J. Jiang, Z. Guo, Y. Ding, Y. Sun, Y. Li, H. Jiang, and H. Chen, “Experimental demonstration of the robust edge states in a split-ring-resonator chain,” Opt. Express **26**, 12891–12902 (2018). [CrossRef] [PubMed]

**23. **C. Ling, M. Xiao, C. Chan, S. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express **23**, 2021–2031 (2015). [CrossRef] [PubMed]

**24. **D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. **102**, 123904 (2009). [CrossRef] [PubMed]

**25. **A. Poddubny, A. Miroshnichenko, A. Slobozhanyuk, and Y. Kivshar, “Topological majorana states in zigzag chains of plasmonic nanoparticles,” ACS Photonics **1**, 101–105 (2014). [CrossRef]

**26. **I. S. Sinev, I. S. Mukhin, A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, A. K. Samusev, and Y. S. Kivshar, “Mapping plasmonic topological states at the nanoscale,” Nanoscale **7**, 11904–11908 (2015). [CrossRef] [PubMed]

**27. **V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. **111**, 3888–3912 (2011). [CrossRef] [PubMed]

**28. **N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. **10**, 2342–2348 (2010). [CrossRef] [PubMed]

**29. **K. C. Huang, M.-K. Seo, T. Sarmiento, Y. Huo, J. S. Harris, and M. L. Brongersma, “Electrically driven subwavelength optical nanocircuits,” Nat. Photonics **8**, 244–249 (2014). [CrossRef]

**30. **A. Tao, P. Sinsermsuksakul, and P. Yang, “Tunable plasmonic lattices of silver nanocrystals,” Nat. Nanotechnol. **2**, 435–440 (2007). [CrossRef]

**31. **S. A. Maier, *Plasmonics: Fundamentals and Applications* (Springer Science & Business Media, 2007).

**32. **S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. **11**, 23–36 (2016). [CrossRef] [PubMed]

**33. **A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge states in optically resonant dielectric structures,” Phys. Rev. Lett. **114**, 123901 (2015). [CrossRef] [PubMed]

**34. **Y. Cheng, C. Zhou, B. Yuan, D. Wu, Q. Wei, and X. Liu, “Ultra-sparse metasurface for high reflection of low-frequency sound based on artificial mie resonances,” Nat. Mater. **14**, 1013–1019 (2015). [CrossRef] [PubMed]

**35. **Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. **105**, 233904 (2010). [CrossRef]

**36. **S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. **89**, 077002 (2002). [CrossRef] [PubMed]

**37. **L. Wang, R.-Y. Zhang, M. Xiao, D. Han, C. T. Chan, and W. Wen, “The existence of topological edge states in honeycomb plasmonic lattices,” New J. Phys. **18**, 103029 (2016). [CrossRef]