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Abstract
We propose a new scheme to realize topological photonic states with low group velocity (vg) and zero group velocity dispersion (zero-GVD) based on a sandwiched photonic crystal (PC) waveguide, which are composed of finite sized PCs with different topological phases. In our proposed sandwiched heterostructure, two coupled topological edge states (CTESs) can be found, one of which can be applied to slow light by modifying the radii of the dielectric rods. The slow light characteristics of CTES, including zero-GVD, large average group index and normalized delay-bandwidth product, are discussed in detail based on finite element method (FEM) simulation. Besides, the robustness of CTES with slow light is verified, when introducing random disorders. Time-domain simulation results demonstrate the dispersionless transport of CETS in zero-GVD region. Our findings pave a way of topological slow light, enrich the topological PC research, and have new application in optical buffers and optical delay lines.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the past few decades, researchers have been working on how to control the speed of light effectively in dispersive optical media. To enhance the interaction between light and matter, reducing the speed of light pulses to very low levels, called slow light, is considered a promising solution for optical caching and optical communication. Slow light can be achieved in several ways, including stimulated Brillouin scattering [1,2], coupled-cavity waveguides [3–5], and photonic crystals (PCs) [6,7]. PCs are not only small and easy to achieve high-density integration [8,9], but also can work at room temperature and achieve large bandwidth and low dispersion [10,11]. Compared with other ways to achieve slow light, the PC waveguide has greater application potential [12].
When topological phases were discovered in condensed matter physics [13–18], the concept of topological insulator (TI) was introduced into photonics and phononics [19–22]. Topologically protected edge states of TI benefit from the backscattering immunity and good robustness [23–25], and provide a good platform for realizing many potential applications, including optical waveguides [26–29], topological lasers [30,31] and so on. However, the edge state of topological insulator usually behaves as the fast optical mode, which brings challenges to the realization of topological slow light waveguides with high group index [32]. Y Oshimi et al. proposed a boundary modification method based on topological valley photonic crystals, which supports topological kink mode with group index exceeding 100 [33]. Other methods are also achieved for topological slow light, by using the coupling effect in magneto-optical photonic crystal waveguide [34–36] and winding edge state many times around the Brillouin zone [37,38]. Thus, introducing topological protection into slow-light devices would greatly improve the performance being insensitive to undesired backscattering [39].
In this work, we propose a new method to achieve topological slow light with zero group velocity dispersion (GVD), based on a sandwiched photonic crystal. This sandwiched heterostructure are composed of three photonic crystals with different topological phases, which generates two coupled topological edge states (CTESs). By decreasing the radii of the dielectric cylinders, the CTES of high frequency can achieve slow light with zero-GVD, large average group index and normalized delay-bandwidth product. Finite element method (FEM) simulation results show the characteristics and original of slow light in detail. Besides, the robustness of CTES with slow light is verified, and dispersionless transport in zero-GVD region is demonstrated based on the time-domain simulation. Our study paves a new way for topological slow light waveguides, being applied in devices in optical buffers and optical delay lines.
2. Sandwiched heterostructure and flat band of CTESs
Figure 1(a) shows the original PC with square lattice geometry, and each unit cell consists of four dielectric rods. The lattice constant is a, and the spacing between the nearest adjacent dielectric rods is a/2. Radius of the rod is r, and the relative dielectric permittivity is ε=11.7. Here we just consider TM mode [40]. Δ is the moving distance of dielectric rods along coordinate axis direction (x or y direction), and Δ=0 represents the original PC. Figure 1(b) shows the band structure of the original PC for the TM polarization mode. The first band and second band are degenerate at the X point with no bandgap. Δ>0 (Δ<0) represents the expanded (shrunken) unit structure, which is indicated by blue (red) arrows in Fig. 1(a), respectively. According to the generalized two-dimensional Su-Schrieffer-Heeger (SSH) model [41–43], Δ could control the inter-cell and intra-cell coupling strength to obtain different topological phases.
Fig. 1. (a) The 2D square photonic crystal with primitive unit cell in the inset. (b) Band structure TM mode for Δ=0. (c) Band structure for Δ=0.125a (left) and Δ=-0.125a (right), which represents expanded and shrunken lattice, respectively. (d) The electric field distributions Ez of the first band (lower panels) and the second band (upper panels) in the unit cell with Δ= ± 0.125a at X point. Odd mode is marked as px mode and even mode is marked as s mode.
We calculate the band structure of TM polarization mode when Δ=0.125a and Δ=-0.125a in Fig. 1(c). In both cases, PCs have the same band structure, and there is a full bandgap (blue region) between the first and second bands in the frequency range of 0.2214∼0.3347(2πc/a), but topological properties are opposite. In a further way, we calculate Ez field at high symmetry points in terms of s mode and px mode as shown in Fig. 1(d), when s mode has mirror symmetric distribution (even mode) and px mode has mirror antisymmetric distribution (odd mode). It is found when Δ changes from 0.125a to -0.125a, the parities of the first and second bands indicates that the bands are inverted, and topological phase transition occurs.
To characterize the topological properties of PCs, 2D Zak phase Z = (Zx, Zy) is introduced as the topological invariant based on the 2D bulk polarization:
where j = x or y. ${\hat{A}_j}({k_x},{k_y}) = i\left\langle {u({\bf k})} \right|{\partial _{k\textrm{j}}}|{u({\bf k})} \rangle $ is the Berry connection with the periodic Bloch function $|u({\mathbf k})$. The 2D Zak phase can also be calculated by the 2D bulk polarization via Zj =2πPj, which is related to the parities of Bloch mode at the Brillion zone center and boundary [44]. When the artificial atoms have mirror symmetry along the j direction, Zj is quantized to 0 or π. Meanwhile, since Bloch mode at X and Y points have the same symmetry, Zx and Zy have equal values. Obviously, we can get P = (0, 0) and Z = (0, 0), when Δ=-0.125a, which indicates ordinary insulator (OI) with trivial phase. At the same time, we can get P = (1/2, 1/2) and Z = (π, π) when Δ=0.125a, which indicates the topological insulator (TI) with nontrivial phase.We combine PC1(TI, Δ>0) and PC2(OI, Δ<0) together, and there exists a topological edge state (TES) [19,20] due to opposite topological phase. Figure 2(a) shows one unit of PC1/PC2 structure, and the periodicity is along the x direction. Δ for PC1(PC2) is 0.125a(-0.125a), and other geometric parameters are the same with those of Fig. 1. We calculate the projected band diagram in Fig. 2(b), and an edge state (indicated by the solid red line) appears in the bandgap. The simulated |E| fields of TES with A and B points under the frequency 0.2927(2πc/a) are shown in of Fig. 2(c). Electric field amplitude rapidly decays in the bulk region along y direction, which demonstrates the localized characteristic of TES. Besides, the direction of energy transport at A point is opposite to that of B point, as shown by the energy flux in the right panel of Fig. 2(c). As well known, there are requirements, including linear dispersion, wideband, and low group velocity dispersion (GVD(=d2k/dω2)), for achieving slow light in PC. However, the dispersion of TES in PC1/PC2 structure is not monotonic, so it is not suitable for slow light waveguide.
Fig. 2. (a) Lattice composed by PC1 and PC2. (b) The bulk band and dispersion curve of TES in the projected band structure. (c) |E| field and energy flux of TES with A and B points under the frequency 0.2927(2πc/a).
To achieve slow light waveguide based on TES, we propose a sandwiched heterostructure, which is composed of PC1, PC2 and PC3, as shown in Fig. 3(a). Here, PC1 and PC3 are both in nontrivial phase with Z = (π, π), and Δ = 0.125a; PC2 are in trivial phase with Z = (0, 0), and Δ = -0.125a. The radius r of dielectric rod for PC1 and PC2 is r(=0.125a), which is the same with that of Fig. 2. The radius of dielectric rod for PC3 is indicated by r’ as shown in the inset of Fig. 3(a). As reported in our previous work, the two coupled topological edge states (CTESs) can be found, due to the coupling between TESs at the two interfaces (PC1/PC2 and PC2/PC3) [45]. The dispersions of the two CTESs can be controlled by changing r’ of PC3. When r’ is 0.125a, two CTESs (denoted by red and black lines) appear in band gap, and cross each other at kx = 0.8π/a as shown in Fig. 3(b1). Near the crossing point, we choose points 1 and 2 at two CTESs as shown in Fig. 3(b1). |E| fields of CTESs at points 1 and 2 are shown in Fig. 3(c), which show that the electric field is highly localized at interfaces and rapidly decays in the bulk region. Besides, the energy fluxes at points 1 and 2 are plotted with black arrows in Fig. 3(c), and the blue arrows can indicate the directions of the energy fluxes. It is found that the energy flux transports leftward and rightward for the CTESs at points 1 and 2, respectively. Thus, the CTESs which cross with each other, are not suitable for slowing waveguide, when r’ is 0.125a.
Fig. 3. (a) Schematics of PC1/PC2/PC3 sandwiched structure. The project bands with r’=0.125a (b1), r’=0.120a (b2), r’=0.115a (b3), r’=0.110a (b4), r’=0.105a (b5), and r’=0.100a (b6). |E| field and energy flux of CTESs at points 1(c1) and 2(c2). |E| field and energy flux of CTES1 under the frequencies ω1 = 0.329(2πc/a) (d1), ω2 = 0.332(2πc/a) (d2), ω3 = 0.335(2πc/a) (d3), ω4 = 0.338(2πc/a) (d4).
When the radius r’ of dielectric rod for PC3 decreases from 0.12a to 0.1a, the two CTESs anti-cross with each other and the gap between the two CTESs increases as shown in Figs. 3(b2)–(b6). For clarity, the CTES of high frequency is named as CTES1; and low frequency one is named as CTES2. With decreasing r’, the dispersion of CTES1 becomes flat. When r’=0.1a, the dispersion of CTES1 is flat and linear, which ensures that there is a constant group index over a wide frequency range. Thus, CTES1 is suitable for slow light, when r’=0.1a. We choose four frequencies (ω1 = 0.329(2πc/a), ω2 = 0.332(2πc/a), ω3 = 0.335(2πc/a), and ω4 = 0.338(2πc/a)) of CTES1, which are indicated by black points in Fig. 3(b6). |E| fields and the energy fluxes under these frequencies are plotted in Figs. 3(d1)–(d4), respectively. The directions of energy fluxes are indicated by blue arrows, illustrating the CTES1 mode propagates along the positive x-direction. When r’=0.125a, the unidirectional energy flux is located between the energy flux vortexes and leads to lack of the energy transport between the top and lower energy flux vortexes in Fig. 3(c1). When r’=0.1a, the unidirectional energy flux propagates along the lower interface and energy flux vortexes are both above the lower interface as shown in Fig. 3(d). The anticlockwise vortex which is at the center of the cell and the clockwise one which is at the boundary stagger in turn, leading to stronger energy transport in eight-shaped loops. The eight-shaped transport loops have longer journey than that of the unidirectional energy flux, and the long journey of energy flux makes the vg of CTES1 mode small. Thus, it provides a possible way for achieving slow light by using CTES1 mode. Besides, the dispersion of CTES2 mode is not monotonic, part of which enters bulk band. So CTES2 is not suitable for slow light waveguide.
3. Slow light characteristics and the robustness of CTES1
Next, we discuss the slow light characteristics for the CTES1, including group index ng and GVD. Based on the results of Figs. 3(b3)–(b6), we calculate ng and GVD, when r’ is chosen as 0.115a, 0.110a, 0.105a, and 0.1a, respectively. As shown in Figs. 4(a) and (b), ng could be larger than 12.5 in the considerable range, and group velocity vg is smaller than 0.08c. Besides, the range for CTES1 increases from [0.307(2πc/a), 0.316(2πc/a)] to [0.327(2πc/a), 0.339(2πc/a)], when r’ deceases from 0.115 to 0.1a, and ng reaches maximum at the boundary of Brillouin zone. Figure 4(c) shows that there are always points, where GVD equals zero (zero-GVD). Near the zero-GVD region, the CTES1 shows very small group velocity dispersion, which is suitable for slow light waveguide. Besides, the zero-GVD region can be tuned by changing r’.
For further discussion about the slow light region of CTES1, we calculate the average group index ${\bar{n}_g}$ [41,42], which is defined as:
Based on the simulation results of Fig. 4, we calculate the band width Δω, the central frequency ω0, the average group index . and NDBP, as shown in Table 1. As r’ increases from 0.100a to 0.115a, the band width Δω decreases from 0.00704(2πc/a) to 0.00434(2πc/a), the central frequency ω0 shifts from 0.3338(2πc/a) to 0.3115(2πc/a), the average group index ${\bar{n}_g}$ decreases from 12.85 to 11.33, the average group velocity ${\bar{v}_g}\textrm{ = }\left( {\frac{c}{{{n_g}}}} \right)$ increases gradually from 0.0778c to 0.0882c and NDBP decreases from 0.2694 to 0.1565. Obviously, when r’ is chosen as 0.1a, the performance of slow light band for CTES1 is best.
Table 1. Band width Δω, the central frequency ω0, the average group index ${\bar{{\boldsymbol n}}_{\boldsymbol g}}$, the average group velocity ${\bar{{\boldsymbol v}}_{\boldsymbol g}}$ and NDBP under different r’ of PC1/PC2/PC3 sandwiched structure
Considering the actual waveguide based on PC1/PC2/PC3 sandwiched structure with r’=0.1a, we set a point source which is indicated by a yellow star in Fig. 5(a1), and simulate the Ez field. The frequency of point source is 0.3322(2πc/a), which is zero-GVD and corresponding to the point in Fig. 4(b). As shown in Fig. 5(a1), the Ez field is centered at the interfaces of PC1/PC2 and PC2/PC3, and is very weak in bulk regions of PC1 and PC3. The energy flux is plotted with black arrows in Fig. 5(a2), and it illustrates the CTES1 mode propagates along the positive x-direction. The energy flux shows eight-shaped transport loops indicated by red arrows, and propagation path increases. Thus, ng of CTES1 with 0.3322(2πc/a) reaches large.
Fig. 5. (a1) Ez field and (a2) energy flux in the waveguide of PC1/PC2/PC3 sandwiched structure with frequency 0.3322(2πc/a). The source is denoted by a yellow star. |E| field at t = 300a/c (b1)((c1)), 400a/c(b2)((c2)), 500a/c (b3)((c3)) without (with) disorders. The disorders are introduced in the red dotted box and the details are shown in inset. The Gaussian pulse envelopes at I(xi = 0) and O(xo = 30a) points without disorders (d) and with disorders (e).
In order to verify the robustness of the slow light for CTES1 mode, some disorders are introduced around the interfaces of PC1/PC2 and PC2/PC3 as shown in the inset of Fig. 5(c1)–5(c3), and we modified the unit cells by changing the position and radius of the dielectric rods randomly in the red dotted box. The positions of some rods are randomized around their initial positions by distances from [-0.04a, 0.04a] in any direction and the radii changes of the dielectric rod are chosen from [-0.001a, 0.001a]. And we set the source as a Gaussian pulse centered at 0.3322(2πc/a), which is corresponding to zero-GVD, and conduct the time-domain simulation. The CTES1 mode propagates from the leftward to the rightward, and the |E| field is calculated at different times. Figures 5(b1), 5(b2), 5(b3) (5(c1), 5(c2), 5(c3)) show the |E| field at t = (300a/c, 400a/c, 500a/c) without (with) disorders, respectively. There is almost no change after introducing disorders, which verifies the robustness of topological slow light for CTES1 mode. Figure 5(d) shows the Gaussian pulse envelopes |E|2 at I(xi =0) point and O(xo =30a) point without disorder, and the time delay Δt is 384a/c. Thus, the average group index ${\bar{n}_0} = \frac{c}{{{{\bar{v}}_0}}} = \frac{{c \times \varDelta t}}{{{x_o} - {x_i}}} = 12.8$, while the correspond result in ${\bar{v}_0}$ is 0.078c being almost the same with the result of Table 1(0.0778c). Besides, the Gaussian pulse envelopes at I(xi = 0) point is almost the same with that at O(xo = 30a) point, and the Gaussian pulse propagates without distortion, which confirms zero-GVD slow light. Figure 5(e) plots the Gaussian pulse envelopes |E|2 at I(xi = 0) point and O(xo = 30a) point with disorders, and the time delay Δt is 376a/c, which is almost the same with those of Fig. 5(d). Thus, the robustness of the slow light is proved.
4. Conclusions
In conclusion, we have proposed a sandwiched heterostructure, which is composed of photonic crystals with different topological phases. By changing the radii of the dielectric rods, the dispersions of CTESs are modified, and topological slow light can be achieved by using the CTES of high frequency. Based on the FEM simulation results, the slow light characteristics are discussed in detail, and large average group index ${\bar{n}_g}$ (=12.85) and NDBP (=0.2694) can be obtained. The energy flux is also plotted, which can explain the origin of slow light mode. Besides, the time-domain simulation with frequency 0.3322(2πc/a) is conducted, which confirms zero-GVD slow light. Importantly, FEM simluation result also shows the CTES mode is topologically protected and immune to disorders. This study provides an idea for building topological flat band, which can be applied to slow light waveguide.
Funding
College Students Innovation and Entrepreneurship Training Program of Jiangnan university (2021356Y); 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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