In the past few decades the novel concept of topological states has attracted intensive research interest, and has been exploited widely in many fields of modern physics [1, 2] and materials science [3, 4], due to unique boundary states and the related propagation properties. Inspired by these encouraging ideas, topological photonics has emerged as a hot research topic to manipulate the behavior of light, promising a new variety of high-performance photonic devices [5–7]. Tremendous amounts of research [3–8] have shown that photonic topological protection can be realized in photonic crystals (PCs), waveguides, nanocavities, coupled resonators, metamaterials, etc. In PCs, topological states manifest as interface states [9–11] in periodic as well as quasiperiodic composite structures, which are topologically protected and hence robust against defects, edges, or disorder owing to experimental nonuniformity or finite size [12, 13]. Thanks to the structural flexibility and experimental advances, topological PCs provide an excellent platform for potential applications in enhancing nonlinearity [14–16], lasing [17, 18], sensing [19], waveguiding [20], etc.

Xiao et al. [21] and Wang et al. [22] theoretically studied the topological interface states in one-dimensional (1D) photonic systems, and illustrated the conditions for the optical interface states in a composite structure consisting of two PCs with inversion symmetry. Optical interface states were demonstrated later by Gao et al. [23, 24] in 1D composite structures fabricated by an electron-beam-evaporation technique. Recently, the controllability of interface states in a 1D composite system composed of two binary-layered PCs was researched by our group [25]. This tunability has a direct application in narrow-band filters.

In this paper, we systematically characterize the robustness of the interface states in 1D composite structures consisting of two binary layered PCs, with and without inversion symmetry, employing the transfer-matrix method. The inversion-symmetric PC’s interface states are well defined, deep inside of the photonic band gaps, and insensitive to system size. Thereafter, the layer thickness of the PCs is combined with random noise to analyze the robustness of the interface states, confirming that the inversion-symmetric PC’s interface states are the most robust. Overall, the robustness of the non-inversion-symmetric PC’s interface states has a linear dependence on the positions of the interface states inside the band gaps, which yields practical insight into the development of photonic devices by fine tuning the interface states.

The transfer-matrix method (TMM) is a common method to model 1D photonic structures [26]. It involves two matrices: one for the propagation of light in each medium (propagation matrix), and another for the transmission across the boundary (matching matrix) [25]. The propagation matrix relating to the change of electric fields in the forward and backward directions (E_+j, E_−j) when light passes through a homogeneous j^th layer is expressed as

(1)Ej+Ej−=exp(−ikjdj)00exp(ikjdj)Ej+1+Ej+1−=Pjkj,djEj+1+Ej+1−,

where k_j = 2π / λ_j is the value of the wave vector with wavelength λ_j, and d_j is the thickness of the j^th layer. The matching matrix M_j→j+1, representing the electric fields across the interface when light passes from the j^th layer to the (j + 1)^th layer, is expressed as

(2)Ej+Ej−=zj+1+zj2zj+1zj+1−zj2zj+1zj+1−zj2zj+1zj+1+zj2zj+1Ej+1+Ej+1−=Mjzj,zj+1Ej+1+Ej+1−,

where zj= μ j / ε j is the impedance of the j^th layer [21].

The top inset of Fig. 1 depicts the composite photonic structure with PC I on the left and PC II on the right, in which a boundary is generated in the middle and indicated by the vertical green dashed line. The boundary layer is controlled by the boundary factor x, varying from 0 to 1. For x = 0 or 1, the composite structure is a simple PC with an AB or BA configuration. For x = 0.5, the unit cell of PC I is 0.5d_a – d_b – 0.5d_a and the unit cell of PC II is 0.5d_b – d_a – 0.5d_b; thus they both have inversion symmetry.

Figure 1.Composite structure. Top inset: the composite structure, consisting of photonic crystal (PC) I and PC II. The unit cells for PC I and PC II are enclosed in the cyan dotted and red dashed boxes respectively. (a) Band diagram of the individual PC. (b) and (c) Reflectance of PC I and PC II respectively, with 50 unit cells, for x = 0.5. (d) Reflectance of the composite structure. The black arrows indicate the interface states.

The transfer matrix of the unit cell for the ABA (from left to right) configuration, i.e. PC I, can be described as

(3)TABA1−x=Pka,da⋅1−x⋅Ma→b⋅Pkb,db⋅Mb→a⋅Pka,da⋅x= a1 +b1 i −c1 +d1 i −c1 −d1 i a1 −b1 i ,

where the matrix elements are

(4)a1=coskadacoskbdb−12zbza+zazbsinkadasinkbdbb1=−sinkadacoskbdb+za2+zb2coskadasinkbdb2zazbc1=za2−zb2sinkbdbsinkada1−2x2zazbd1=−za2−zb2sinkbdbcoskada1−2x2zazb.

Here z_a(z_b) and d_a(d_b) are respectively the impedance and layer thickness of the A (B) layer.

Similarly, the transfer matrix of the unit cell for the BAB configuration, i.e. PC II, can be described as

(5)TBABx=Pkb,db⋅x⋅Mb→a⋅Pka,da⋅Ma→b⋅Pkb,db⋅1−x= a2 +b2 i c2 +d2 i c2 −d2 i a2 −b2 i ,

where the matrix elements are

(6)a2=coskadacoskbdb−12zbza+zazbsinkadasinkbdbb2=−sinkbdbcoskada+za2+zb2coskbdbsinkada2zazbc2=zb2−za2sinkadasinkbdb1−2x2zazbd2=−zb2−za2sin(kada)coskbdb(1−2x)2zazb.

Note that a₁ = a₂ = a. The determinants of the transfer matrixes should be equal to 1, due to energy conservation [26],

(7)detTABA1−x=a2+b12−c12−d12=1detTBABx=a2+b22−c22−d22=1.

Specifically, when the individual PC has inversion symmetry, i.e. x = 0.5, c₁ and c₂ are zero.

The dispersion relation of PC I (and also for PC II) can be calculated using the Bloch theorem with the following equation [27, 28]:

(8)cosKΛ=coskadacoskbdb−12kakb+kbkasinkadasinkbdb,

where K is the Bloch wave number, and Λ = d_a + d_b is the thickness of the unit cell. It is clear from Eq. (8) that the photonic bands and band gaps correspond to |cos(KΛ)| < 1 where K is real and |cos(KΛ)| > 1 where K is complex, respectively. Since Eq. (8) does not depend on the boundary factor x, PC I and PC II will exhibit the same dispersion and band gaps, independent of x. Figure 1(a) demonstrates the dispersion relation of the first three band gaps (indicated by the yellow columns) for an infinite PC with the following physical parameters [25, 29]: d_a = 166 nm, d_b = 34 nm, and n_a/b = n_a0/b0 (1 + n1 / λ²), where n_a0 = 1.4659, n_b0 = 1.2741, and n₁ = 0.0054.

The transfer matrix T_PCI for PC I in air with N unit cells and matching matrixes (Mair→a= z0+za −z0+za −z0+za z0+za/2za and Ma→air= z0+za z0−za z0−za z0+za/2z0, where z₀ is the impedance of air) at the air–PC boundary can be described as

where the matrix elements are

(10)A1=a−a2−1N+a+a2−1N2B1=−a−a2−1N−a+a2−1Nb1+d1z02+b1−d1za24a2−1z0zaC1=−a−a2−1N−a+a2−1Nc12a2−1D1=−a−a2−1N−a+a2−1Nb1+d1z02+−b1+d1za24a2−1z0za.

Furthermore, the transmission (t) and reflection (r) coefficients can be obtained from

(11)1r=TPCI1−x⋅t0.

Similarly, the transfer matrix for PC II with N unit cells in the air is

(12)TPCIIx=Μair→b⋅ T BAB xN⋅Mb→air= A2 +B2 i C2 +D2 i C2 −D2 i A2 −B2 i ,

where the matrix elements are

(13)A2=a−a2−1N+a+a2−1N2B2=−a−a2−1N−a+a2−1Nb2+d2z02+b2−d2zb24a2−1z0zbC2=−a−a2−1N−a+a2−1Nc22a2−1D2=−a−a2−1N−a+a2−1Nb2+d2z02+−b2+d2zb24a2−1z0zb.

Note that A₁ = A₂ = A and the matrixes of the PC I and PC II should satisfy energy conservation:

(14)detTPCI=A2+B12−C12−D12=1detTPCII=A2+B22−C22−D22=1.

From Eqs. (9), (11), and (12) we can obtain the reflection coefficients, and hence the reflectance of PC I and PC II respectively

(15)rPCI1−x=−C1−D1iA+B1i,RPCI1−x=rPCI1−x⋅rPCI∗1−xrPCIIx=C2−D2iA+B2i,RPCIIx=rPCIIx⋅rPCII∗x.

For x = 0.5 with N = 50 and the same parameters as those used in the dispersion calculation, the reflectances of PC I and PC II are shown in Figs. 1(b) and 1(c) respectively.

Finally, the transfer matrix for the 1D composite structure can be described as

(16)Tcomx=TPCI1−x⋅TPCIIx= A+B1 i −C1 +D1 i −C1 −D1 i A−B1 i A+B2 i C2 +D2 i C2 −D2 i A−B2 i .

The reflection coefficient is

(17)rcomx=−C1−D1iA+B2i+A−B1iC2−D2iA+B1iA+B2i+−C1+D1iC2−D2i.

On the basis of Eq. (15), the reflectance of the composite structure can be expressed as

(18)Rcomx=rcomx⋅rcom∗x=RPCIx+RPCIIx−rPCI ∗xrPCII ∗x+rPCIxrPCIIx1−rPCIxrPCIIx1−rPCI ∗xrPCII ∗x,

where rPCIx=C1−D1iA+B1i and RPCI (x) = rPCI (x) ∙ r_*PCI (x).

The interface states, visible as reflection dips inside the band gaps, are clearly observed in the middle of the odd band gaps, as indicated by the black arrows in Fig. 1(d) for the first and third band gaps. Similar interface states can also be observed for 0 < x < 1, although the positions of the interface states are no longer in the middles of the band gaps, but shift continuously inside the band gaps towards the band edges with variation of x [25]. Below we will explore more characteristics of these interface states, especially the robustness against imperfections or defects, for component PCs with and without inversion symmetry.

3.1. Dependence on the System Size

We first study the dependence of the interface states on the number of unit cells N used in PC I and PC II. Fig. 2(a) provides the reflectance covering only the first band gap of the composite structure composed of non-inversion-symmetric PCs (x = 0.3), for different numbers of unit cells. The interface states shift as N increases and only converge roughly when N is large. On the contrary, the inversion-symmetric PC’s interface states, i.e., x = 0.5, remain unchanged at wave number 1 / λ₀ = 1.71452 μm₋₁, as visualized in Figs. 2(b) and 2(c) for different unit cells. Figure 2(c) summarizes the shift of interface states with N increasing from 20 to 130 under various boundary factors, signifying that interface states will vary with N and only reach the bulk states when N > 50, except for the inversion symmetric case (x = 0.5). We will explain this phenomenon for a finite-size composite structure using the TMM in the following section.

Figure 2.Dependence on N. (a) and (b) Reflectance near the first band gap of the composite structure for different numbers of unit cells, for x = 0.3 and 0.5 respectively. The black short-dash-dot lines indicate the interface states wave number for N = 130. (c) Interface-state wave number as a function of the number of unit cells N, for various x as labeled.

3.2. Conditions for Interface States

Considering two PCs as components, interface states will exist in the composite structure when the sum of reflection phases of the two individual PCs is equal to zero, i.e. φ₁ + φ₂ = 0 [25]. The reflection phases are evaluated at the interface of the composite structure (top inset of Fig. 1). We calculate the reflection coefficients for incidence from the right-hand side for PC I, i.e. r_PCI(x) = |r_PCI|exp(iφ1), and from the left-hand side for PC II, i.e. r_PCII(x) = |r_PCII|exp(iφ2). For the interface states to appear, the result of r_PCI(x) ∙ r_PCII(x) should be a positive real number:

(19)rPCIx⋅rPCIIx=rPCIxrPCIIxexpiφ1+φ2=rPCIxrPCIIx=rPCI∗xrPCII∗x.

When x = 0.5, c₁ and c₂ are zero per Eqs. (4) and (6), and hence C₁ = 0 and C₂ = 0, leading to

(20)rPCIx⋅rPCIIx=−D1D2A2−B1B2+AB1 +B2 i.

Additionally, according to Eq. (19) the imaginary part of r_PCI(x) ∙ r_PCII(x) is required to be zero at the interface states, leading to

(21)B1=−B2.

Figure 3(a) shows clearly that |B₁| = |B₂| at the interface states, and that they are nonzero. Furthermore, combining Eqs. (14), (20), and (21) yields rPCIx⋅rPCIIx=−D1D2A2+B12. and

Figure 3.Reflectance (blue solid curves), transfer-matrix elements (|B| and |D|), and |rPCI| − |rPCII| of the composite structure in Fig. 1, for (a)–(c) x = 0.5 and (d)–(f) x = 0.3, respectively.

(22)D1=−D2

at the interface states, as shown in Fig. 3(b). According to Eqs. (10) and (13), the conditions given by Eqs. (21) and (22) can be simplified as

(23)−b1+d1z02+b1−d1za2zb=b2+d2z02+b2−d2zb2za−b1+d1z02−b1−d1za2zb=b2+d2z02−b2−d2zb2za,

relating the physical parameters of PC I and PC II. Particularly, these conditions for interface states are independent of N, verifying that the interface states are independent of the system size when the component PCs have inversion symmetry.

For non-inversion-symmetric cases, i.e. x ≠ 0.5, c₁ and c₂ are nonzero (and thus C₁ ≠ 0 and C₂ ≠ 0), leading to B₁ ≠ −B₂ and D₁ ≠ −D₂ at the interface states, as shown in Figs. 3(d) and 3(e) for x = 0.3. As a result, the interface states will vary with the number of unit cells and converge to the bulk state only for large N, as plotted in Fig. 2(c).

According to Eqs. (18) and (19), the reflectance of interface states in the composite structure can be described as

(24)Rcom_inter=rPCIx−rPCIIx21−rPCIxrPCIIx2.

When x = 0.5, combining Eqs. (15), (21), and (22) we have r_PCI (x) = r^*_PCII (x), so |r_PCI(x)| − |r_PCII(x)| = 0 at the interface states [Fig. 3(c)]; hence the reflectance of the interface states must be zero, independent of the number of the unit cells N. However, when x ≠ 0.5, |r_PCI(x)| − |r_PCII(x)| ≠ 0 at the interface states; hence the reflectance of interface states is nonzero, as plotted in Fig. 3(f) for x = 0.3.

3.3. Robustness of interface states

For practical applications, randomness is inevitable in the preparation of photonic structures. The layer thickness is most susceptible to minor errors due to fabrication uncertainties, even as it has a crucial role in regulating the interface states in PCs. Accordingly, the robustness of the interface states against random thickness is significant, for both simulations and experiments. To study the robustness of the interface states of the 1D composite structures described above, we add Gaussian noise to the layer thickness. N = 50 is enough to resemble the bulk states for both PC I and PC II in the model, as shown in Fig. 2(c). In addition, the total number of layers for each material (A and B) is 99 in the composite structure.

The layer thicknesses of materials A and B are written as

(25)dai=da+δai,dbi=db+δbi,

where δa_i and δb_i are the variations due to the random Gaussian noise, with the constraint below:

(26)σ δai/σ δbi=da/db.

Here σ(δa_i) is the standard deviation, defined as

(27)σδai=1n∑ i=1 nδai−δai 2,

where n is the number of δa_i and <δa_i> is the average value. δb_i follows the same notations. We calculate 500 realizations with n = 49,500 for each noise level.

The robustness of interface states against different noise levels is investigated for both inversion (Figs. 4 and 5) and non-inversion symmetry (Figs. 6 and 7). Gaussian-distributed values of δa_i / d_a for different noise levels and boundary factors are listed in Figs. 4(a), 5(a), 6(a), and 7(a). δb_i / db has the same distribution as δa_i / d_a. The Poisson-like distributions for each of the reflection dips, representing the interface states under the same conditions, are listed in Figs. 4(b), 5(b), 6(b), and 7(b). These dips concentrate at small reflectance values, approaching the zero reflection dip for ideal interface states. However, reflection dips are much larger than zero for high noise levels [Figs. 5(b) and 7(b)]. To identify the presence of the interface states, we define a criterion for the interface-state reflection cutoff level Ro, below which interface states may appear. Two such cutoffs are determined as

Figure 4.Interface states under conditions of x = 0.5 and σ(δa_i / d_a) = 0.03000. (a) The distribution of δa_i / d_a. (b) The distribution of interface states for the reflectance dip. (c) The cumulated sum of the reflectance dip. The blue dotted and red dashed lines are cutoffs for 1.8 SD and 2.0 SD respectively. (d)−(f) Realizations of the reflectance of the composite structure at different reflectance dips corresponding to the arrows and letters (d, e, and f) in (b).

Figure 5.Interface states under conditions of x = 0.5 and σ(δa_i / d_a) = 0.04820. (a) The distribution of δa_i / d_a. (b) The distribution of interface states for the reflectance dip. (c) The cumulated sum of the reflectance dip. The blue dotted and red dashed lines are cutoffs for 1.8 SD and 2.0 SD respectively. (d)−(f) Realizations of the reflectance of the composite structure at different reflectance dips corresponding to the arrows and letters (d, e, and f) in (b).

Figure 6.Interface states under conditions of x = 0.3 and σ(δa_i / d_a) = 0.01506. (a) The distribution of δa_i / d_a. (b) The distribution of interface states for the reflectance dip. (c) The cumulated sum of the reflectance dip. The blue dotted and red dashed lines are cutoffs for 1.8 SD and 2.0 SD respectively. (d)−(f) Realizations of the reflectance of the composite structure at different reflectance dips corresponding to the arrows and letters (d, e, and f) in (b).

Figure 7.Interface states under conditions of x = 0.3 and σ(δa_i / d_a) = 0.03300. (a) The distribution of δa_i / d_a. (b) The distribution of interface states for the reflectance dip. (c) The cumulated sum of the reflectance dip. The blue dotted and red dashed lines are cutoffs for 1.8 SD and 2.0 SD respectively. (d)−(f) Realizations of the reflectance of the composite structure at different reflectance dips corresponding to the arrows and letters (d, e, and f) in (b).

(28)R01.8=mean+1.8SD and R02.0=mean+2.0SD,

where mean = <R₀> is the average reflectance value and SD=1m∑ i=1 mR0−R02 is the standard error of the 500 realizations. A larger cutoff value yields a shallower reflection dip. It is worth noting that the 2.0 SD cutoff contains more data than that of the 1.8 SD cutoff. The reflectance spectrum of the composite structure corresponding to three different realizations, marked by the arrows plus letters (d, e, and f) in Figs. 4(b), 5(b), 6(b), and 7(b), are listed in Figs. 4(d)–4(f), 5(d), 5(d)–5(f), 6(d), 6(d)–6(f), and 7(d)–7(f) respectively. For those four cases, the interface states are obvious for small σ(δa_i), but are insufficiently explained for large σ(δa_i). For more objective confirmation, we propose a second criterion, by which interface states will be regarded as robust against a given noise level σ(δa_i) if the cumulative sum of the reflectance dips is more than 95%, for a cutoff given by Eq. (28).

For x = 0.5, when σ(δa_i / d_a) = 0.03000 only the 2.0 SD cutoff satisfies the requirement for the cumulative sum of the reflectance dips to be more than 95% [Fig. 4(c)]. Therefore, the interface states are regarded as robust for the 2.0 SD cutoff but not for the 1.8 SD cutoff. When σ(δa_i / d_a) = 0.04820, neither cutoff satisfies the second criterion [Fig. 5(c)]; thus the interface states are regarded as not robust. In addition to the interface states, extra dips emerge inside the band gap, as shown in Figs. 5(d)–5(f), which could also be used as evidence against the robustness of the interface states. As for the non-symmetric cases, e.g. x = 0.3, the simulation results are similar according to Figs. 6 and 7. The interface states are robust for the 2.0 SD cutoff with a low noise level [σ(δa_i / d_a) = 0.01506], yet not robust for either cutoff with a high noise level [σ(δa_i / d_a) = 0.03300].

In terms of both cutoff levels and cumulative-sum criteria, we map out the robust interface states for the composite structures of various x in Fig. 8. It is illustrated that more interface states are identified as robust for the 2.0 SD cutoff than for 1.8 SD. Importantly, the inversion-symmetric case (x = 0.5) is more robust than the non-inversion-symmetric cases (x ≠ 0.5). Notably, the robustness decreases as the interface states move towards the band edges. This phenomenon is well presented in Fig. 8(c), which shows the robustness versus the shift of interface states using x = 0.5 as a reference. The interface states for the symmetric case are right at the center of the band gap, and are robust for noise levels up to approximately 0.03. For the non-inversion-symmetric case, the robustness decreases linearly and the interface states become unstable/not robust for any noise at the band edges. Consequently, the robustness is highly dependent on the location of the states inside the band gap, i.e. the interface states become more stable toward the center. Above all, the dependence is almost linear, suggesting that the non-inversion-symmetric PC’s interface states could be analyzed by linear expansions about the inversion-symmetric case.

Figure 8.Results for the average wave number of the interface states at the first band gap, for cutoffs at (a) 1.8 SD and (b) 2.0 SD, for different x as shown in the top legend. (c) σ(δa_i / d_a) versus the absolute shift of the interface states relative to that of x = 0.5, for the 2.0 SD cutoff.

In summary, the characteristics of the interface states in composite photonic structures consisting of two PCs have been illuminated by means of the transfer-matrix method. Interface states could be realized at the centers of the photonic band gaps for the inversion-symmetric case, but off-center toward the band edges for the non-inversion-symmetric case. We further examined the robustness of the interface states against noise by adding randomness to the layer thickness, and found that the interface states for the inversion-symmetric case were robust against random noise, as they were deep inside the band gaps and topologically protected while they were more sensitive to random noise for the non-inversion-symmetric case. More importantly, the robustness of the interface states scaled linearly with the relative location inside the band gap with respect to that of the inversion-symmetric case, and became unstable near or at the band edges. This work may be helpful for corresponding experimental fabrication, and has potential value in interface-state-relevant applications such as narrow-band filters.

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, which may be obtained from the authors upon reasonable request.

National Natural Science Foundation of China (grant numbers: 11874232, 12174211, 61905127); Hong Kong RGC grants (grant numbers: AoE P-02/12, C6013-18G).