1. Introduction

Topological edge states originated from the topological electronics to explain the integer quantum Hall effect [1,2], which is related to the topological invariant of the system, i.e., Chen number [3,4]. Through the researches, people found the bulk-boundary correspondence, that is, an edge state appear at the interface when two materials with different topological invariants are in contact [5–7], and this edge state shows chirality in quantum Hall effect [8]. With the discovery of topological phase in electronic systems, the concept of topology had been introduced into photonics in order to design and control the behavior of light.

Most of the researches on topological photonics are focused on two-dimensional systems [9,10]. Besides, one-dimensional (1D) systems possessing topological invariant have also attracted a great deal of attention. In 1D topological system, the topological phase is related to the chiral symmetry [11]. Among them, Su-Schrieffer-Heeger (SSH) model is the most representative 1D topological model [12]. Therefore, finding the optical correspondence of SSH model has become a research hotspot.

In 2009, Malkova et al. first observed the topological edge modes of SSH model in the photonic superlattice [13]. Since then, the research on SSH model mainly focused on two aspects: on one hand, realizing SSH model in different electromagnetic systems, such as microwave photonic crystals [14], coupled waveguide arrays [15,16], pure dielectric cavities [17], waveguide ring resonators [18,19], plasmonic superlattices [20], multilayer graphene [21], and so on; on the other hand, promoting and applicating of SSH model. For example, in 2017, St-Jean et al. realized a topological laser based on the edge mode of a 1D topological structure [22]; Kitagawa et al. implemented a topologically nontrivial discrete-time quantum walk in experiment for the first time by single photons [23]; within the study of topological invariant of quantum walk in orbital angular momentum space, topologically protected states in quantum walk has been experimentally realized [24]; Kraus et al. realized the unitary photonic topological pumps by 1D coupled waveguide arrays, and experimentally observed the topologically protected edge states [25].

Among previous works, SSH models are mainly discussed based on coupled mode theory [15,26–28], quasistatic point dipole approximation [29,30], and topological band theory [20]. However, the limitation of propagating constant has not been considered. We find that when dealing with surface modes of one-dimensional multi-layer structure, the coupled mode theory is not always consistent with Maxwell's equation.

We first briefly introduce the optical SSH model defined by a finite coupled optical waveguide array as shown in Fig. 1(a). It is composed of N coupled waveguides, where ${c_1}$ and ${c_2}$ are two alternating coupling coefficients of the nearest neighbors, and ${\beta _0}$ is the propagation constant of every isolated waveguide. Following the coupled mode theory in the tight-binding approximation, the Hamiltonian of the optical field in the corresponding infinite array can be written as

 

Fig. 1. (a) Schematics of a 1D finite SSH model with N waveguides. (b) Band structure with N=24, c₁=1, c₂=3, and W=1. Gray areas indicate the bulk bands. (c) The eigenmode profiles of two edge modes (near-zero modes) with N=24, where n refers to the site of waveguide. (d) Diagrammatic sketch of the multilayer metal-vacuum periodic structure.

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According to Eq. (1), there are two bands distributed in the region $\beta \in{\pm} [|{{c_1} - {c_2}} |,|{{c_1} + {c_2}} |]$, shown in the gray regions of Fig. 1(b). It is a gap between them. However, if the waveguide array is finite, the nontrivial topological edge modes appear in the case of ${{{c_2}} / {{c_1}}} > 1$, which is indicated by the winding number W=1. Figure 1(b) shows the mode diagram of an array of 24 waveguides with ${c_1} = 1$ and ${c_2} = 3$, which is marked by circles. Actually, there are 24 modes, which equals to the number of waveguides. Among them, 22 modes are distributed in the upper and lower energy bands (gray regions) which are defined as extended modes, and 2 modes (blue and red circle) appear in the mid-gap which is called edge modes or zero modes. The spatial distribution of the edge modes is shown in Fig. 1(c). It is clear that the field of these two edge modes is mainly localized at two outmost waveguides, and they can be distinguished due to the symmetric or asymmetric manners. These edge modes have been demonstrated in the finite zigzag chains of plasmonic and dielectric nanoparticles [31,32,17].

In this paper, we analyze the surface modes in a 1D photonic crystals to carry out an optical analogy of SSH model. The surface mode in 1D layered structure was the content in the text book [33]. Especially, the surface modes near the isolated interface separating metal and vacuum as well as that in the metal layer were the contents that beginners must master. However, when 1D structure is extended to multi-layer or many layers, the research on its surface modes becomes less and less. There are mainly two reasons: 1) the inevitable loss of metal limits the application of multilayer surface modes; 2) the calculation of surface modes in multilayer structure is complicated. We also find that the coupled mode theory fails under some conditions in our system. Here we will study in detail the surface modes of multilayer periodic structures and examine the surface modes from the viewpoint of topology. Since the surface modes are related to metal, we are talking about photonic crystals composed of metal and vacuum alternately.

This paper is arranged as follows: In Sec.2, we introduce the 1D photonic crystals containing metal, and compare it with the waveguide array. In Sec. 3, we give the method to calculate the surface modes in 1D multilayer structure. In Sec.4, we calculate the surface modes in the 1D photonic crystals containing metal, and analyze the properties of the edge modes and the extended modes. In Sec.5, we discuss the robustness of edge modes. Finally, we draw the conclusion in Sec. 6.

2. 1D photonic crystals and the principle

Here we consider a 1D photonic crystal containing metal as shown in Fig. 1(d). Layer A marked by green refers to metal with permeability ${\varepsilon _a}$ and thickness ${d_a}$; Layer B marked by white refers to vacuum with permeability ${\varepsilon _b} = {\varepsilon _0}$ and thickness ${d_b}$. The number of layers in the whole structure is odd to ensure that the two outermost layers are metal. The whole structure is placed in the vacuum.

Assume that the photonic crystal has N layers (N is odd), and there are N+1 (even) interfaces. All interfaces are the same because they all separate the metal from vacuum. In this setting, the photonic crystal shown in Fig. 1(d) is an analogy to the waveguide array shown in Fig. 1(a).

For each isolated interface, it supports one TM polarized surface mode which is characterized by the propagation constant ${k_{{\parallel} 0}} = {k_0}\sqrt {{{{\varepsilon _a}{\varepsilon _b}} / {({{\varepsilon_a} + {\varepsilon_b}} )}}}$. The field of the surface mode is concentrated near the interface, and decays exponentially away from the interface, as shown in Fig. 2(a). Therefore, the surface mode of each isolated interface ${k_{{\parallel} 0}}$ can be equivalent to the waveguide mode in the isolated waveguide ${\beta _0}$.

 

Fig. 2. (a) The surface mode profile in the isolated interface separating semi-infinite metal and vacuum; (b) scheme of one metal layer in vacuum, its surface mode can be seen as the coupling between two isolated interface; (c) the profile of surface modes of metal layer in vacuum, the parameters of the structure are ${\varepsilon _a} ={-} 3$ and ${d_a} = 0.15\lambda$. A and B indicates metal and vacuum, respectively. Two dotted vertical lines in (c) refer to the interfaces of metal layer.

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Similar to the evanescent coupling between adjacent waveguides, the coupling between adjacent interfaces is also evanescent here. Therefore, the tight-binding approximation holds for the surface mode in the 1D photonic crystals in principle here.

Take a single layer A as example, it can be regarded as the coupling of two interfaces, shown in Fig. 2(b), and can produce the well-known symmetric and anti-symmetric surface mode, as shown in the Fig. 2(c). We set the metal layer with ${\varepsilon _a} ={-} 3$ and ${d_a} = 0.15\lambda$. The propagation constants of the symmetric and antisymmetric surface modes are ${k_{{\parallel} s}} = 1.37717{k_0}$ and ${k_{{\parallel} a}} = 1.12552{k_0}$, respectively, which deviate from that of the isolated interface ${k_{{\parallel} 0}} = 1.2247{k_0}$.

Now we extend to the case of multilayer structure, shown in Fig. 1(d). As the surface mode in isolated interface possessing the propagation constant ${k_{{\parallel} 0}}$ is equivalent to that of a waveguide, we can similarly define the coupling coefficient between the two interfaces of one metal layer, i.e., ${C_m}$, which is similar to ${c_1}$ in the waveguide array; and the coupling coefficient between two adjacent interfaces separated by vacuum, i.e., ${C_v}$, similar to ${c_2}$. So, the surface modes of the 1D photonic crystals containing metal can be seen as the SSH model, which is the coupled interface array with the alternating coupling strength ${C_m}$ and ${C_v}$.

Next, we will perform the numerical calculation to check the surface modes in such 1D photonic crystals. Although it is an analogy to the SSH model, it cannot be calculated by the coupled mode theory directly, but can only be resolved by Maxwell equations combined with boundary conditions. However, with the increase of layers, the difficulty of solving is also increasing rapidly because it needs more equations obtained by boundary conditions. Hence, we combine the transfer matrix method with the residue theorem to simplify the calculation. This method has the advantage of easily extending to structure with arbitrary layers.

3. Method to calculate the surface mode in multilayer structures

We firstly introduce the transfer matrix method [34], which is a most commonly used method to calculate the traveling wave transmission in multilayer structures. However, we apply it to the surface mode analysis combining with the residue theorem in our work. Consider a 1D structure containing N layers, where the principal axis is along the z-axis, and the origin is at the leftmost interface. From left to right, there are layers 1, 2, 3, …, j, …, N, respectively. For the j-th layer, the permittivity and the permeability are ${\varepsilon _j}$ and ${\mu _j}$, respectively, and the coordinate of the left interface is ${z_{j - 1}}$, while the right is ${z_j}$, so that the thickness of j-th layer is ${d_j} = {z_j} - {z_{j - 1}}$.

Setting the x-z plane as the incident plane, we consider a TM-polarized traveling wave entering from left with ${k_\parallel }$ which is the component of wave vector paralleling to interface. The magnetic field is set to along y-axis. According to boundary conditions, the magnetic field and the x-component of electric field satisfy the continuity condition crossing the interfaces, so that the two-component wave function vector ${\Psi _j}({z,\omega } )$ was taken to describe the electromagnetic field in j-th layer as

The matrix ${M_j}({\Delta z,\omega ,{k_\parallel }} )$ connects the electromagnetic fields in the same j-th layer by ${\Psi _j}({{z_{j - 1}} + \Delta z,\omega ,{k_\parallel }} )= {M_j}({\Delta z,\omega ,{k_\parallel }} ){\Psi _j}({{z_{j - 1}},\omega ,{k_\parallel }} )$, with the definition

The wave vector component parallel to the interface ${k_\parallel }$ and the component along z-axis in j-th layer ${k_{jz}}$ satisfy $k_\parallel ^2 + k_{jz}^2 = {\varepsilon _j}{\mu _j}{{{\omega ^2}} / {{c^2}}}$.

Similarly, the matrix $Q({{z_{j - 1}} + \Delta z,\omega ,{k_\parallel }} )$ connects the field in j-th layer ${\Psi _j}({{z_{j - 1}} + \Delta z,\omega ,{k_\parallel }} )$ to that at the leftmost interface ${\Psi _1}({{z_0},\omega } )$ through

The electromagnetic field at the leftmost interface (z₀=0) is the superposition of incident and reflective fields, which is in the form of

The reflection coefficient r and transmission coefficient t in Eqs. (7) and (8) can be expressed as

Therefore, the distribution of electromagnetic field in multilayer structure can be obtained by Eq. (5) when r and t are known.

The transfer matrix method mentioned above is used for input-output analysis, which needs incident field. However, the surface mode is a special kind of electromagnetic mode with discrete wave vector (propagation constant), which has no concept of incidence, reflection and transmission.

According to our previous work concerning atomic spontaneous decay [35], the discrete propagation constant of surface mode is determined by the singularity of the reflection coefficient Eq. (9). When the wave vector component along the interface ${k_\parallel }$ exceeds the wave number ${\omega / c}$, the traveling wave becomes into the evanescent field. Furthermore, the reflection coefficient will diverge at certain wave vector component, which is just the propagation constant of the surface mode ${k_{{\parallel} SPP}}$. Since the singularity is of first order, we can use the residue theorem to take the residue of reflection coefficient at the propagation constant. And then the spatial distribution of surface mode can be obtained by replacing the reflection coefficient in Eq. (5) with its residue.

Let's rewrite the reflection coefficient r as the following formation

When ${k_\parallel }$ equals to the propagation constants of surface modes ${k_{{\parallel} SPP}}$, the denominator of reflection coefficient $D({{k_\parallel }} )$ equals to zero. As ${k_{{\parallel} SPP}}$ is first-order singularity, the residue of reflection coefficient is

So, we can get the surface mode distribution in the 1D multilayer as follows

For ${k_\parallel }\textrm{ = }{k_{{\parallel} SPP}}$, all elements of the transfer matrix $Q({{k_\parallel }} )$ are real numbers. Therefore, by taking the imaginary part in the above formula, the incident part (i.e., the term corresponding to constant 1) is eliminated naturally, making it conform to the physical description of the surface mode. The fields on the left and right regions of the structure can be obtained in similar ways.

We adopt this method to repeat the surface modes of one-layered [36] and three-layered structures [37], and found that the results are exactly the same, which verifies the reliability of such method.

4. Surface modes in 1D photonic crystals containing metal

In this section, we numerically calculate the surface modes of 1D PC containing metal based on the method mentioned above. We will compare the results with that of SSH model, and study the property and existence condition of the edge modes.

As mentioned in Sec. 2, for the photonic crystals containing metal, the surface mode of an isolated interface is equivalent to a waveguide. Two interfaces of one metal layer construct two identical waveguides in a unit cell. Meanwhile, two adjacent interfaces separated by vacuum form the waveguides between cells. For the periodic metal-vacuum structure with finite layers, these two kinds of coupling appear alternately, which is similar to the SSH model of waveguide array. It will constitute the SSH model of surface modes.

We take the photonic crystals with 9 layers as example. The permittivity of the metal is ${\varepsilon _a} ={-} 3$. The thicknesses of the metal and vacuum layer are and ${d_b} = 0.15\lambda$, respectively. And the propagation constant of the isolated interface is ${k_{{\parallel} 0}} = 1.2247{k_0}$. With the method mentioned above, the propagation constants ${k_{{\parallel} SPP}}$ according to $D({{k_\parallel }} )= 0$ and the profiles of the corresponding surface modes are shown in Fig. 3(a).

 

Fig. 3. (a) Intensity of electric field ${|{{E_x}} |^2}$ and the corresponding propagation constants of the surface modes in the photonic crystals with (AB)⁴A, ${\varepsilon _a} ={-} 3$ and ${d_a} = {d_b} = 0.15\lambda$. (b) Electric field ${E_x}$ of the edge mode. The dotted vertical lines refer to the outmost interfaces of photonic crystal.

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According to SSH model, the number of eigenstates is the same with the number of waveguides. However, from Fig. 3(a), there are only 6 surface modes for 10 interfaces, which is different from the SSH model. Among them, there are 2 edge modes with the nearly degenerated eigenvalues which are ${k_{{\parallel} SPP}} = 1.2242{k_0}$ and ${k_{{\parallel} SPP}} = 1.2253{k_0}$; the remaining 4 modes are extended modes whose propagation constants are all larger than that of isolate interface ${k_{{\parallel} 0}} = 1.2247{k_0}$, and their corresponding ${|{{E_x}} |^2}$ are distributed in the shape of Bloch waves across the whole structure, shown in Fig. 3(a).

From the corresponding ${|{{E_x}} |^2}$ distribution along the z-axis, 2 edge modes are nearly the same, which are marked in red and black. Their electric field is mainly concentrated on the two outermost interfaces of the structure (referred by two dotted vertical lines), and decays exponentially with the distance from the interface. Except near the third interface from the left and from the right, the field strength can be ignored in other places. Also, the propagation constants of 2 edge modes are nearly degenerate, and tend to the propagation constant of the isolated interface ${k_{{\parallel} 0}} = 1.2247{k_0}$. In Fig. 3(b), we extract these two edge modes, and plot their ${E_x}$ distribution along z axis. Now the two edge modes can be distinguished, because they are antisymmetric and symmetric corresponding to the center point. We also find that the degree of degeneracy of the two edge modes propagation constants increases with the number of layers. These properties about the edge modes are the same as that of SSH model.

It is necessary to find out why several extended modes are missing. According to the standard SSH model, there should be two band separated by a gap. These two energy bands are distributed in the range of ${k_{{\parallel} 0}} \pm [|{{C_m} - {C_v}} |,|{{C_m} + {C_v}} |]$ [38]. Reference [39] pointed out that the coupling coefficient between two waveguides can be defined by $C = {{({{k_{{\parallel} s}} - {k_{{\parallel} a}}} )} / 2}$, in which ${k_{{\parallel} s}}$ and ${k_{{\parallel} a}}$ are the propagation constants of the symmetric and antisymmetric modes, respectively.

For the photonic crystals shown in Fig. 3(a), ${C_m}$ can be gotten from Fig. 2(b), which is ${C_m}\textrm{ = }{{({{k_{{\parallel} s}} - {k_{{\parallel} a}}} )} / 2} = 0.1258{k_0}$. To get ${C_v}$, we need to analyze the structure that two semi-infinite metal slabs with ${\varepsilon _a} ={-} 3$ separated by vacuum with ${d_b} = 0.15\lambda$. The result shows that there is only one antisymmetric surface mode with ${k_{{\parallel} a}} = 1.7248{k_0}$. So, we get ${C_v}\textrm{ = }{{({{k_{{\parallel} 0}} - {k_{{\parallel} a}}} )} / 2} ={-} 0.2500{k_0}$. The lower band should be not only lower than ${k_{{\parallel} 0}} = 1.2247{k_0}$, but also should be lower than ${k_0}$ here. However, when ${k_\parallel } < {k_0}$, the field is not the surface mode but the traveling wave. This is the reason of missing extended modes.

Therefore, the coupled mode theory to treat the waveguide array did not consider the limitation of propagating constant. We also find that when dealing with surface modes of one-dimensional multi-layer structure, the coupled mode theory is inconsistent with Maxwell's equation, that is, the range of the upper energy band obtained is different. For example, the upper band obtained from coupled mode theory should be $[1.3489{k_0},1.6005{k_0}]$, but the result calculated from Maxwell theory is larger than $1.6005{k_0}$, which is confirmed by Fig. 3(a).

Although the coupled mode theory cannot accurately describe the surface modes of one-dimensional multilayer structures, it is qualitatively effective. For example, the propagating constant of the edge modes is ${k_{{\parallel} 0}}$, and that of the extended modes are in the upper and lower bands. From the qualitative results of coupled mode theory, we can understand why some extended modes disappear because their propagating constants are less than ${k_0}$.

There are two ways to change the surface modes here, one is the permittivity of the metal, the other is the thickness.

When the permittivity of metal is different, the electric field distributions of the corresponding surface modes are shown in Fig. 4. In Fig. 4(a), the permittivity of metal is ${\varepsilon _a} ={-} 2$, and the propagation constants of the two edge modes are ${k_{{\parallel} SPP}} = 1.4086{k_0}$ and ${k_{{\parallel} SPP}} = 1.4200{k_0}$, respectively. In Fig. 4(b), the permittivity of metal is ${\varepsilon _a} ={-} 4$, and the propagation constants of the two edge modes are ${k_{{\parallel} SPP}} = 1.1546{k_0}$ and ${k_{{\parallel} SPP}} = 1.1548{k_0}$. These two cases show that when the thicknesses of metal and vacuum layers are both fixed, the existence of the two edge modes is insensitive to the permittivity of metal. Comparing Fig. 4 and Fig. 3(a), with the decrease of the absolute permittivity of metal, the ratio of the maximum strength of the extended modes to the edge modes increases, and the degree of localization on the leftmost and rightmost interface is also increased for the edge modes. The reason is that the propagation constant of the isolated interface ${k_{{\parallel} 0}} = {k_0}\sqrt {{{{\varepsilon _a}} / {({{\varepsilon_a} + 1} )}}}$ is increasing with the decrease of the absolute permittivity of metal, so does the imaginary part of ${k_{jz}}$, which enhance the localization of the field. Therefore, reducing the absolute permittivity of metal can increase the concentration of energy on the two outermost interfaces.

 

Fig. 4. Electric field intensity ${|{{E_x}} |^2}$ of surface modes in photonic crystals with (AB)⁴A and ${d_a} = {d_b} = 0.15\lambda$. (a) the permittivity of metal is ${\varepsilon _a} ={-} 2$; (b) the permittivity of metal is ${\varepsilon _a} ={-} 4$. The dotted vertical lines refer to the outmost interfaces of photonic crystal.

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When we focus on the missing extended modes for the cases of ${\varepsilon _a} ={-} 2$ and ${\varepsilon _a} ={-} 4$, we found that all missing modes fall in the lower band whose propagating constants are lower than ${k_0}$.

According to the bulk-boundary correspondence of SSH model, the existence of edge modes depends on the two alternate coupling strengths, that is the edge modes exist for the case of weak coupling ending, i.e., ${C_m} < {C_v}$. As ${C_m}$ relates to the thickness of metal layer, the existence of edge modes relates to the thicknesses of metal and vacuum layers, i.e., ${d_a}$ and ${d_b}$.

In order to show the condition of edge modes more intuitively, we give the phase diagram in the case of ${\varepsilon _a} ={-} 3$ and (AB)⁴A in Fig. 5(a), in which the thickness of metal and vacuum layer are both considered within the region of ${d_a}\textrm{,}{d_b} \in [0.05\lambda ,0.5\lambda ]$. The red region of parameters refers to the existence of edge mode, while the green region does not. Noted that the zigzag edge in Fig. 5(a) is caused by the step size of $0.025\lambda$ during calculation.

 

Fig. 5. Multilayer metal-vacuum periodic structure: (a) Relation between the existence of edge modes and the thickness of metal layer ${d_a}$ and vacuum layer ${d_b}$, where the red (green) area represents the existence (disappearance) of edge modes; Distribution of electric field ${|{{E_x}} |^2}$ on z-axis with (b) ${d_a} = 0.075\lambda$, ${d_b} = 0.15\lambda$ (c) ${d_a} = 0.3\lambda$, ${d_b} = 0.15\lambda$. The parameters of the structure are (AB)⁴A and ${\varepsilon _a} ={-} 3$. The dotted vertical lines in (b) and (c) refer to the outmost interfaces of photonic crystal.

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We fix the thickness of vacuum layer as ${d_b} = 0.15\lambda$, and check the influence of thickness of metal ${d_a}$ on the existence of edge modes. We first consider an example of ${d_a} = 0.075\lambda$ and ${d_b} = 0.15\lambda$, which falls in the left green region in Fig. 5(a). According to the coupled waveguide theory, we get ${C_m} = 0.4135{k_0}$, ${C_v} ={-} 0.2500{k_0}$, i.e., ${C_m} > |{{C_v}} |$, which is the strong coupling ending. Therefore, the disappearance of edge mode is consistent with the predict of coupled mode theory (Fig. 5(b)). In this case, there are still 6 extended modes, in which 5 modes’ propagation constants are larger than ${k_{{\parallel} 0}} = 1.2247{k_0}$, and one is smaller than ${k_{{\parallel} 0}} = 1.2247{k_0}$. The missing 4 extended modes is due to the limitation of propagation constant of surface mode, i.e., ${k_{{\parallel} SPP}} > {k_0}$.

With the increasing thickness of metal, ${C_m}$ decreases but ${C_v}$ keeps unchanged because we fixed $\; {d_b}$. So, with the increase of ${d_a}$, the edge modes exist as ${C_m}$ is smaller than $|{{C_v}} |$. An example is the case of Fig. 3, in which ${d_a} = 0.15\lambda$, ${C_m} = 0.1258{k_0}$. ${C_m} < |{{C_v}} |$ refers to the weak coupling ending, the existence of edge states is consistent with the coupled mode theory.

According to SSH mode, if ${C_m} < |{{C_v}} |$, the edge modes will exist. However, according to our numerical simulation, edge modes disappear when the thickness of metal layers increases further, shown in the right green region in Fig. 5(a). We take the case of ${d_a} = 0.3\lambda$ as an example, where ${C_m} = 0.0168{k_0}$. Although ${C_m} < |{{C_v}} |$ here, edge modes still disappear. The corresponding surface mode distribution and propagation constants are shown in Fig. 5(c). There are only 4 surface modes and their propagation constants are close to each other. These modes show the obvious Bloch wave profiles, but has no field on the two outmost interfaces, which is much different from that in Fig. 5(b). Thus, we conclude that the coupled mode theory can only be applied to the mode analysis of photonic crystal with thin layers. When the layer is very thick, the coupled mode theory will fail. This is a difference of surface modes in 1D multilayer from the SSH model. Similar discrepancy in the bulk-boundary correspondence in non-Hermitian topological systems is also one of the research hotspots in recent [40,41].

5. Robustness of edge modes

Finally, we introduce two types of perturbation to study the robustness of the edge modes. One is the frequency perturbation of oscillators, that is, the diagonal fluctuation, which destroys the chiral symmetry; the other is the coupling strength perturbation between the lattice sites, i.e., the off-diagonal fluctuation, which maintains the chiral symmetry [38].

Based on the case of Fig. 3, we add perturbations to permittivity of some metal layers in the photonic crystals, and the results are shown in Fig. 6. In Fig. 6(a), the permittivity of the fifth layer (metal) is added by 1% perturbation, i.e., ${\varepsilon _{a5}} = {\varepsilon _a} + 0.01{\varepsilon _a}$, and the two edge modes distributed on the outermost interfaces are with ${k_{{\parallel} SPP}} = 1.2217{k_0}$ and ${k_{{\parallel} SPP}} = 1.2248{k_0}$. The electric field intensity of the two edge modes is not symmetrical anymore, and mainly concentrates on the left outmost interface; while the intensities of extended mode in the middle layers is less and symmetric. In Fig. 6(b), we add 1% and 0.5% perturbation to the permittivity of the fifth and seventh metal layers respectively, and the propagation constants of the two edge modes are ${k_{{\parallel} SPP}} = 1.2231{k_0}$ and ${k_{{\parallel} SPP}} = 1.2249{k_0}$. Although the propagation constants of the edge modes of the two examples is still close to ${k_{{\parallel} 0}} = 1.2247{k_0}$, the spatial distribution of the edge states is obviously changed, and the amplitudes of the field strength on both sides of the edges are no longer equal. This is because changing permittivity is equivalent to changing frequency of oscillators, breaking the chiral symmetry. Even there is only 1% perturbation, it can obviously change the electric field distribution of edge modes. Therefore, the perturbation in permittivity of metal will destroy edge modes, and the edge modes are not robust.

 

Fig. 6. Electric field intensity ${|{{E_x}} |^2}$ of surface modes in photonic crystals with (AB)⁴A, ${\varepsilon _a} ={-} 3$ and ${d_a} = {d_b} = 0.15\lambda$. (a) 1% perturbation was added to the metal permittivity of the 5th-layer. (b) 1% and 0.5% perturbations were added to the permittivity of the 5th- and 7th-layer metal respectively. The dotted vertical lines refer to the outmost interfaces of photonic crystal.

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When we add perturbations to the coupling strength ${C_m}$ and ${C_v}$ in the structure, that is, the thickness of some metal and vacuum layers, and the results are revealed in Fig. 7. In Fig. 7(a), 1% perturbation is applied on the seventh metal layer, i.e., ${d_{a7}} = {d_a} - 0.01{d_a}$, we obtain two edge modes with propagation constants as ${k_{{\parallel} SPP}} = 1.2239{k_0}$ and ${k_{{\parallel} SPP}} = 1.2256{k_0}$. From this, we can see that the electric field intensity distribution of the two edge modes almost have no change, showing the robustness to perturbation. If we keep increasing this perturbation, the distribution of the two edge modes will be obviously asymmetrical. The maximum allowable perturbation value is about 3%, which is shown in Fig. 7(c). Then, we add 2% and 1% perturbation to the 6th and 8th vacuum layers, respectively, in Fig. 7(b). Similarly, with perturbations, the change is mainly in the electric field distribution of extended modes in middle layers, and the propagation constants and the electric field intensity distribution of the edge modes hardly changes. Compared with Fig. 3, for the perturbations on thickness, only the electric field distribution of extended modes becomes asymmetrical. Consequently, as an off-diagonal perturbation, the thickness perturbation does not destroy edge modes and keeps the chirality symmetry, which shows that the edge modes have good robustness here.

 

Fig. 7. Electric field intensity ${|{{E_x}} |^2}$ of surface modes in photonic crystals with (AB)⁴A, ${\varepsilon _a} ={-} 3$ and ${d_a} = {d_b} = 0.15\lambda$. (a) 1% perturbation were added to the thicknesses of the 7th-layer metal. (b) 2% and 1% perturbations were added to the thicknesses of the 6th- and 8th-layer vacuum respectively. (c) 3% perturbation were added to the thicknesses of the 7th-layer metal. The dotted vertical lines refer to the outmost interfaces of photonic crystal.

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In the preparation of photonic crystals, the perturbation of permittivity can be ignored, but the perturbation of thickness often occurs. Therefore, the edge modes discussed here is robust in experiment.

6. Conclusion

The surface modes of 1D photonic crystal containing metal can construct an optical analogy of SSH model. When thickness of layers is small and satisfies the appropriate range, there are two edge modes whose field is mainly concentrated on the two outermost interfaces. Their propagation constants are nearly degenerated and are close to that of isolated interface. The existence of such edge modes mainly depends on the thickness of metal layers and vacuum layers.

However, the surface modes in photonic crystals are not completely consistent with SSH model treated by the coupled mode theory, especially in the number of surface modes and the existence conditions of edge modes. This is because the propagation constant of surface mode must be greater than ${k_0}$, and the coupled mode theory is not suitable for large thickness. Although the surface mode is strictly determined by Maxwell’s equation and boundary conditions, they can still be viewed from the perspective of topology.

In addition, the perturbations of the thickness of vacuum and metal layers have little influence on the edge modes, whether the distribution of the electric field or the propagation constant. But the perturbation of permittivity can destroy the electric field of edge modes. This is because that the perturbation of permittivity corresponds to the diagonal fluctuation; while the perturbation of thickness corresponds to the off-diagonal fluctuation which holds the chiral symmetry. Therefore, the robustness of our edge modes agrees with the SSH model very well.

The topologically protected edge modes can be used to couple with atoms or quantum dots in order to obtain some new phenomenon of interaction between atoms and light field or atoms and atoms. It can be seen from the above that changing the number of layers does not affect the existence and distribution of edge modes. With the increase of the number of layers, the two interfaces connected by the edge state move away from each other. This provides a new way to couple two atoms or quantum dots in a long distance through edge modes. The advantage of our result is that the limitation of propagation constant leads to the reduction of number of extended mode but holds the edge mode. When an atom or quantum dot interacts with the outmost interface, it can couple to all surface modes. The reduction of the number of extended modes improves the coupling efficiency between atom and edge modes. Due to the inevitable loss of metal, our results will also be affected by metal loss. In general, the metal loss makes the propagation constant of surface mode complex. In our model, it means that the surface mode decays as it propagates along the interface. However, we do not care about the propagation along the interface, but focus on the evanescent coupling between the outermost two interfaces on the z-axis through the concept of topological edge mode. It has nothing to do with whether the propagation constant is real or complex. Therefore, the tiny metal loss will not affect our results qualitatively when the number of layers is not large.

Our results not only provide a general platform for the study of the robustness topological edge states, but also have potential applications in information transmission, power transfer and so on.