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
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 . Over the past decades, a number of topological states have been successfully emulated in photonic systems, such as magneto-optical photonic crystals , coupled resonator optical waveguides [3,4], photonic Floquet topological insulators , 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 . Tamm states, one of the surface Bloch states localized at the interfaces of truncated crystalline materials, have been sufficiently studied in superlattices , photonic crystals  and photonic lattices . 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 . 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 , harmonic wave generation , and fluorescence enhancement .
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 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.
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:36]: − tanh(kdw/2) = km ϵd/kd ϵm, where ϵm,d are the dielectric constants, and 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)
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 , the band m ≡ m1 + m2 and band m − 1 will cross at the frequency satisfying . When the gap is open, the frequency derived by 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 , at TPI and , 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 , as an analogy of the Berry phase , 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:
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.,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 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)
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.
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  and Zak  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.
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 . 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 = nπ, n ∈N+, the resonance frequency will come back to the starting position.
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.
4.1. 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:
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:
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:
In the same manner, one can conclude that:
Expanding the expression, we have
After some mathematics, one could arrive at
An important case is that when η0 = ηN+1, because of and , the reflectivity can be reduced as
When the structure is composite of two PCs, one can also regard the two structures with two separated homogeneous layers: , where
4.2. Bloch function
We start from the expression TAEN = MTAEN+1, thus we have MEEN = EN+1 with the definition . 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 ,where
It is worth mentioning that and equal to zero simultaneously at the frequency point at which . 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 . Specially, the Bloch magnetic fields in origin satisfy u(z = 0) ∝ 1 − Γ. Due to the relation , the field at the right limit of interface labeled ’1’ satisfies . From which one can get the electromagnetic field in slab B:
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., and are adopted due to βA > βB; thus and , 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 and dB2=302.8 nm for PC2 to share the same band gap with PC1. For convenience, the tuning process 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 , due to ηA > ηB, one can obtain 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.
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.
The authors declare that there are no conflicts of interest related to this article.
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