Abstract

The Zak phase and topological plasmonic Tamm states in plasmonic crystals based on periodic metal-insulator-metal waveguides are systematically investigated. We reveal that robust topological interfacial states against structural defects exist when the Zak phase between two adjoining plasmonic lattices are different in a common band gap. A kind of efficient admittance-based transfer matrix method is proposed to calculate and optimize the configuration with inverse symmetry. The topologically protected states are favorable for the spatial confinement and enhancement of electromagnetic fields, which open a new avenue for topological photonic applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

An important aspect in the development of nano-optics is analogous studies of photons with the electrons in solid-state materials. Topological insulator, a compelling and peculiar material that possesses non-trivial topological bounded states protected by time-reversal symmetry, has drawn considerable attentions due to its elusive properties of one-way spin-polarized transport [1]. Over the past decades, a number of topological states have been successfully emulated in photonic systems, such as magneto-optical photonic crystals [2], coupled resonator optical waveguides [3,4], photonic Floquet topological insulators [5], topological crystalline insulators [6,7], topological insulator lasers [8–11]. Moreover, topological states in plasmonic systems also attract growing interests in the last few years. The topological plasmonic interfacial modes, as a successful extension of the famous Su-Schrieffer-Heeger (SSH) model for polyacetylene-type solitons to plasmonic nanoparticles chains [12–15] and plasmonics waveguide arrays [16–18], exist when the nearby structures possess different Zak phases. Nowadays, more complex plasmonic structures composed of metallic patterns [19,20] and graphene superlattices [21,22] are demonstrated to construct topologically protected one-way edge states. The nontrival topologically protected states have opened up a fascinating avenue for topological photonic applications.

On the other hand, plasmonic crystals (PCs), which can be recognized as photonic crystals composed of plasmonic materials, can also induce an energy band structure analogous to superlattices and photonic crystals. Meanwhile, the electromagnetic surface states such as Tamm states would exist due to the strong Bragg scattering [23]. Tamm states, one of the surface Bloch states localized at the interfaces of truncated crystalline materials, have been sufficiently studied in superlattices [24], photonic crystals [25] and photonic lattices [26]. Unlike conventional Tamm states, the Plasmon Tamm states (PTSs) formed by MIM Bragg reflector (BR) and metal exist in subwavelength unit cells [27,28]. Attributing to strong confinement of surface plasmon polaritons(SPPs), the electromagnetic fields can be enhanced by three orders of magnitude. However, the high index core is required to contact with the metal end according to the general phase-matching condition in these configurations, as a result, the Zak phase can assume any value as a function of the band index due to the lack of spatial inverse symmetry [29]. Fortunately, some research groups have found that the topologically protected interfacial state can exist in 1D photonic crystals [30–32]. Nevertheless, large scale and relatively high operation light intensity limit the applications of traditional 1D photonic crystals in nanoscale all-optical circuits, such as fluorescence enhancement and bistable switches. Inspired by these works in 1D photonic crystals, a topologic plasmonic Tamm states (TPTSs) can be designed in the plasmonic materials with spatial inverse symmetry to overcome these drawbacks of photonic crystals.

In this paper, analogous to the SSH model in polyacetylene, the TPTSs are proposed in connected plasmonic crystals (PCs). The topological transition points are found in PCs and regarded as parent states to construct a topologically protected interfacial state via geometrical parameters tuning. Regarding to the symmetric PCs, an efficient admittance-based transfer matrix method (TMM) is proposed to calculate and optimize the structure, we find that the symmetric PCs can be regarded as homogenous layer with effective phase delay and equivalent characterised admittance. Through calculating the reflective spectra by the proposed method, the TPTSs are found to be robust against the inserted defect of the PCs. Our proposed TPTSs provide an efficient way to trap and enhance the plasmon wave, which makes it a promising optical device for integrated photonic circuits and able to provide significant applications, such as all-optical bistable switches [33], harmonic wave generation [34], and fluorescence enhancement [35].

2. Results and discussion

Our proposed scheme composed of two connected PCs, i.e., PC1 and PC2, is indicated in Fig. 1(a). The PCs consist of A/B/A layered metal-insulator-metal (MIM) waveguides with different unit cells as marked by the dashed frame in the inset of Fig. 1(a). The dielectric core has a width of w = 40 nm. Here, the cores of layer A and B are chosen as air and SiO2, respectively. And the thicknesses of layer A and B are defined as 2dA1, dB1 in PC1 and 2dA2, dB2 in PC2, respectively. And the periods of PC1 and PC2 are Λ1 = 2d A1 + d B1 and Λ2 = 2dA2 + dB2. For the sake of simplicity, the metal is silver characterized by a Drude dielectric function ϵ(ω)=ϵωp2/(ω2+iγω) for optical communication frequencies, with ϵ = 3.7, ωp = 9.1 eV, and γ = 0.018 eV, and the refractive index of SiO2 is set as 1.5. The plasmon wave propagates along z-direction in MIM waveguide.

 figure: Fig. 1

Fig. 1 (a) Sketch of two adjoining one dimensional plasmonic crystals (PCs) composed of symmetric ABA layers. The insets show the side views of PC1 and PC2, of which the unit cells Λ1 = 2dA1 + dB1 and Λ2 = 2dA2 + dB2 are highlighted by blue and red dashed frames, respectively. Here the width w = 40 nm, and the refractive index nA = 1 and nB = 1.5, respectively. (b) The band structure of the PCs as a function of dA when δA + δB = 3π is fixed, where δA and δB are the phase delay in slab A and slab B for a unit cell, respectively. The blank zones indicate the stopband. TPI and TPII are two topological phase points, and the integer pairs in parentheses are δA and δB in the unit of π, respectively. Moreover, the dashed lines show the parameters adopt to construct edge states. (c, d) The dispersion relation (c) and mode distributions (d) of the edge modes, where the magnetic field distributions H are calculated at kx = 0.

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Let us start by considering the propagating behavior of SPPs in a homogeneous MIM waveguide. According to the transmission line theory, the characteristic admittance of MIM waveguides can be expressed as:

η=I/V=cHxdx/+Eydy=ωϵdϵ0dxβw,
where w is the width in y-direction, ϵ0ϵd is the permittivity of the dielectric core, dx denotes the unit length in x-direction, and β denotes the propagation constants of the MIM structure, which can be solved analytically through the dispersion relations of the odd TM mode (Ez is odd) [36]: − tanh(kdw/2) = km ϵd/kd ϵm, where ϵm,d are the dielectric constants, andkm,d=β2ϵm,dk02 denote the transverse propagation constants of the metal and the dielectric, respectively.

With the characteristic admittance, the band structure of a binary PC shown in Fig. 1(a) can be easily found from the effective phase delay in a unit cell (see methods)

cosδ=cosδAcosδB12(ηA/ηB+ηB/ηA)sinδAsinδB,
where δA = 2βAdA, δB = βBdB and δ = KΛ are the actual phase delay in slab A, slab B and a unit cell, while ηA and ηB are the characteristic admittance of slab A and B. K and Λ = 2dA + dB are the Bloch wavevector and period of the PC, respectively. In order to construct a topological nontrivial interface state, one needs to make sure the band topologies of the two PCs are different inside a common band gap. A commonly used way to achieve this goal is tuning the system parameters across a topological transition point. Therefore, the topological phase transition states are usually used as parent states to construct topologically protected edge states. The critical parameters of topological phase transition point in PCs could be adjusted to the crossing of band structure according to the condition:
α=δAδB=m1m2.m1,m2+

It is worth noting that the optical loss should be omitted when calculating the band structure in order to realize isolated energy bands. For the telecom optical wavelength at 1550 nm (≈0.8 eV), we have βA = 5.891 µm−1 and βB = 8.868 µm−1, respectively. According to the band crossing theory [30], the band mm1 + m2 and band m − 1 will cross at the frequency ω˜ satisfying δA(ω˜)+δB(ω˜)=mπ. When the gap is open, the frequency derived by δA(ω˜)+δB(ω˜)=mπ labels the midgap positions of the mth gap. At the lowest crossing frequency, the dependence of the band structure on the thickness of slab A is shown in Fig. 1(b) for fixed δA + δB = 3π. One can find that there are four crossing points at the operation frequency. At the crossing points, (δA, δB) equals (0, 3π), (π, 2π), (2π, π), and (3π, 0), respectively. Among them, the first and last crossing points are meaningless because the length of the slab A or B is zero. Noting that an inverted band gap appears when the parameters went through either of the remaining crossing points [1,37], therefore the two remaining crossing points are assumed to be topological transition points (labeled by TPI and TPII), which can be regarded as the 1D counterpart of Dirac points. The critical parameters are dAI=π/2βA=266.6nm, dBI=2π/βB=708.5nm at TPI and dAII=π/βA=533.3nm, dBII=π/βB=354.3nm at TPII, respectively. As a result, sin δA = sin δB = 0 and cos δ = cos δA cos δB = −1 are fulfilled at the crossing points. Accordingly, the band 2 and band 3 will cross at the boundary of the Brillouin zone. It is expected that the PC constructed from the parameters sandwiched by the two topological phase points is topologically distinct from the remanning parameter space.

Based on the aforementioned theory, the parameters of PC1 are chosen as dB1 = 2dA1, i.e., dB1 = 3π/(βA + βB) = 638.6 nm and dA1=319.3 nm without loss of generality as marked in Fig. 1(b). The band gap of PC1 is 0.038 eV. Considering the band edge depends linearly on the thicknesses near the topological transition points, one can easily obtain the parameters dA2=210 nm and dB2=738.8 nm for PC2 to possess the same band gap width. Apparently, the band gap is shared by PC1 and PC2, which serves as a good platform to support edge states. When PC1 and PC2 are connected, for example, on the left side of a boundary is the semi-infinite PC1 while on the right side is the semi-infinite PC2. This photonic boundary is translation invariant along the x-direction and its corresponding band structure is given in Fig. 1(c). An edge state expanding from kx = 0 to π/a appears in the common gap. Figure 1(d) plots the amplitude of magnetic fields of one representative edge state with normal incident wave (kx = 0). The magnetic field localizes around the boundary and decays exponentially away from the boundary.

The Zak phase [29], as an analogy of the Berry phase [38], has been used to classify the band topology for studying their edge states. For each isolated band n, the Zak phase can be defined as the formula:

θnZak=π/Λπ/Λ[iunitcelldzun,q*(z)qun,q(z)]dq,
where iunitcelldzun,q*(z)qun,q(z) is the Berry connection, and un,q(z + Λ) = un,q(z) is the periodic Bloch magnetic field eigenfunction of a state on the nth band with wave vector q, i.e., Hy;n,q = un,q(z) exp (iqz). The function un,q(z) can be obtained analytically from the admittance based transfer-matrix method (see methods). Due to the inversion symmetry of the structure, the Zak phase is quantized at either 0 or π when the origin coincides with either of the inversion centers. In the paper, the boundary of the unit cell is chosen as the origin for calculating Zak phases.

It is generally known that the reflectivity satisfies rleftrright = 1 for the presence of an interface state between two adjacent semi-infinite structures, which indicates that the sign of reflection phases for the left and right parts must be opposite. If there is no band crossing, the Zak phases are related to the signs of the reflection phases φ(n), i.e.,

sgn[φ(n)]=(1)nexp(im=0n1θmZak),n+
where exp(iθ0Zak)=sgn[1ηA2/ηB2] denotes the Zak phase of the lowest band. This surface bulk correspondence means whether there is an interface state depends on the summations of Zak phases below the gap. More specifically, if two PCs with different summations of Zak phases below the nth gap are connected, it is expected that there is an edge state localized near the interface in the nth gap. In Figs. 2(a) and 2(b), we show the band structures of two PCs from the 1st to the 4th gaps with normal incident plasmon wave, i.e., kx = 0. For convenience, the tuning process dA1dAIdA2 is considered. The Zak phases of bands 0 and 1 remain unchanged while the Zak phases of bands 2 and 3 switch, which comes from a corresponding sign change for the reflection phases in the 3th gap after the perturbation. Thus the PCs undergo a topological phase transition in the parameter tuning process. To determine the location of the interface state, we recall the general condition rleftrright = 1, which is governed by φPC1 = −φPC2 in our configuration. In the symmetric configuration, the reflectivity of N layer ABA structure can be reduced as (see methods)
rPC=|r|exp(iφPC)=η0/ηη/η02icot(Nδ)+η0/η+η/η0,
where η0 = ηN+1 is the admittance of input and output MIM waveguide, is the total phase delay in the structure. And η is the equivalent admittance of unit cell, is derived from M12 or M21:
η=sinδ/{1/ηBsinδB(cos2δA2ηB2/ηA2sin2δA2)+1/ηAsinδAcosδB}.

The calculated reflection phases are shown in Fig. 2(c). One can find that there is only one interface state in the third gap when the photon energy is 0.8 eV. It is worth mentioning that a symmetric unit cell can always be regarded as a homogenous layer with effective phase δ and the equivalent admittance η, which lays the foundation for the further calculations of composite structures.

 figure: Fig. 2

Fig. 2 (a, b) The band structure and Zak phase of plasmonic crystals with parameters (a) d A1 = 319.3 nm, and dB2 = 2dA1 = 638.6 nm for PC1, and (b) dA2 = 210 nm, and dB2 = 738.8 nm for PC2, respectively. The Zak phase of each individual band is labeled in green, and the numbers of the bands and gaps are listed with red and black labels. The magenta strip represents the gap with reflection phase φPC > 0, while the cyan strip represents the gap with φPC < 0. Moreover, the symmetries of the eight band edge states, K, L, M, N, O, P, Q, R are labeled with red (antisymmetric) and indigo (symmetric) circles, respectively. (c) The black and blue broken curves represent the reflection phase φPC1 and the negative value of the reflection phase −φPC2 of the plasmonic crystals consisting of 50 periods calculated by Eq. (6). The intersection point denotes the location of the interface states. (d) The Bloch magnetic field eigenfunctions of the band-edge states at z = 0. The zero amplitude at the origin represents the Bloch magnetic fields are antisymmetric, while the non-zero values indicate the the Bloch magnetic fields are symmetric in the plasmonic crystals.

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The calculation of Zak phase by Eq. (4) needs to integrate the Berry connection over the whole Brillouin zone, which is very tedious and cumbersome. Fortunately, the Zak phase in an isolated band can be also determined by the symmetries of the eigenstates at high symmetry points in Brillouin zone, i.e., K = 0 and K = ±π/Λ. Considering that the amplitude of the Bloch magnetic fields for the band-edge states at the origin (z = 0) is either zero or maximum, one can divide the edge states into two categories: antisymmetric states with zero amplitude (marked by red circles) or symmetric states with maximum amplitude (marked by indigo circles). The Bloch magnetic fields of the eight band edge states sandwiched by band 2 and band 3 are calculated by the admittance based transfer-matrix method (see methods). Figure 2(d) shows the functions |un,q(z = 0)| for the four edge states in PC1 at points K, L, M, and N labeled in Fig. 2(a) and in PC2 at points O, P, Q, and R labeled in Fig. 2(b). Due to the mode orthogonality, one can find that the symmetry of the two edge states sandwiching a gap is always different. According to the admirable result from Kohn [39] and Zak [29] for 1D systems, the Zak phase is zero if the two edge states have the same symmetry in the nth band, otherwise, the Zak phase is π. From the eigenfunctions, one can find the edge states K, L, N, O, Q, and R are antisymmetric states, yet the remaining states P and M are symmetric states. Thus the values of the Zak phases are 0(π) for band 2(3) in PC1, while they switch in PC2, indicating the appearance of a topological phase transition.

In experiments, the TPTSs shown in Fig. 1(a) can be excited by impinging surface plasmon polaritons(SPPs) from either side of the MIM waveguides. When the incident SPPs are injected from the left side, the energy is delivered across the PC1 and confined at the interface between PC1 and PC2 simultaneously. Due to the different roles PC1 and PC2 play in exciting the topological edge states, the numbers of periods N1 and N2 are usually unequal. In the process, PC2 acts purely as a reflector, one can choose a large number, e.g., N2 = 8 to maximally reflect the incoming SPPs. However, PC1 works as an energy delivery as well as a reflector. To achieve the balance between energy confinement and Ohmic loss, N1 can not be too small or too large. The optimized number of periods can be derived from the minimum of reflection spectra. When TPTSs are excited, electromagnetic fields are trapped and significantly Ohm damped. Therefore, a minimum in reflection spectrum reveals the excitation of TPTSs. The reflection of the system PC1/PC2 can be expressed as functions of the phase delay(δ1, δ2), characteristics admittance(η1, η2) and layer numbers(N1, N2) find that the optimized layer number of PC1 is N1 = 4. In Figs. 3(a)-3(c), the reflection spectra of 4 unit cells of PC1, 8 unit cells of PC2, and their composites are plotted, where band gaps 2 and 3 are included. The spectra are calculated by TMM and confirmed by finite element method (marked by crosses). There is a sharp dip with a minimum of 0.6% and FWHM of 24.1 nm near the photon energy E=0.8 eV, which is in the 3rd gap of PC1 and PC2 simultaneously. As a contrast, no edge mode exists in the 2nd band despite there is similar reflectance of PC1 and PC2 in gaps 2 and 3, this can be interpreted directly via the topologies below gap 2 are the same between PC1 and PC2. In Fig. 3(b), one can see that the spectra is flat with reflectivity over 75% near the operation frequency, indicating that 8 unit cells of PC2 are good enough to act as a well-behaved reflector in the MIM waveguide. However, the spectrum is dispersive with reflectivity of only 47% near the operation frequency in Fig. 3(a), indicating the balance of the two functions of PC1. The TPTS has a ignorable blue-shift(~ 1.2 nm) compared with the designed operation wavelength attributing to the imperfect reflector of PC1, which makes it a well-behaved narrow-band absorber at designed wavelength.

 figure: Fig. 3

Fig. 3 (a-c) The reflection spectra of 4 unit cells of PC1 (a), 8 unit cells of PC2 (b) and a system composed of 4 unit cells of PC1 connecting with 8 unit cells of PC2 (c). The solid lines indicate the spectra calculated by the TMM, while the cross marks indicate the spectra simulated by FEM. (d,e) The electric and magnetic field amplitude distributions inside the MIM structure. The contour plots show the normalized field amplitude along the central axis. The magenta dash line shows the position of interface between PC1 and PC2.

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In Figs. 3(d) and 3(e), we plot the field amplitude distribution at the resonant photon wavelength of 1550 nm. Both electric field |E| and magnetic field |H| are normalized to incident wave. The electromagnetic fields are trapped and enhanced near the interface between PC1 and PC2. The insets show the distributions of |E| and |H| along the center of MIM waveguide. The fields in PC1 and PC2 are approximately symmetric with the PC1/PC2 interface. Specifically, one can find that the symmetric axis of electric field is in PC1, while it is in PC2 for magnetic field. Due to the larger Ohm loss in PC2, the field in PC1 is larger than its symmetric positions in PC2. As a result, the maximal values of electric and magnetic fields are always in PC1. The maximal |E| and |H| in PC1 occur on the left and right of the first ϵA/ϵB interface near the PC1/PC2 interface with an enhancement factor of 2.7 and 3, respectively. Moreover, the position for each local maximum of |E| is also related to local minimum of |H| and vice versa. The phenomenon originates from the simple relation rE = −rH, so there is a phase different of π between reflected electric and magnetic fields. As a result, constructive interference for one quantity will indicate destructive interference for the other quantity.

Since the Zak-Berry phase is a ’global’ or ’topological’ property of the entire band, the edge mode protected by Zak phase is expected robust against structural perturbations. Next, we introduce the geometric perturbations by inserting a MIM waveguide between PC1 and PC2. For simplicity, the core of the defective MIM waveguide is chose as air. In our configuration, the defect is equivalent to prolong the thickness of the part A connecting PC1 and PC2. The thickness of slab A sandwiched by slab B without perturbation at the interface is dA1 + dA2. After the defect is inserted, the total thickness of slab A increases to d = dA1 + dA2 + Δd. The reflectance spectra for Δd = 0 nm, 100 nm, 300 nm, and 450 nm are shown in Fig. 4(a). The TPTS red-shifts as the increase of Δd. Nevertheless, a new resonance dip appears when Δd ≳ 300 nm, which certifies that the interface state always exists in the considered band gap. Incidentally, we find the field spot of protected states are always in the defect layer, which make it a promising all-optical bistable switch [33]. The dependence of the resonance photon energy of edge mode on the thickness of defect is shown in Fig. 4(b). From calculating the phase delay in defect layer Δδ = βAΔd, one can find that the second mode appears if π/2 ≤ Δδ ≤ 3π/2. Actually, this is a periodic function of Δδ with a period of π. Specially, when Δδ = , nN+, the resonance frequency will come back to the starting position.

 figure: Fig. 4

Fig. 4 (a) The reflection spectra of PC1/defect/PC2 composites, the thickness Δd are 0, 100, 300, and 450 nm, respectively. (b) The resonance energy of TPTSs as a function of the thickness of defect. The upper scale denotes the phase delay Δδ in the defect.

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3. Conclusion

A TPTS configuration is proposed by two adjacent semi-infinite MIM PCs. By using the proposed admittance based TMM, we find that the unit cell of symmetric PC can be regarded as a homogeneous layer with effective phase delay and equivalent admittance. On this basis, we have studied the topological properties of PCs and revealed the relationship among their topologies, reflective phase and phase delay in each layer. As such, their topology and the associated edge (interface) states are extremely robust against the structural perturbations. Such topologically protected localized states can be used as a generalization of plasmonic Tamm states, which makes it a promising optical device for integrated photonic circuits and able to provide significant applications, such as optical nonlinear enhancement.

4. Methods

4.1. Admittance based transfer matrix method

In the layered periodic MIM structure, the electric field of transversal magnetic (TM) wave can be expressed by incident and reflected waves as:

Ey=E0+cosh(kdly)exp(iβlz)+E0cosh(kdly)exp(iβlz);
where the first and second terms are incoming and reflected waves in dielectric core of the lth layer, respectively. According to the continuous boundary condition of the electromagnetic field at the interface j, the field can be written as:
TA(EAj+EAj)=TB(EBj+EBj),
where TA=[11ηAηA],TB=[11ηBηB] represent the boundary transition relationship. In the inner of each layer, the phase delay can be characterized by:
Pα=[exp(iβαdα)00exp(iβαdα)].(α=A,B)

If the A/B and B/A interface in a unit cell shown in Fig. 1(b) are marked by green number ’1’ and ’2’ from left to right, the electric field in the beginning and end of the Nth unit cell satisfy:

TAEN=TAPAE1l=TAPATA1TBE1r=TAPATA1TBPBE2l=TAPATA1TBPBTB1TAE2r=TAPATA1TBPBTB1TAPATA1TAEN+1=MAMBMATAEN+1
where
MαTαPαTα1=[cos(βαdα)isin(βαdα)/ηαiηαsin(βαdα)cos(βαdα)]
is the transmission matrix of a homogenous layer, the superscripts represent the left(l) and right(r ) limit of the related interfaces. After some mathematics, one could arrive at TAEN = MTAEN+1, where
M=MAMBMA=[cosδisinδ/ηiηsinδcosδ],
in which the phase delay in unit cell satisfies
cosδ=cosδAcosδB12(ηA/ηB+ηB/ηA)sinδAsinδB
and the equivalent characteristic admittance of the unit cell is
η=±M21M12=sinδ/{1ηBsinδB(cos2δA2ηB2ηA2sin2δA2)+1/ηAsinδAcosδB}.

One can find the expression of transmission matrix of unit cell is the same as the transmission matrix of a homogeneous layer. Thus the unit cell can be regarded as ’homogenous’ with equivalent phase delay and admittance. As a result, the phase delay of the unit cell represents the dispersion relation of the PCs, i.e., cos δ = cos (K Λ).

When the PCs are composed of N periods, the total transmission can be written as:

MN=MN=[cos(Nδ)isin(Nδ)/ηiηsin(Nδ)cos(Nδ)][M^11M^12M^21M^22].

In the same manner, one can conclude that:

T0E0l=TAE0r=MNTAEN+1l=MNTN+1EN+1r.

Expanding the expression, we have

(E0+E0)=T01MNTN+1(EN+1+0).

After some mathematics, one could arrive at

rPC=E0E0+=M^11η0+M^12η0ηN+1M^21M^22ηN+1M^11η0+M^12η0ηN+1+M^21+M^22ηN+1.

An important case is that when η0 = ηN+1, because of M^11=M^22=cos(Nδ) and ηM^12=M^21/η=isin(Nδ), the reflectivity can be reduced as

rPC=isin(Nδ)(η0/ηη/η0)2cos(Nδ)isin(Nδ)(η0/η+η/η0)=η0/ηη/η02icot(Nδ)+η0/η+η/η0

When the structure is composite of two PCs, one can also regard the two structures with two separated homogeneous layers: M=MN1MN2=[M˜11M˜12M˜21M˜22], where

M˜11=cos(N1δ1)cos(N2δ2)η2/η1sin(N1δ1)sin(N2δ2),M˜12=i/η1sin(N1δ1)cos(N2δ2)i/η2cos(N1δ1)sin(N2δ2),M˜21=iη1sin(N1δ1)cos(N2δ2)iη2cos(N1δ1)sin(N2δ2),M˜22=cos(N1δ1)cos(N2δ2)η1/η2sin(N1δ1)sin(N2δ2);
where N1,2, δ1,2 and η1,2 are the layer numbers, phase delay and equivalent characteristic admittance of PC1 and PC2, respectively. Similarly, when an extra MIM waveguide is inserted into the PCs, the transition matrix must be substituted by M = MN1 MdMN2, where Md=[cosΔδi/ηdsinΔδiηdsinΔδcosΔδ] is the transmission matrix of the defect layer. Finally, the reflectance of the composite structure can be expressed as:
rPC=M˜11η0+M˜12η0ηN+1M˜21M˜22ηN+1M˜11η0+M˜12η0ηN+1+M˜21+M˜22ηN+1.

4.2. Bloch function

We start from the expression TAEN = MTAEN+1, thus we have MEEN = EN+1 with the definition META1M1TA. If the periodic boundary condition is applied to a unit cell, the Bloch electric field satisfies MEEN = exp(iKΛ)EN. This is a typical eigenvalue problem, and the eigenfunction at origin can be chosen as E0(1[exp(iKΛ)M11E]/M12E)(1Γ),where

M11E=exp(iδA)[cosδB+i2(ηAηB+ηBηA)sinδB],M12E=i2sinδB(ηAηBηBηA).

It is worth mentioning that M12E and exp(iKΛ)M11E equal to zero simultaneously at the frequency point ω˜ at which sin(δB(ω˜))=0. If one isolated band (excluding the 0th band) contains these frequency points , then the Zak phase of the band must be π in the configuration.

In the first period, we consider the electromagnetic field along the central axis(y = 0), the field in slab A is straightforward: E (z ) = exp(iβAz ) + Γ exp(−iβAz ) and dxwHx(z)=ηAexp(iβAz)ηAΓexp(iβAz). Specially, the Bloch magnetic fields in origin satisfy u(z = 0) ∝ 1 − Γ. Due to the relation TAE0=MATBE1r, the field at the right limit of interface labeled ’1’ satisfies E1r=(E1r+E1r)=TB1MA1TAE0. From which one can get the electromagnetic field in slab B:

Ey(z)=E1r+exp[iβB(zdA)]+E1rexp[iβB(zdA)]
and
dxwHx(z)=ηBE1r+exp[iβB(zdA)]ηBE1rexp[iβB(zdA)];
therefore, the Bloch magnetic field eigenfunctions can be summarized as
u(z)=wdxexp(iKz)×{ηAexp(iβAz)ηAΓexp(iβAz),x[nΛdA,nΛ+dA]ηBE1r+exp[iβB(zdA)]ηBE1rexp[iβB(zdA)].x[nΛ+dA,(n+1)ΛdA]

Appendix Topological plasmonic Tamm states supported by HfO2/SiO2/HfO2 waveguides

In the main paper, we have explored the plasmonic crystals (PCs) with air/SiO2/air cores, indicating ϵA < ϵB, this hypothesis is very compatible with the current fabrication process. But in principle, the case ϵA > ϵB is also available such as adopting the PCs with HfO2/SiO2/HfO2 cores. In the same manner, to verify the existence of such an topological interface states, we consider two adjoining one dimensional PCs with parameters across the topological transition point, and a silver-air-silver waveguide are adopted as the input and output waveguides just as the case in the main paper. For the telecom optical wavelength at 1550 nm (≈0.8 eV), we have βA = 11.884 µm −1, βB = 8.868 µm −1 and β0 = βN+1 = 5.891 µm −1, respectively. Specifically, TPII, i.e., δA(ω˜)=2π and δB(ω˜)=π are adopted due to βA > βB; thus dAII=2π/βA/2=264.4nm and dBII=π/βB=354.3nm, respectively. The parameters of PC1 are chosen as dB1 = 2dA1, i.e., dB1 = 3π/(βA + βB) = 454.2 nm and dA1=227.1 nm without loss of generality. The band structure and Zak phase of plasmonic crystals consist of HfO2/SiO2/HfO2 cores are shown in Fig. 5. The Considering the band edge is linear dependent on thicknesses near the topological transition points, one can easily obtain the parameters dA2=2dAIIdA1=151.4nm and dB2=302.8 nm for PC2 to share the same band gap with PC1. For convenience, the tuning process dA1dAIIdA2 is considered. The Zak phases of bands 0 and 1 remain unchanged while the Zak phases of bands 2 and 3 switch, which comes from a corresponding sign change for the reflection phases in the 3th gap after the perturbation. Thus the PCs undergo a topological phase transition in the parameter tuning process. Compared with the PCs with air/SiO2/air cores, the Zak phase of the 0th and 1st band is π(0) rather than 0(π). The reason for this is that the Zak phase of the 0th band is charactered by exp(iθ0Zak)=sgn[1ηA2/ηB2], due to ηA > ηB, one can obtain θ0Zak=π immediately. Due to the sign of reflection phase in gap 2 is the same as discussed in main paper, thus the Zak phase of the 0th and 1st band must be opposite with the case ηA < ηB. Moreover, the Zak phase of the band 2 is different attributing to the topological phase transition, thus a topologically protected interface state exists at the interfaces between PC1 and PC2. For comparison, the reflection spectra of the configurations are shown in Fig. 6. One can find a similar TPTS at considered wavelength in the configurations.

 figure: Fig. 5

Fig. 5 (a, b) The band structure and Zak phase of plasmonic crystals consist of HfO2/SiO2/HfO2 cores with parameters (a) dA1 = 227.1 nm, and dB2 = 2dA1 = 454.2 nm for PC1, and (b) dA2 = 151.4 nm, and dB2 = 302.8 nm for PC2, respectively. The Zak phase of each individual band is labeled in green, and the numbers of the bands and gaps are listed with red and black labels. The magenta strip represents the gap with reflection phase φPC > 0, while the cyan strip represents the gap with φPC < 0. Moreover, the symmetries of the eight band edge states, K, L, M, N, O, P, Q, R are labeled with red (antisymmetric) and indigo (symmetric) circles, respectively. (c) The black and blue broken curves represent the reflection phase φPC1 and the negative value of the reflection phase −φPC2 of the plasmonic crystals consisting of 50 periods. The intersection point denotes the location of the interface states. (d) The Bloch magnetic field eigenfunctions of the band-edge states at z = 0. The zero amplitude at the origin represent the Bloch magnetic fields are antisymmetric, while the non-zero values indicate the the Bloch magnetic fields are symmetric in the plasmonic crystals.

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 figure: Fig. 6

Fig. 6 (a) The reflection spectra of 4 unit cells of PC1 (a), 8 unit cells of PC2 (b) and a system composed of 4 unit cells of PC1 connecting with 8 unit cells of PC2 (c). The solid lines indicate the spectra calculated by the TMM, while the cross marks indicate the spectra simulated by FEM. (d,e) The electric and magnetic field amplitude distributions inside the MIM structure. The contour plots show the normalized field amplitude along the central axis. The magenta dash line shows the position of interface between PC1 and PC2.

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Funding

National key R & D Program of China (2017YFA0303800, 2017YFA0305100); Program for Changjiang Scholars and Innovative Research Team in University (IRT13_R29); National Natural Science Foundation of China (NSFC) (11604283, 91750204, 11774185, 11374006, 11504184, 61775106); 111 Project (B07013); Tianjin Natural Science Foundation (18JCQNJC02100); Key Research Projects of Henan Provincial Department of Education (16A140048); Fundamental Research Funds for the Central Universities; Nanhu Scholars Program for Young Scholars of XYNU.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006). [CrossRef]   [PubMed]  

2. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009). [CrossRef]   [PubMed]  

3. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011). [CrossRef]  

4. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013). [CrossRef]  

5. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013). [CrossRef]   [PubMed]  

6. L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011). [CrossRef]   [PubMed]  

7. L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015). [CrossRef]   [PubMed]  

8. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018). [CrossRef]  

9. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018). [CrossRef]  

10. B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017). [CrossRef]   [PubMed]  

11. H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018). [CrossRef]   [PubMed]  

12. 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]  

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

14. S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018). [CrossRef]  

15. D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017). [CrossRef]  

16. Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015). [CrossRef]  

17. L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015). [CrossRef]   [PubMed]  

18. H. Deng, X. Chen, N. C. Panoiu, and F. Ye, “Topological surface plasmons in superlattices with changing sign of the average permittivity,” Opt. Lett. 41, 4281–4284 (2016). [CrossRef]   [PubMed]  

19. F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016). [CrossRef]   [PubMed]  

20. X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017). [CrossRef]   [PubMed]  

21. D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017). [CrossRef]   [PubMed]  

22. D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017). [CrossRef]   [PubMed]  

23. M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007). [CrossRef]  

24. I. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sov. Union 1, 733 (1932).

25. T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008). [CrossRef]   [PubMed]  

26. S. Longhi, “Zak phase of photons in optical waveguide lattices,” Opt. Lett. 38, 3716–3719 (2016). [CrossRef]  

27. Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014). [CrossRef]  

28. L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016). [CrossRef]   [PubMed]  

29. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989). [CrossRef]   [PubMed]  

30. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

31. K. H. Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and K. H. Fung, “Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals,” Opt. Lett. 41, 1644–1647 (2016). [CrossRef]   [PubMed]  

32. W. Gao, M. Xiao, B. Chen, E. Y. B. Pun, C. T. Chan, and W. Y. Tam, “Controlling interface states in 1d photonic crystals by tuning bulk geometric phases,” Opt. Lett. 42, 1500–1503 (2017). [CrossRef]   [PubMed]  

33. Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013). [CrossRef]  

34. B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018). [CrossRef]  

35. S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015). [CrossRef]  

36. S. A. Maier, Plasmonics: Fundamentals and Applications (SpringerUS, 2007).

37. L. Xu, H. Wang, Y. Xu, H. Chen, and J. Jiang, “Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals,” Opt. Express 24, 18059–18071 (2016). [CrossRef]   [PubMed]  

38. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984). [CrossRef]  

39. W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809–821 (1959). [CrossRef]  

References

  • View by:

  1. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
    [Crossref] [PubMed]
  2. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
    [Crossref] [PubMed]
  3. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
    [Crossref]
  4. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
    [Crossref]
  5. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
    [Crossref] [PubMed]
  6. L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
    [Crossref] [PubMed]
  7. L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
    [Crossref] [PubMed]
  8. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
    [Crossref]
  9. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
    [Crossref]
  10. B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
    [Crossref] [PubMed]
  11. H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
    [Crossref] [PubMed]
  12. 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]
  13. C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
    [Crossref] [PubMed]
  14. S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
    [Crossref]
  15. D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
    [Crossref]
  16. Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
    [Crossref]
  17. L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
    [Crossref] [PubMed]
  18. H. Deng, X. Chen, N. C. Panoiu, and F. Ye, “Topological surface plasmons in superlattices with changing sign of the average permittivity,” Opt. Lett. 41, 4281–4284 (2016).
    [Crossref] [PubMed]
  19. F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
    [Crossref] [PubMed]
  20. X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
    [Crossref] [PubMed]
  21. D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
    [Crossref] [PubMed]
  22. D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
    [Crossref] [PubMed]
  23. M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
    [Crossref]
  24. I. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sov. Union 1, 733 (1932).
  25. T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
    [Crossref] [PubMed]
  26. S. Longhi, “Zak phase of photons in optical waveguide lattices,” Opt. Lett. 38, 3716–3719 (2016).
    [Crossref]
  27. Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
    [Crossref]
  28. L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
    [Crossref] [PubMed]
  29. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
    [Crossref] [PubMed]
  30. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
  31. K. H. Choi, C. W. Ling, K. F. Lee, Y. H. Tsang, and K. H. Fung, “Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals,” Opt. Lett. 41, 1644–1647 (2016).
    [Crossref] [PubMed]
  32. W. Gao, M. Xiao, B. Chen, E. Y. B. Pun, C. T. Chan, and W. Y. Tam, “Controlling interface states in 1d photonic crystals by tuning bulk geometric phases,” Opt. Lett. 42, 1500–1503 (2017).
    [Crossref] [PubMed]
  33. Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
    [Crossref]
  34. B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
    [Crossref]
  35. S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
    [Crossref]
  36. S. A. Maier, Plasmonics: Fundamentals and Applications (SpringerUS, 2007).
  37. L. Xu, H. Wang, Y. Xu, H. Chen, and J. Jiang, “Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals,” Opt. Express 24, 18059–18071 (2016).
    [Crossref] [PubMed]
  38. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984).
    [Crossref]
  39. W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809–821 (1959).
    [Crossref]

2018 (5)

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

2017 (6)

W. Gao, M. Xiao, B. Chen, E. Y. B. Pun, C. T. Chan, and W. Y. Tam, “Controlling interface states in 1d photonic crystals by tuning bulk geometric phases,” Opt. Lett. 42, 1500–1503 (2017).
[Crossref] [PubMed]

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

2016 (6)

2015 (5)

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

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

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
[Crossref] [PubMed]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

2014 (3)

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]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

2013 (3)

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

2011 (2)

L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
[Crossref] [PubMed]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

2008 (1)

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

2007 (1)

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

2006 (1)

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

1989 (1)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984).
[Crossref]

1959 (1)

W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809–821 (1959).
[Crossref]

1932 (1)

I. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sov. Union 1, 733 (1932).

Abram, R. A.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Afinogenov, B. I.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Amili, A. El

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Badugu, R.

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

Bahari, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Bandres, M. A.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Baryshev, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Bernevig, B. A.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984).
[Crossref]

Bessonov, V. O.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Brand, S.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Cai, W.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Chamberlain, J. M.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Chan, C. T.

Chen, B.

Chen, H.

L. Xu, H. Wang, Y. Xu, H. Chen, and J. Jiang, “Accidental degeneracy in photonic bands and topological phase transitions in two-dimensional core-shell dielectric photonic crystals,” Opt. Express 24, 18059–18071 (2016).
[Crossref] [PubMed]

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Chen, X.

Cheng, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Choi, K. H.

Chong, Y.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Chong, Y. D.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

Choudhury, S. D.

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

Christensen, T.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

Christodoulides, D. N.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Davis, T. J.

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Deng, H.

Dorofeenko, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

El-Ganainy, R.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Fainman, Y.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

Fang, N. X.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

Fedyanin, A. A.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Feng, L.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Fu, L.

L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
[Crossref] [PubMed]

Fung, K. H.

Gao, F.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Gao, W.

Gao, Z.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

García de Abajo, F. J.

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

Ge, L.

Giannini, V.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

Gómez, D. E.

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

Goto, T.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Granovsky, A. B.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Hafezi, M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Han, D.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
[Crossref] [PubMed]

Harari, G.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Hu, X.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

Huang, Y.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

Hughes, T. L.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Huidobro, P. A.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

Hwang, Y.

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

Inoue, M.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Iorsh, I.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Jiang, J.

Jin, D.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

Joannopoulos, J. D.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Kaliteevski, M.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Kanté, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Kavokin, A. V.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Khajavikhan, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

Kivshar, Y.

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]

Kohn, W.

W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809–821 (1959).
[Crossref]

Lakowicz, J. R.

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

Lee, K. F.

Li, T.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Lin, J.

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

Lin, X.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Ling, C. W.

Lisyansky, A. A.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Longhi, S.

Lu, L.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Lukin, M. D.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Lukyanchuk, B.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Lumer, Y.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Luo, W.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (SpringerUS, 2007).

Malzard, S.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Meng, Y.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

Merzlikin, A. M.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Miao, P.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

Miroshnichenko, A.

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]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

Ndao, A.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Niu, L.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Pan, D.

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

Pan, Y.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Panoiu, N. C.

Parto, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Pocock, S. R.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

Poddubny, A.

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]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Popkova, A. A.

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Pun, E. Y. B.

Qi, J.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Rechtsman, M. C.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Ren, J.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Roberts, A.

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

Schomerus, H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Segev, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Shelykh, I. A.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

Shi, X.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Slobozhanyuk, A.

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]

Soljacic, M.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Tam, W. Y.

Tamm, I.

I. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sov. Union 1, 733 (1932).

Taylor, J. M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Teimourpour, M. H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Tian, J.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

Tsang, Y. H.

Vallini, F.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

Vinogradov, A. P.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

Wang, H.

Wang, L.

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
[Crossref] [PubMed]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Wang, P.

Wang, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Wen, W.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23, 21585–21595 (2015).
[Crossref] [PubMed]

Wittek, S.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

Wu, L.-H.

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

Wu, X.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

Xiang, H.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

Xiang, Y.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Xiao, M.

Xiao, X.

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

Xu, H.

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Xu, J.

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Xu, L.

Xu, Y.

Yang, Z.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Ye, F.

Ying, C.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Ying, C.-F.

Yu, R.

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

Yu, S. F.

Zak, J.

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref] [PubMed]

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Zhang, B.

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

Zhang, S.-C.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Zhang, X.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014).
[Crossref]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

Zhang, Z. Q.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

Zhao, H.

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

Zhu, S.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Accounts Chem. Res. (1)

S. D. Choudhury, R. Badugu, and J. R. Lakowicz, “Directing fluorescence with plasmonic and photonic structures,” Accounts Chem. Res. 48, 2171–2180 (2015).
[Crossref]

ACS Photonics (3)

S. R. Pocock, X. Xiao, P. A. Huidobro, and V. Giannini, “Topological plasmonic chain with retardation and radiative effects,” ACS Photonics 5, 2271–2279 (2018).
[Crossref]

D. E. Gómez, Y. Hwang, J. Lin, T. J. Davis, and A. Roberts, “Plasmonic edge states: An electrostatic eigenmode description,” ACS Photonics 4, 1607–1614 (2017).
[Crossref]

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]

AIP Adv. (1)

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Adv. 3, 012106 (2013).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser & Photonics Rev. (1)

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser & Photonics Rev. 9, 392–398 (2015).
[Crossref]

Nat. Commun. (4)

F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D. Joannopoulos, M. Soljačić, H. Chen, L. Lu, Y. Chong, and B. Zhang, “Probing topological protection using a designer surface plasmon structure,” Nat. Commun. 7, 11619 (2016).
[Crossref] [PubMed]

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8, 1304 (2017).
[Crossref] [PubMed]

H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9, 981 (2018).
[Crossref] [PubMed]

D. Pan, R. Yu, H. Xu, and F. J. García de Abajo, “Topologically protected dirac plasmons in a graphene superlattice,” Nat. Commun. 8, 1243 (2017).
[Crossref] [PubMed]

Nat. Photon. (1)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

Nat. Phys. (1)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Nature (2)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
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Opt. Express (3)

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Phys. Rev. B (2)

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric bragg mirror,” Phys. Rev. B 76, 165415 (2007).
[Crossref]

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B. Lukyanchuk, and A. A. Fedyanin, “Phase matching with Tamm plasmons for enhanced second- and third-harmonic generation,” Phys. Rev. B 97, 115438 (2018).
[Crossref]

Phys. Rev. Lett. (5)

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[Crossref] [PubMed]

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
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D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118, 245301 (2017).
[Crossref] [PubMed]

L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
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L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

Phys. Rev. X (1)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

Phys. Z. Sov. Union (1)

I. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sov. Union 1, 733 (1932).

Proc. Royal Soc. Lond. (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Royal Soc. Lond. 392, 45–57 (1984).
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Sci. Rep. (1)

L. Niu, Y. Xiang, W. Luo, W. Cai, J. Qi, X. Zhang, and J. Xu, “Nanofocusing of the free-space optical energy with plasmonic tamm states,” Sci. Rep. 6, 39125 (2016).
[Crossref] [PubMed]

Science (4)

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: Theory,” Science 359, aar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359, aar4005 (2018).
[Crossref]

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636–640 (2017).
[Crossref] [PubMed]

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Other (1)

S. A. Maier, Plasmonics: Fundamentals and Applications (SpringerUS, 2007).

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Figures (6)

Fig. 1
Fig. 1 (a) Sketch of two adjoining one dimensional plasmonic crystals (PCs) composed of symmetric ABA layers. The insets show the side views of PC1 and PC2, of which the unit cells Λ1 = 2dA1 + dB1 and Λ2 = 2dA2 + dB2 are highlighted by blue and red dashed frames, respectively. Here the width w = 40 nm, and the refractive index nA = 1 and nB = 1.5, respectively. (b) The band structure of the PCs as a function of dA when δA + δB = 3π is fixed, where δA and δB are the phase delay in slab A and slab B for a unit cell, respectively. The blank zones indicate the stopband. TPI and TPII are two topological phase points, and the integer pairs in parentheses are δA and δB in the unit of π, respectively. Moreover, the dashed lines show the parameters adopt to construct edge states. (c, d) The dispersion relation (c) and mode distributions (d) of the edge modes, where the magnetic field distributions H are calculated at kx = 0.
Fig. 2
Fig. 2 (a, b) The band structure and Zak phase of plasmonic crystals with parameters (a) d A1 = 319.3 nm, and dB2 = 2dA1 = 638.6 nm for PC1, and (b) dA2 = 210 nm, and dB2 = 738.8 nm for PC2, respectively. The Zak phase of each individual band is labeled in green, and the numbers of the bands and gaps are listed with red and black labels. The magenta strip represents the gap with reflection phase φPC > 0, while the cyan strip represents the gap with φPC < 0. Moreover, the symmetries of the eight band edge states, K, L, M, N, O, P, Q, R are labeled with red (antisymmetric) and indigo (symmetric) circles, respectively. (c) The black and blue broken curves represent the reflection phase φPC1 and the negative value of the reflection phase −φPC2 of the plasmonic crystals consisting of 50 periods calculated by Eq. (6). The intersection point denotes the location of the interface states. (d) The Bloch magnetic field eigenfunctions of the band-edge states at z = 0. The zero amplitude at the origin represents the Bloch magnetic fields are antisymmetric, while the non-zero values indicate the the Bloch magnetic fields are symmetric in the plasmonic crystals.
Fig. 3
Fig. 3 (a-c) The reflection spectra of 4 unit cells of PC1 (a), 8 unit cells of PC2 (b) and a system composed of 4 unit cells of PC1 connecting with 8 unit cells of PC2 (c). The solid lines indicate the spectra calculated by the TMM, while the cross marks indicate the spectra simulated by FEM. (d,e) The electric and magnetic field amplitude distributions inside the MIM structure. The contour plots show the normalized field amplitude along the central axis. The magenta dash line shows the position of interface between PC1 and PC2.
Fig. 4
Fig. 4 (a) The reflection spectra of PC1/defect/PC2 composites, the thickness Δd are 0, 100, 300, and 450 nm, respectively. (b) The resonance energy of TPTSs as a function of the thickness of defect. The upper scale denotes the phase delay Δδ in the defect.
Fig. 5
Fig. 5 (a, b) The band structure and Zak phase of plasmonic crystals consist of HfO2/SiO2/HfO2 cores with parameters (a) dA1 = 227.1 nm, and dB2 = 2dA1 = 454.2 nm for PC1, and (b) dA2 = 151.4 nm, and dB2 = 302.8 nm for PC2, respectively. The Zak phase of each individual band is labeled in green, and the numbers of the bands and gaps are listed with red and black labels. The magenta strip represents the gap with reflection phase φPC > 0, while the cyan strip represents the gap with φPC < 0. Moreover, the symmetries of the eight band edge states, K, L, M, N, O, P, Q, R are labeled with red (antisymmetric) and indigo (symmetric) circles, respectively. (c) The black and blue broken curves represent the reflection phase φPC1 and the negative value of the reflection phase −φPC2 of the plasmonic crystals consisting of 50 periods. The intersection point denotes the location of the interface states. (d) The Bloch magnetic field eigenfunctions of the band-edge states at z = 0. The zero amplitude at the origin represent the Bloch magnetic fields are antisymmetric, while the non-zero values indicate the the Bloch magnetic fields are symmetric in the plasmonic crystals.
Fig. 6
Fig. 6 (a) The reflection spectra of 4 unit cells of PC1 (a), 8 unit cells of PC2 (b) and a system composed of 4 unit cells of PC1 connecting with 8 unit cells of PC2 (c). The solid lines indicate the spectra calculated by the TMM, while the cross marks indicate the spectra simulated by FEM. (d,e) The electric and magnetic field amplitude distributions inside the MIM structure. The contour plots show the normalized field amplitude along the central axis. The magenta dash line shows the position of interface between PC1 and PC2.

Equations (26)

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η = I / V = c H x d x / + E y d y = ω ϵ d ϵ 0 d x β w ,
cos δ = cos δ A cos δ B 1 2 ( η A / η B + η B / η A ) sin δ A sin δ B ,
α = δ A δ B = m 1 m 2 . m 1 , m 2 +
θ n Zak = π / Λ π / Λ [ i u n i t c e l l d z u n , q * ( z ) q u n , q ( z ) ] d q ,
sgn [ φ ( n ) ] = ( 1 ) n exp ( i m = 0 n 1 θ m Zak ) , n +
r PC = | r | exp ( i φ PC ) = η 0 / η η / η 0 2 i cot ( N δ ) + η 0 / η + η / η 0 ,
η = sin δ / { 1 / η B sin δ B ( cos 2 δ A 2 η B 2 / η A 2 sin 2 δ A 2 ) + 1 / η A sin δ A cos δ B } .
E y = E 0 + cosh ( k d l y ) exp ( i β l z ) + E 0 cosh ( k d l y ) exp ( i β l z ) ;
T A ( E A j + E A j ) = T B ( E B j + E B j ) ,
P α = [ exp ( i β α d α ) 0 0 exp ( i β α d α ) ] . ( α = A , B )
T A E N = T A P A E 1 l = T A P A T A 1 T B E 1 r = T A P A T A 1 T B P B E 2 l = T A P A T A 1 T B P B T B 1 T A E 2 r = T A P A T A 1 T B P B T B 1 T A P A T A 1 T A E N + 1 = M A M B M A T A E N + 1
M α T α P α T α 1 = [ cos ( β α d α ) i sin ( β α d α ) / η α i η α sin ( β α d α ) cos ( β α d α ) ]
M = M A M B M A = [ cos δ i sin δ / η i η sin δ cos δ ] ,
cos δ = cos δ A cos δ B 1 2 ( η A / η B + η B / η A ) sin δ A sin δ B
η = ± M 21 M 12 = sin δ / { 1 η B sin δ B ( cos 2 δ A 2 η B 2 η A 2 sin 2 δ A 2 ) + 1 / η A sin δ A cos δ B } .
M N = M N = [ cos ( N δ ) i sin ( N δ ) / η i η sin ( N δ ) cos ( N δ ) ] [ M ^ 11 M ^ 12 M ^ 21 M ^ 22 ] .
T 0 E 0 l = T A E 0 r = M N T A E N + 1 l = M N T N + 1 E N + 1 r .
( E 0 + E 0 ) = T 0 1 M N T N + 1 ( E N + 1 + 0 ) .
r PC = E 0 E 0 + = M ^ 11 η 0 + M ^ 12 η 0 η N + 1 M ^ 21 M ^ 22 η N + 1 M ^ 11 η 0 + M ^ 12 η 0 η N + 1 + M ^ 21 + M ^ 22 η N + 1 .
r PC = i sin ( N δ ) ( η 0 / η η / η 0 ) 2 cos ( N δ ) i sin ( N δ ) ( η 0 / η + η / η 0 ) = η 0 / η η / η 0 2 i cot ( N δ ) + η 0 / η + η / η 0
M ˜ 11 = cos ( N 1 δ 1 ) cos ( N 2 δ 2 ) η 2 / η 1 sin ( N 1 δ 1 ) sin ( N 2 δ 2 ) , M ˜ 12 = i / η 1 sin ( N 1 δ 1 ) cos ( N 2 δ 2 ) i / η 2 cos ( N 1 δ 1 ) sin ( N 2 δ 2 ) , M ˜ 21 = i η 1 sin ( N 1 δ 1 ) cos ( N 2 δ 2 ) i η 2 cos ( N 1 δ 1 ) sin ( N 2 δ 2 ) , M ˜ 22 = cos ( N 1 δ 1 ) cos ( N 2 δ 2 ) η 1 / η 2 sin ( N 1 δ 1 ) sin ( N 2 δ 2 ) ;
r PC = M ˜ 11 η 0 + M ˜ 12 η 0 η N + 1 M ˜ 21 M ˜ 22 η N + 1 M ˜ 11 η 0 + M ˜ 12 η 0 η N + 1 + M ˜ 21 + M ˜ 22 η N + 1 .
M 11 E = exp ( i δ A ) [ cos δ B + i 2 ( η A η B + η B η A ) sin δ B ] , M 12 E = i 2 sin δ B ( η A η B η B η A ) .
E y ( z ) = E 1 r + exp [ i β B ( z d A ) ] + E 1 r exp [ i β B ( z d A ) ]
d x w H x ( z ) = η B E 1 r + exp [ i β B ( z d A ) ] η B E 1 r exp [ i β B ( z d A ) ] ;
u ( z ) = w d x exp ( i K z ) × { η A exp ( i β A z ) η A Γ exp ( i β A z ) , x [ n Λ d A , n Λ + d A ] η B E 1 r + exp [ i β B ( z d A ) ] η B E 1 r exp [ i β B ( z d A ) ] . x [ n Λ + d A , ( n + 1 ) Λ d A ]

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