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Abstract
We propose a new kind of reconfigurable topological valley photonic crystal (TVPC), and a novel topological waveguide can be formed by constructing a domain wall between two TVPCs with opposite valley-Chern indices. The topological waveguide mode in the composite TVPC has large group refractive index. A topologically protected coupled waveguide cavity system is then designed by introducing a hexagonal ring cavity at the center of the straight domain wall of a combined TVPC, in which a narrow plasmon induced transparency window rises at 3.8848 GHz with a Q-factor of 1387 and a maximum group refractive index as high as 186. We propose a notch filter with a resonant frequency of 3.8852 GHz and a very high Q-factor of 10224. By changing the refractive index of liquid crystals via an external voltage applied between two parallel metal plates, the filter can be switched between band-pass and band-stop based on the reconfigurable topological interface state.
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1. Introduction
The valley used in valleytronics is a binary degree of freedom usually occurring in a two-dimensional honeycomb lattice [1–7]. In analog to spin in spintronics [8], topological transmission based on the valley degree of freedom is a new way to transmit information and energy, which does not require a strong spin-orbit interaction demanded by spintronic topological transport [9]. It can be actively controlled in a variety of two-dimensional materials through the electric field which determines the sign of Berry curvature at the valley K (K’) of the Brillouin zone. The emerging field of topological photonics opens new opportunities to realize novel topological effects in new ways [10–18]. In order to achieve more kinds of light field control and to further reduce the requirements of material properties and external conditions by topological non-trivial states, the valley degree of freedom is introduced into photonic crystals [19,20]. One of their attractive applications is topological valley transmission, which can realize the robust propagation of confined photons.
Photonic crystal waveguides offer wavelength flexibility, because their working frequencies are defined by their lattice constants. However, the performance of conventional photonic crystal waveguides and cavities is easily affected by defects and impurities caused by actual fabrication errors. The optical waveguides based on topological interface states can overcome these disadvantages and immune fabrication errors [21–24]. They have been applied to splitters [25], logic gates [26,27] and lasers [28,29] with significantly improved device performance. On the other hand, strong localization can be achieved in optical cavities. Therefore, a coupled waveguide cavity system formed by combining a waveguide and a cavity together will possess strong localization and novel transmission property, which have great potential to work as Fano resonance, band-pass/band-stop filter, refractive index sensor, optical switch, logic gates, delay lines, nonlinear devices, slow-light device, etc. Coupled waveguide cavity systems based on photonic crystals have been widely used in filters [30,31], sensors [32,33], lasers [34,35] and other micro-nano integrated photonics devices. However, the work about topologically protected coupled waveguide cavity systems is rare so far.
Electromagnetically induced transparency (EIT) effect has great application prospects for slow light applications, super Raman scattering, Kerr effect, optical storage, quantum switch and sensing technology [36–40]. However, the experimental conditions to achieve the quantum EIT effect are rigorous and the selectable materials are very limited. In recent years, plasmon induced transparency (PIT) devices based on metamaterials have attracted wide interest due to their flexible design and simple implementation conditions. Benefiting from destructive interference between bright and dark modes, a variety of optical structures have been proposed to realize optical analogies of EIT [41–51] and PIT [43,44]. However, even though the ohmic losses in most plasmonic metals are quite low, it is difficult to achieve the induced transparency peak with high quality factor due to the high radiation losses [52].
2. Design and research method
In this paper, we propose a new kind of reconfigurable topological valley photonic crystal (TVPC) based on liquid crystal. The interface states with TE and TM polarizations can be selectively excited. Then a novel waveguide is designed in a composite TVPC by constructing a domain wall between two TVPCs with opposite valley-Chern indices. The structure of 25 supercells has the largest group refractive index among four calculated structures of different number of supercells. The PIT effect is realized by introducing a cavity defect mode into the domain wall with a narrow transparent window located at 3.8848 GHz and a Q-factor equal to 1387. The maximum group refractive index is as high as 186. Then we propose an ultra-narrow-band notch filter, whose transmission valley locates at 3.8852 GHz with a very high Q-factor equal to 10224. By changing the director orientation of liquid crystals, the filter can be switched between band-pass and band-stop with the reconfigurable topological interface state. Because the proposed composite TVPCs possess both strong localization and robust transmission, they have great potential to work as PIT window, Fano resonance, band-pass/band-stop filter, refractive index sensor, optical switch, logic gates, delay lines, nonlinear devices, slow-light device, etc. Our structures provide a very meaningful guide for the design of reconfigurable topological valley photonic crystals by combining topological waveguide and cavity into a composite TVPC. We performed numerical simulations using COMSOL Multiphysics based on the finite element method.
As shown in Fig. 1(a) and (c), both valley photonic crystals have a triangular lattice, and their unit cells are composed of a metal rod suspended between two parallel metal plates. Two metallic cover plates are used to mimic the PEC boundaries and prevent the TM modes from leaking into air. Air gaps usually have two purposes. One is to control the orientation of the liquid crystal by applying a voltage between the upper and lower metal plates. The other is to apply the near-field scanning technique in the experimental implementation [53]. The air gap will decrease the effective relative dielectric permittivity of the TVPC and lead to higher eigenfrequencies. Figure 1(b) and (d) show the energy band diagrams of these two kinds of photonic crystals with cylindrical lattice points and equilateral triangular lattice points, respectively. Because the geometry of the triangle breaks the inversion symmetry, a bandgap (3.78-3.94 GHz) appears near the valley K (K’) point. Field distributions of the eigenmodes near the valley K point (corresponding to the energy bands with labels 1 to 4 in Fig. 1(d)) are shown in Fig. 1(e). The Poynting vectors of the TE eigenmodes of band 1 and band 3 rotate counterclockwise and clockwise along the triangle respectively, which shows that the valley degree of freedom of photons corresponds to the orbital angular momentum in analogy to the valley degree of freedom in the electronic system. While the Poynting vectors of the TM eigenmodes of band 2 and band 4 rotate counterclockwise and clockwise along the triangle, respectively. These mean that polarization can be used as a degree of freedom independent of the valley degrees of freedom. The valley-Chern index can be calculated by the following equations:
where F is the Berry curvature, ${\nabla _k} = ({{\partial_{kx}},{\partial_{ky}}} )$, $\overrightarrow A (k)$ is the Berry connection, ${u_k}(r)$ is the normalized Bloch wave functions calculated from numerical simulations, and C is the valley-Chern index. For TE and TM polarizations, we calculate the valley-Chern indices [15,16,54,55] CK = 1/2 and CK’ = -1/2.
Fig. 1. (a) A unit cell of the cylindrical lattice. The lattice constant is d =36.8 mm. The air gaps g = 1.1 mm are filled with a foam spacer. h = 34.6 mm, r = 0.1725d, l = 15.35 mm; (b) The bulk band structure of the cylindrical lattice; (c) A unit cell of the triangular rod lattice; (d) The bulk band structure of the triangular rod lattice; (e) The simulated field distributions of eigenmodes for corresponding bands 1 to 4, respectively. The black arrows represent Poynting vectors.
3. Results and discussions
Figure 2 illustrates a composite TVPC with a straight domain wall formed by two triangular lattices of perfect-electric-conductor (PEC) regular triangular rods with opposite valley-Chern indices and without touching upper and lower parallel PEC plates. A liquid crystal (E7) is filled between the two plates, which is a nematic liquid crystal with positive dielectric anisotropy. In the beginning, the liquid crystal director is parallel to the cover plates and perpendicular to the polarization of the incident TM wave (ordinary light). When a strong enough electric field is applied along z direction, the liquid crystal director is then parallel to the polarization of the incident TM wave (extraordinary light). In this work, we only consider the initial and the final states of the liquid crystal.
We then construct a composite structure with a Z-shaped domain wall between two TVPCs of opposite valley-Chern indices. Horizontal (vertical) phased-array dipoles are placed along the z (y) direction to emit a single TM (TE) mode at Port 1 [56]. As shown in Fig. 3(a), the triangle direction of the lower TVPC is opposite to that of the upper TVPC, so the valley-Chern index of the lower region is opposite to that of the upper region, i.e., CK = -1/2 and CK’ = 1/2 for the upper TVPC, and CK = 1/2 and CK’ = -1/2 for the lower TVPC. Due to the different valley-Chern indices between the upper and lower regions, there are two topological interface states [12–16,54–56] locked to the K valley and the K’ valley for different polarizations as shown in Fig. 3(e). These two topological boundary states (A and B) can be regarded as waveguide modes. The frequency range of the topologically protected waveguide modes is 3.74-3.97 GHz, and the stopband of the PC without the domain wall is 3.78-3.94 GHz. The other intrinsic modes in the bandgap (3.74-3.97 GHz) are ignored, because they only exist at the edges of the upper and lower sides of the combined structure. Figure 3(f) shows the transmission spectra of the two kinds of combined structures with the Z-shaped or straight domain wall. Figure 3(a-d) show the robustness of topological interface state in the absence of inter-valley scattering. The periodic peaks in the transmission spectra are due to the coherent scattering at each unit cell of the composite photonic crystal, which acts as a one-dimensional grating. The n layers of supercells arranged along the normal incidence direction can be regarded as n-1 weakly coupled Fabry-Perot cavities, to a certain extent. As shown in Fig. 3(f), the resonance peak of the original Fabry-Perot cavity is split into n-1. If the backscattered light waves from two adjacent supercells are in-phase, the constructive interference will result in a minimum transmitted amplitude. On the contrary, the destructive interference will produce a maximum amplitude. Figure 3(g) shows that clear oscillation behavior remains even when the period becomes very small. The frequency interval between neighbouring peaks becomes smaller as the period becomes larger, suggesting a strongly dispersive characteristic. The group refractive index is calculated based on the equivalent medium theory. After we calculate the group refractive indices of different composite TVPCs composed of 10, 15, 20 and 25 supercells respectively, we find that the structure of 25 supercells has the largest group refractive index (ng = 17) among these four structures. Which can be compared with the conventional photonic crystal waveguide [57] with large group refractive index. These results demonstrate that topological waveguide modes with large group refractive index can be realized in combined TVPCs.
Fig. 3. (a-d) The simulated field distributions of the structure spliced by two TVPCs of opposite valley-Chern indices with different domain walls, f = 3.86 GHz, nbg = 1.59, where (a) corresponds to Hz of the TE mode with a Z-shaped domain wall, (b) corresponds to Hz of the TE mode with a straight domain wall, (c) corresponds to Ez of the TM mode with a Z-shaped domain wall, and (d) corresponds to Ez of the TM mode with a straight domain wall; (e) Band diagram of the spliced structure with the straight domain wall and the field distributions of its strip-like supercell; (f) The transmission spectra of the two kinds of combined structures. The black and red curves are for the combined structure with the Z-shaped domain wall, and the blue and pink curves are for the structure with the straight domain wall; (g) The transmission spectra of the composite structure of the straight domain wall for different supercell cycles n.
As shown in the Fig. 4(a-d), We calculate the refraction of different interface states. The small transmission is mainly due to the reflection by the armchair output boundary of the composite TVPC. When we change the output boundary of the combined TVPC with a Z-typed waveguide from an armchair type to a zigzag type, as shown in Fig. 4(f), the transmittance of the Z-shaped waveguide increases by about 20%. This is consistent with a previous work [56], which means that strong backscattering will occur for the armchair output boundary.
Fig. 4. Topologically protected refraction of different interface states into an empty region. (a-b) The refraction of TE interface states through the armchair output boundary and the zigzag output boundary with a straight domain wall; (c-d) The refraction of TM interface states through the armchair output boundary and the zigzag output boundary with a straight domain wall; (e) The refraction of TM interface state through the zigzag output boundary with a Z-shaped domain wall; (e) The refraction of TM interface state through the zigzag output boundary with a Z-shaped domain wall; (f) The transmission spectra of these two kinds of output boundaries.
Then, Fano resonance and plasmon induced transparency (PIT) are realized by introducing defect modes into composite TVPCs. A hexagonal ring cavity is formed at the center of the straight domain wall of a combined TVPC (Structure 1) by removing six rods, as shown in the Fig. 5(a). There is an obvious transparent window in the absorption valley of Structure 1, as shown in Fig. 5(i). According to the hybridization principle of plasmons [58], the plasmon response of the ring cavity can be seen as an interaction between plasmon response of a rod and that of a cavity. As shown in the Fig. 5(e, f), a hexagonal cavity is formed at the center of the straight domain wall of a combined TVPC (Structure 2) by removing seven rods. As illustrated in Fig. 5(i), the transmission spectrum of Structure 2 has basically a broad Lorentz transmission Valley, which is a dipole mode of the cavity (bright mode). It is worth noting that there is a Fano resonance peak on the left side of the transmission valley, whose field distribution is presented in Fig. 5(e). Next, we introduce a line defect along the straight domain wall and a ring cavity with a rod at its center of a combined TVPC as shown in Fig. 5(g). This special design ensures the demonstration of plasmon resonance of the ring cavity, that is to say, a quadrupole plasmon resonance (dark mode) due to the coupling of a cavity mode and a rod mode. The transmission spectrum of Structure 3 has a very narrow valley at 3.8852 GHz with a Q-factor of 10224. The Q-factor is much higher than that (Q = 636) in previous work [59], which can be used as a notch filter. As illustrated by the black solid line in Fig. 5(i), the destructive interference in Structure 3 between the bright mode by Structure 1 and the dark mode by Structure 2 gives rise to a narrow transparent window, thus the absorption around the resonance frequency disappears. In our work, we mainly focus on the PIT effect of TM modes, the PIT effect for TE mode is also expectable by selecting a similar and suitable structure.
Fig. 5. (a) Schematic diagram of Structure 1; The simulated field distributions (b) Structure 1, f = 3.8792 GHz, nbg = 1.59; (c) Structure 1, f = 3.8848 GHz, nbg = 1.59; (d) Structure 1, f = 3.8892 GHz, nbg = 1.59; (e) The Fano resonance, Structure 2, f = 3.8602 GHz, nbg = 1.59; (f) Structure 2, f = 3.875 GHz, nbg = 1.59; (g) Structure 3, f = 3.8852 GHz, nbg = 1.59; (h) Sample 1, f = 3.8848 GHz, nbg = 1.74; (i) The transmission spectra of the four kinds of composite TVPCs. (j) The black solid line and the red dotted line are the group refractive indices ng, and the imaginary parts Im(ne) of the effective refractive indices for Structure 1, respectively.
In order to further analyze the physical mechanism of the PIT effect, the electric field distributions at the resonance frequencies 3.8792 GHz (Left valley), 3.8848 GHz (Transmission peak) and 3.8892 GHz (Right valley) of Structure 3 are calculated, as shown in Fig. 5(b-d), respectively. The left and right valleys are mainly due to the coupling between the waveguide and the cavity, forming a stable standing wave, so that the light wave is localized in the input waveguide and the ring cavity, and cannot be transmitted. It is obvious from Fig. 5(c) that the energy at the transmission peak is coupled from the input waveguide into the ring cavity, and then coupled to output waveguide. The topologically protected waveguide makes the optical energy coupled into the cavity with low loss, and the robustness of the topology will greatly reduce the requirements for the machining accuracy of the waveguide and cavity. Therefore, compared with the traditional waveguide coupled cavity system, our design has great advantages in practical applications. As shown in Fig. 5(j), the imaginary part of the effective refractive index is very small, and the maximum group refractive index is as high as 186 in the vicinity of the transparent window, which is much larger than that in the previous work [51], demonstrating that the composite TVPCs have great potential for slow light applications. The reconfigurable topological interface state can be realized by filling a liquid crystal (E7) in the composite structure, whose orientation can be adjusted by applying an external voltage between two parallel metal plates. Without the external voltage, the orientation of E7 liquid crystal molecules is parallel to the propagation direction of light, and the background refractive index is nbg = no = 1.59. When the applied voltage is over a threshold, the orientation of E7 liquid crystal molecules can be perpendicular to the propagation direction of light, nbg = ne = 1.74. The red solid line in Fig. 5(i) represents the transmission spectrum of the Sample 1 shown in Fig. 5(h). When the background refractive index increases from 1.59 to 1.74, the transparency peak disappears. That is to say, if the transparent peak is used as a band-pass filter, the composite TVPC can be switched between a band-pass filter and a band-stop filter.
Recently, many researches have been done using similar valley photonic crystals [18,60–62]. Wang et al. proposed a tunable valley photonic crystal by using metal rods embedded in a liquid crystal [60], Our designs have a similar structure but different parameters and physics. The different size ratio of the triangular rod to the unit cell causes a completely different band structure and Berry curvature. Our topological indices are 1/2 and -1/2, while their topological indices are 1 and -1. Their topological interface states propagate unidirectionally, but ours do not. Their excitation source is a chiral source, while our excitation source is a TM / TE wave. Ma et al. proposed topological photonic crystals based on a triangular lattice with a tripod in each unit cell [18]. Our design is quite close to theirs, but the tripod is replaced by a regular triangle rod. In our design, we find that the TVPC composed of regular triangular rods can be regarded as one-dimensional grating to a certain extent, wherein Fabry-Perot-like oscillations are obtained due to the coherent scattering. The topological waveguide mode in the composite TVPC has large group refractive index. A topologically protected coupled waveguide cavity system is then designed. We realized Fano resonance in Structure 2 by introducing a hexagonal cavity at the center of the straight domain wall of a combined TVPC. The coupling between the bright mode of Structure 2 and the dark mode of Structure 3 mode results in a transparent window, and the PIT phenomenon is realized.
As shown in Fig. 6, a switchable waveguide is designed by introducing a Z-shaped domain wall into a combined TVPC and filling with a liquid crystal (E7). When the background refractive index increases from 1.59 to 1.74, the topological interface state at 3.86 GHz disappears. This is because the band gap shifts from (3.74-3.97 GHz) to (3.45-3.60 GHz).
Fig. 6. The simulated field distributions of TE mode in the combined TVPC with a Z-shaped domain wall. f = 3.86 GHz, (a) nbg = 1.59; (b) nbg = 1.74.
4. Conclusions
In summary, we propose a new kind of reconfigurable topological valley photonic crystal (TVPC) based on a liquid crystal. Then a novel waveguide is designed in a composite TVPC by constructing a domain wall between two TVPCs with opposite valley-Chern indices. The structure of 25 supercells has a group refractive index as large as 17. Then we propose a topologically protected coupled waveguide cavity system by introducing a cavity defect mode into the domain wall, a narrow transparent window appears at 3.8848 GHz with a Q-factor of 1387 with a maximum group refractive index as high as 186. We propose an ultra-narrow-band notch filter with a resonant frequency of 3.8852 GHz. The notch filter has a very high Q-factor of 10224. By changing the orientation direction of liquid crystal molecules, the filter can be switched between band-pass and band-stop with the reconfigurable topological interface state. The proposed composite TVPCs are expected to work as PIT window, Fano resonance, band-pass/band-stop filter, refractive index sensor, optical switch, logic gates, delay lines, nonlinear devices, slow-light device, etc. Our structure provides a very meaningful guide for the design of reconfigurable topological valley photonic crystal.
Funding
Program for Changjiang Scholars and Innovative Research Team in University (IRT_13R29); 111 Project (B07013); National Natural Science Foundation of China (11674182, 12074201, 91750204); National Key Research and Development Program of China (2017YFA0303800, 2020YFB1805800).
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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