Coupled topological edge states in one-dimensional all-dielectric heterostructures

Tongtong Wei; Tongtong Wei; Yueke Wang; Yueke Wang

doi:10.1364/OE.470783

1. Introduction

The emergence of the concept of topological photonic crystals (PCs) has inspired a great interest [1,2]. The topological edge state (TES) can appear in the topological PCs, which shows remarkable advantages in integrated optics, such as topological protection [3,4] and structural disorders immunity [5]. Various systems have been proposed to achieve TES in PCs, such as one-dimensional (1D) PCs [6–12], 2D PCs with broken symmetry [13], and 3D gyroid PCs [4,14,15]. In consideration of design and fabrication processes, more researchers are attracted to investigate the TES in 1D PCs. The TES can induce photons to transmit through the PCs despite being in the bandgap. Chan [16] et al. proposed the Zak phase to explain the origin of TES in the 1D PC heterostructure. The TES can be applied to perfect absorbers [8,11], the enhancement of nonlinear optical responses [9,10] and the topological protection of quantum correlation [17–19].

In previous reports, photonic crystals waveguide can be achieved by introducing a chain of coupled cavities proposed by Yariv et al [20]. or by forming series of coupled point or line defects [21–29], known as coupled-resonator optical waveguide. Besides, Feng et al. proposed a coupled-plasmon-resonant waveguide system based on multilayer metal-dielectric nanofilm structure [28], which can be applied to Dense Wavelength Division Multiplex (DWDM). Ding et al. proposed to use a coupled-cavity waveguide system to realize a multi-channel bistable switch [29]. Large-scale coupled nanocavities were reported by Notomi et al., based on line-defect cavity in a 2D triangular-lattice air-hole PCs to achieve ultrahigh-qulity factor [30]. It is reasonable to consider the TESs as a series of resonators in 1D PCs, and thus we provide a study of a new type coupled-resonator optical waveguide design, coupled-topological-edge-state waveguide (CTESW) in 1D heterostructures.

Here, we propose to construct a CTESW system, based on the 1D all-dielectric heterostructures composed of stacked binary PCs with opposite topological properties. The TESs can appear at each interface between the binary PCs, and the multiple transmission peaks can be found due to the excitation of CTESW modes, which originate from the coupling between a sequence of TESs. Physical mechanism of CTESW modes has been analyzed by the coupled mode theory (CMT) [29]. Besides, the CTESW modes can be tuned by changing the geometric parameters and the incident angles (under TE and TM polarization). In particular, the structure we proposed has a hopeful application in DWDM filters, which can achieve 50 GHz channel spacing.

2. Structure design and theories

Figure 1(a) shows the schematic of the 1D all-dielectric structure consisting of q periods of PC-1/PC-2. PC-1 and PC-2 are both stacked by Si(A) and SiO₂(B) layers. The configuration of PC-1(PC-2) is A_0.5B₁A_0.5 (B_0.5A₁B_0.5), both with p periods. The thickness d_A (d_B) of A (B) is 100 nm (240 nm), and the refractive indexes of Si and SiO₂ are n_A (= 3.48) [31] and n_B (= 1.44) [32], respectively. The outside of the whole structure is air. The sizes of PC-1 and PC-2 are identical, i.e. d_PC-1 = p*(d_A/2 + d_B + d_A/2) = d_PC-2 = p*(d_B/2 + d_A + d_B/2), and thus the band structures of the two PCs are also the same (as shown in Fig. 1(b)). The band structure can be obtained from the following expression [16]:

(1)$$\cos ({{K_B}\Lambda } )= \cos {k_A}{d_A}\cos {k_B}{d_B} - \frac{1}{2}\left( {\frac{{{z_A}}}{{{z_B}}} + \frac{{{z_B}}}{{{z_A}}}} \right)\sin {k_A}{d_A}\sin {k_B}{d_B}$$

where ${k_i} = \frac{{\omega {n_i}}}{c};\;{n_i} = \sqrt {{\mu _i}{\varepsilon _i}} ,\;;\;{z_i} = \frac{1}{{{n_i}}}({i = \textrm{A},\; \; \textrm{B}} );$ $\mathrm{\Lambda} \; ({ = {d_\textrm{A}} + {d_B}} )$ is thickness of the unit cell; and K_B is the Bloch wave vector. However, the configuration (A_0.5B₁A_0.5) is different from the configuration (B_0.5A₁B_0.5) based on topological symmetry [33], according to the band Zak phase of the two PCs. We have numbered the bands and bandgaps. For the m^th band, the Zak phase can be defined as [34–37]:

(2)$$\theta _m^{Zak} = \int_{ - \pi /\Lambda }^{\pi /\Lambda } {\left[ {i\int\limits_{unit\;cell} {dz\varepsilon (z)u_{m,{K_B}}^\ast (z)} {\partial_{{K_B}}}{u_{m,{K_B}}}(z)} \right]} d{K_B}$$

where $\varepsilon (z )$ refers to the dielectric function, ${u_{m,{K_B}}}$ is the periodic-in-cell part of the Bloch electric field eigenfunction of a state on the m^th band with wave vector K_B, and $[i\int_{unit\; cell} {dz\varepsilon (z )\;\,u_{m,\,{K_B}}^\ast \,(z)\,{\partial _{{K_B}}}\,{u_{m,\,{K_B}}}\,(z)\,]}$ is the Berry connection. We have numbered bandgaps in green, each isolated band in orange and the Zak phase of every band in red ‘$\pi $’ or ‘0’. Here, the Zak phase of 1^st band can be obtained by $\textrm{exp}({i\theta_1^{Zak}} )= \textrm{sgn}[{1 - ({{\varepsilon_A}{\mu_B}} )/({{\varepsilon_B}{\mu_A}} )} ],$ where μ_A, μ_B and ε_A, ε_B are the relative permeability and permittivity, respectively. For non-magnetic dielectric materials (μ_A = μ_B = 1), when the inversion center of the PC has a high refractive index (ε_A > ε_B), $\theta _1^{Zak}$ is π; on the contrary (ε_A<ε_B), $\theta _1^{Zak}$ is 0. The Zak phase at higher band can be determined by the condition sin(2πn_id_i/λ) = 0, where λ is the wavelength. The Zak phase $\theta _m^{Zak}$(m > 1) will be π if λ falls in the band; otherwise, it will be 0 [38]. The topological properties of n^th bandgap can be expressed by [16]:

(3)$$sgn[{{\xi^n}} ]= {({ - 1} )^n}{({ - 1} )^l}exp\left( {i\sum\limits_m^{} {\theta_m^{Zak}} } \right)$$

where the integer l is the number of crossing points under the n^th bandgap, and $\mathop \sum \limits_m \theta _m^{Zak}$ means the sum of the $\theta _m^{Zak}$ under the n^th bandgap. The positive (+1) and negative (-1) topological properties of each gap are denoted by pink and blue in Fig. 1(b), respectively. From the previous researches, a TES is located at the interface between two PCs, which have different topological properties in the overlapped gaps [33]. According to Fig. 1(b), we find that the topological property of PC-1 is opposite to that of PC-2 in the 1^st(2^nd) gap.

Fig. 1. (a) Schematic of the 1D all-dielectric structure consisting of q periods of PC-1/PC-2, in which PC-1(PC-2) consists of p periods of A_0.5B₁A_0.5(B_0.5A₁B_0.5). And the light illuminates from left to right. (b) Energy band structure for PC-1 and PC-2. (c) The transmission spectra of individual PC-1, individual PC-2 and PC-1/PC-2 with p = 4 are denoted by red, blue and black lines, respectively. The sharp TES peak in the 1^st gap is marked by a red pentagram. (d) Calculated electric field distribution of E_y corresponding to the TES marked by red pentagram in (c).

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Following, all numerical simulation results are obtained by the commercial software COMSOL Multiphysics based on finite element method (FEM), and the periodic boundary condition is set in the x direction. In Fig. 1(c), under the normal incidence (θ = 0°) along the positive z axis of TE polarization, the transmission spectra of the individual PC-1, individual PC-2 and PC-1/PC-2 with p = 4 are denoted by red, blue and black lines, respectively. We find the transmission peak of PC-1/PC-2 locates at 192.5 THz (1558.6 nm) in the 1^st gap, corresponding to TES marked by red pentagram in Fig. 1(c). The electric field distribution of E_y at λ = 1558.6 nm is shown in Fig. 1(d), and the electric field is confined around the interface edge between PC-1 and PC-2. Obviously, the transmission peak in Fig. 1(c) originates from the excitation of TES.

Based on the above results, we consider PC-1/PC-2 as a unit and pile up q periods of this unit. There exist 2q-1 interface edges between PC-1(PC-2) and PC-2(PC-1) in the structure (PC-1/PC-2)^q, and the interface edge of PC-1/PC-2 is equivalent to that of PC-2/PC-1. Thus, 2q-1 TESs are identical and can be excited at every interface between PC-1 and PC-2. Due to the coupling between the TESs, the mode splitting of TES happens and 2q-1 CTESW modes can be formed [17,27]. For example, we take (PC-1/PC-2)^q with p = 4 and q = 4 for example, and the transmission spectrum is shown in Fig. 2(a). Obviously, the bandgaps of (PC-1/PC-2)^q with p = 4 and q = 4 are similar with Fig. 1(c). However, there are 7(i.e. 2q-1) transmission peaks in the 1^st gap, respectively. For clarity, a zoom-in of the transmission peaks in the 1^st gap is given in Fig. 2(b). The resonate wavelengths are 1532.2 nm, 1538.1 nm, 1547.2 nm, 1558.6 nm, 1570.9 nm, 1582.0 nm and 1589.8 nm, obtained by FEM, respectively. It is believed that these peaks are attributed to the excitation of CTESW modes, which originate from the coupling between TESs (locating at 1558.6 nm).

Fig. 2. (a) The transmission spectrum of (PC-1/PC-2)^q with p = 4 and q = 4. (b) A zoom-in of the transmission peaks corresponding to the CTESW modes in the 1^st gap of (a). (c) |E| distribution in the structure for the resonant wavelengths λ₁∼λ₇ in (b) calculated using FEM (solid lines) and the envelope of superposition coefficients (black lines with balls) based on Eqs. (4) and (7). The positions of the interfaces between PC-1and PC-2 are also shown by the red mark on the horizontal axis.

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Based on CMT [39], the electric field of CTESW modes can be expressed as a linear superposition of 2q-1 individual TESs. The electric field E(λ_i, z) of CTESW modes with λ(=λ₁, λ₂,…λ_2q-1) may be described as follow:

(4)$$\textbf{E}({\lambda _i}\boldsymbol{,z}) = \sum\limits_{m = 1}^{2q - 1} {A_m^{}({\lambda _i})} \textbf{E}_{\lambda 0}^{}(\boldsymbol{z} - m{z_0}{\textbf{e}_{\boldsymbol{z}}})$$

where A_m(λ_i) is a complex coefficient that determines the amplitude of the TES at the m^th interface edge for CTESW modes with λ_i. E_λ₀(z - mz₀e_z) is the electric fields of the TES centered at mz₀e_z, where mz₀ represents the coordinate of the m^th interface edge, and e_z represents the direction of z axis. λ₀ is the resonate wavelength of an individual TES (1558.6 nm). In the CMT [39], considering the condition of nearest neighboring coupling, the 2q-1 resonate wavelengths of the CTESW modes can be calculated by:

(5)$${\lambda _i} = {\lambda _0}\sqrt {\frac{{{C_{^i}} - \alpha + {C_{^i}}\Delta \alpha }}{{{C_{^i}} - \beta }}}$$

where C_i is linear superposition coefficients, which is calculated as Eq. (6).

(6)$${C_i} ={-} {[2\cos (\frac{{i\pi }}{{2q}})]^{ - 1}}$$

Here, i is 1, 2, … 2q-1. A detailed calculation of the coupling parameters ($\alpha $, $\beta $ and $\Delta \alpha $) can be seen in Appendix. Based on Eq. (5), we can calculate the 2q-1 resonate wavelengths λ_i when p = 4 and q = 4. The resonate wavelengths are 1531.1 nm, 1537.4 nm, 1547.0 nm, 1558.6 nm, 1570.6 nm, 1581.0 nm and 1588.1 nm, obtained by CMT, respectively. These results obtained by CMT have a good agreement with those obtained by FEM. For the i^th CTESW mode, we can also obtain the normalized amplitude a_m(λ_i) (=|A_m(λ_i)|) [26] for the TES at m^th interface edge, which is:

(7)$$a_m^{}({\lambda _i}) = \textrm{|}\sin (\frac{{mi}}{{2q}}\pi )\textrm{|}$$

Figure 2(c) shows the field profiles of normalized |E| for the CTESW modes of λ₁∼λ₇ calculated by FEM (solid lines), it is found the field profiles of normalized |E| for CTESW mode of λ_i is the same with that of λ_2q-i. The electric field normalized |E| of CTESW modes at the interfaces are plotted by black lines with balls, based on by CMT (Eqs. (4) and (7)). Clearly, the results of CMT agree well with those of the FEM. For convenience, the positions of the interfaces between PC-1(PC-2) and PC-2(PC-1) are also shown by the red mark on the horizontal axis in Fig. 2(c). From Eq. (7), we can find that a_m(λ_i) = a_m(λ_2q-i), and thus |E(λ_i, z)|= |E(λ_2q-i, z)| at the interfaces based on Eq. (4).

3. Results and discussion

In Fig. 3, we discuss the dependence of the transmission spectrum on q of the all-dielectric structure (PC-1/PC-2)^q, in which PC-1(PC-2) consists of p periods of A_0.5B₁A_0.5(B_0.5A₁B_0.5). Other parameters are the same with those of Fig. 2. Especially, in Figs. 3(a₁) and 3(b₁) with q = 1, the transmission spectrum of PC-1/PC-2 (black solid line) is the same with that of PC-2/PC-1 (red dashed line). Obviously, the interface edge of PC-1/PC-2 is equivalent to that of PC-2/PC-1. In Figs. 3(a₂)–3(a₅) and 3(b₂)–3(b₅), a number of sharp transmission peaks are observed in the spectrum, and the peak values are always one. The transmission peaks originate from the excitations of CTESW modes, the number of which equals to the number(2q-1) of interface edges. In Figs. 3(a₁)–3(a₅) for p = 4, the range of the resonate wavelengths increases from [1555.9 nm, 1561.3 nm] of (a₁) to [1527.6 nm, 1589.6 nm] of (a₅), the Full Width Half Maximum (FWHM) of the central transmission peak decreases from 1.81 nm of (a₁) to 0.18 nm of (a₅), when q increases from 1 to 10. Similarly, in Figs. 3(b₁)–3(b₅) for p = 8, the range of the resonate wavelengths increases from [1558.62 nm, 1558.66 nm] of (b₁) to [1557.44 nm, 1559.85 nm] of (b₅), the FWHM of the central transmission peak decreases from 2.81 × 10⁻³ nm of (b₁) to 4.39 × 10⁻⁴ nm of (b₅), when q increases from 1 to 10. Compared Figs. 3(a_i) with 3(b_i) under the same q, the range and the FWHM of the transmission peaks are both larger, for the smaller p. It is because the coupling strength is stronger and the mode splitting is larger, with decreasing p. Besides, the central transmission peak locates around 1558.6 nm, which is corresponding to the individual TES. Thus, owing to the CTESW modes, the multiple transmission peaks can be tuned by changing p and q of (PC-1/PC-2)^q. In a further way, we give the quantitative analysis as shown in Figs. 3(c)–3(d), including the quality factor (Q = λ/FWHM) of the central transmission peak (black lines with balls) and the window width of transmission peaks (corresponding to the range of the resonate wavelengths, red lines with balls). Figure 3(c) shows the Q factor increases from 8.61 × 10² to 8.58 × 10³ for p = 4, and Fig. 3(d) shows the Q factor increases from 5.57 × 10⁵ to 5.47 × 10⁶ for p = 8, when q increases from 1 to 10. Figures 3(c)–3(d) show window width increases to 62 nm and 2.4 nm, with q increasing from 1 to 10, respectively. The reason is that more CTESW modes are formed with increasing q, which leads to the window width increasing. When q is larger than 4, the window width changes little.

Fig. 3. (a_i) The transmission spectrum of (PC-1/PC-2)^q with p = 4 and q = 1 in (a₁), 2 in (a₂), 3 in (a₃), 4 in (a₄) and 10 in (a₅). (b_i) The similar results with (a_i) but p = 8. Especially, both (a₁) and (b₁) show the results with q = 1, and the transmission spectrum of PC-1/PC-2(PC-2/PC-1) is plotted by black solid line (red dashed line). (c) The Q factors (black lines with balls) of the central transmission peak and the window width (red lines with balls) of (PC-1/PC-2)^q varying with q when p = 4. (d) The similar results with (c) but p = 8.

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Next, we discuss the influences of p on the transmission properties of (PC-1/PC-2)^q with fixed q = 4 and 8, respectively. Other parameters are the same with those of Fig. 2. In Figs. 4(a₁)–4(a₄) and 4(b₁)–4(b₄), a number of sharp transmission peaks are also observed, the number of which equals to the number(2q-1) of interface edges, when p increases from 3 to 6. In Figs. 4(a₁)–4(a₄) for q = 4, the range of the resonate wavelengths decreases from [1501.2 nm, 1638.2 nm] of (a₁) to [1553.1 nm, 1564.4 nm] of (a₄), and the FWHM decreases from 2.31 nm of (a₁) to 0.018 nm of (a₄), when p increases from 3 to 6. Similarly, in Figs. 4(b₁)–4(b₄) for q = 8, the range decreases from [1498.3 nm, 1642.4 nm] of (b₁) to [1552.5 nm, 1564.6 nm] of (b₄), the FWHM decreases from 1.2 nm of (b₁) to 9 × 10⁻³ nm of (b₄), when p increases from 3 to 6. The coupling distance increases with increasing p, so the coupling strength is weaker and the mode splitting is smaller. Thus, FWHM decreases with increasing p. When p is larger than 8, the coupling distance between CETSs is so large that the coupling strength is too small. Thus, CTESW modes can’t appear. In a further way, Figs. 4(c)–4(d) show the Q factor and the window width varying with p when q = 4(Fig. 4(c)) and q = 8(Fig. 4(d)), respectively. In Fig. 4(c), the Q factor increases from $6.77 \times {10^2}$ to $2.2 \times {10^6}$ with p increases from 3 to 8, but the window width decreases from 137 to 2.25nm. Similarly, in Fig. 4(d), the Q factor increases from $1.3 \times {10^3}$ to $3.9 \times {10^6}$, but the window width decreases from 144 to 2.39 nm. Thus, owing to the CTESW modes, the multiple transmission peaks can be tuned by changing p and q of (PC-1/PC-2)^q.

Fig. 4. (a_i) The transmission spectrum of (PC-1/PC-2)^q with q = 4 and p = 3 in (a₁), 4 in (a₂), 5 in (a₃), and 6 in (a₄). (b_i) The similar results with (a_i) but q = 8. The Q factors and the window width varying with p when q = 4 (c) and q = 8 (d).

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All the above results are obtained under the normal incidence (θ = 0°) of TE polarization. As the resonant wavelength of TES is related to the incident angle under TE and TM polarization [11,12], CTESW modes can also be tuned by the incident angle. Here, other geometric parameters of the structure are the same as those of Fig. 2. The transmission spectra of PC-1 or PC-2 as a function of the wavelength and the incident angle under TE and TM polarization are shown in Figs. 5(a) and 5(d). It is found that the bandgap blue shifts from [1033.3 nm, 2051.4 nm] to [887.6 nm, 1942.2 nm] for TE polarization, and from [1033.3 nm, 2051.4 nm] to [949.2 nm, 1691.6 nm] for TM polarization, when the incidence angle increases from 0° to 60°. The transmission spectra of (PC-1/PC-2)^q with p = 4 and q = 4 as a function of the wavelength and the incident angle under TE polarization are shown in Fig. 5(b). Obviously, the transmission peaks due to CTESW modes blue shift with increasing incident angle, in the photonic bandgap. For clarity, the transmission curves under the different incident angles θ (= 0°, 10°, 20°, 30°, 40°,50°, 60°) under TE polarization are presented in Fig. 5(c). The range of the CTESW modes vary from [1532 nm, 1590 nm] to [1432 nm, 1468 nm] under TE polarization, and the transmission peaks become denser. The transmission spectra of (PC-1/PC-2)^q with p = 4 and q = 4 under TM polarization are shown in Figs. 5(e) and (f). Similarly, the transmission peaks due to CTESW modes blue shift with increasing incident angle, and the range of the CTESW modes varies from [1532 nm, 1590 nm] to [1284 nm, 1360 nm]. The window width of transmission peaks become wider, and the transmission peaks become sparser. Thus, the CTESW modes can be easily tuned by changing incident angle.

Fig. 5. (a) The transmission spectra of PC-1 or PC-2 as a function of the wavelength and the incident angle under TE polarization. (b) The transmission spectra of (PC-1/PC-2)⁴ as a function of the wavelength and the incident angle under TE polarization. (c) The transmission curves under the different incident angles θ (= 0°, 10°, 20°, 30°, 40°,50°, 60°) in (b). (d)-(f) are the similar results with (a)-(c) respectively, but under TM polarization.

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In general, there are 2q-1 transmission peaks in the structure (PC-1/PC-2)^q. As shown in Figs. 3 and 4, the Q factor and window width of the transmission peaks can be tuned by the p and q of (PC-1/PC-2)^q. When q is larger than 4, the window width keeps almost constant. Thus, we can construct DWDM filters by increasing q in near-infrared range. A DWDM filter is designed based on CTESW modes of (PC-1/PC-2)^q with p = 4 and q = 110, and the transmission spectrum from 1545 nm to 1565 nm is shown in Fig. 6. All the transmission peaks can reach one. The FWHM, Q factor, and channel spacing are 0.014nm, 1.1 × 10⁵, and 50 GHz, respectively. It is well known that the larger the channel spacing, the weaker the signals interference with each other. However, the larger the channel spacing, the lower the frequency spectrum utilization. While communication quality and spectrum are both important, so it is often necessary to balance the channel spacing and the frequency spectrum utilization. To solve this problem, the density and the FWHM of the DWDM filter can be tuned by changing p and q in the (PC-1/PC-2)^q. Compared with the previous works [28,30,40], the DWDM constructed by the proposed CTESW is lossless and smaller.

Fig. 6. A DWDM filter of 50 GHz channel spacing using CTESW modes of (PC-1/PC-2)^q with p = 4 and q = 110. The plot shows the transmission spectrum from 1545 to 1565 nm.

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4. Conclusion

Based on basic 1D topological photonic crystal PC-1/PC-2, we propose a CTESW system constructed by stacked q periods of PC-1/PC-2. In the all-dielectric structure, a series of TESs locate at the interface edges of two neighboring PCs. We find that the TESs couple with each other, and thus CTESW modes are achieved. By changing p and q in the (PC-1/PC-2)^q, tunable multiple transmission peaks are obtained. The FEM simulation results agree with those of CMT. Besides, the CTESW modes can also be tuned by changing incident angle from 0° to 60° under TE and TM polarization. Moreover, the structure we proposed has a hopeful application in DWDM filters, whose channel spacing can be tuned by the geometry parameters of the structure. Our method has a good help to balance the relationship between channel spacing and the frequency spectrum utilization in the application of DWDM filters.

Appendix. The detailed calculation of the coupling parameters in CMT

Here, we give the detailed calculation of the coupling parameters ($\alpha $, $\beta $ and $\Delta \alpha $) in CMT as follow. The schematic of the 1D all-dielectric structure consisting of PC-1/PC-2/PC-1 is shown in Fig. 7(a), i.e. there are only two interface edges (PC-1/PC-2 and PC-2/PC-1) in the entire structure. p of PC-1(PC-2) is chosen as 4. The transmission spectrum of PC-1/PC-2/PC-1 is shown in Fig. 7(b). There are two corresponding transmission peaks located at λ₁ and λ₂. The transmission peaks originate from the excitation of CTESW modes due to the coupling between TESs at PC-1/PC-2 and PC-2/PC-2 interfaces respectively. The two peaks are at 1543.5 nm and 1574.0 nm, calculated by FEM. Taking λ₁ and λ₂ into Eq. (8), we can obtain $\alpha $ = 0.0034 and $\beta $ = 0.0232, respectively. Here we assume that $\Delta \alpha $ is negligible [22,29].

(8)$${\lambda _{1,2}} = {\lambda _0}\sqrt {\frac{{1 \pm \alpha + \Delta \alpha }}{{1 \pm \beta }}}$$

Fig. 7. (a) Schematic of the 1D all-dielectric structure consisting of PC-1/PC-2/PC-1, in which PC-1(PC-2) consists of 4 periods of A_0.5B₁A_0.5(B_0.5A₁B_0.5). And the light illuminates from left to right. (b) The transmission spectrum of PC-1/PC-2/PC-1 under normal incidence (θ = 0°).

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Funding

National Natural Science Foundation of China (1148081606193050).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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Coupled topological edge states in one-dimensional all-dielectric heterostructures

Abstract

1. Introduction

2. Structure design and theories

3. Results and discussion

4. Conclusion

Appendix. The detailed calculation of the coupling parameters in CMT

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (8)

Optics Express