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Abstract
Originating from the study of topological photonic crystals (TPCs), analogues of the quantum spin Hall effect have been used as a potential way to control the propagation of electromagnetic waves. Due to the topological robustness of the spin TPCs, the edge states along the interface between the trivial and topological areas are topologically protected and not reflected from structural defects and disorders. Here, on the basis of the time-spatial reversal symmetry and topological defect theory, we demonstrate broadening of the edge state bandwidth in spin TPCs made of regular metallic cross structures by simultaneously deforming the hexagonal honeycomb lattice and adjusting the rotation angle. Due to the simultaneous tuning of the two parameters, the designed spin TPCs possess more flexibility. Topologically protected one-way propagating edge states are observed in the terahertz regime, where electromagnetic waves propagate along sharp corners without backscattering. Our findings offer the potential application for topological devices in terahertz technology and are beneficial for the development of 6G mobile communications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The concept of topology has been introduced to classical systems, including electrons in solid states [1–4], acoustics [5–8], elastic waves [9] and photonics [10–12]. Topological photonics has revolutionized our understanding to control electromagnetic waves, and increasing attention has been paid to topological edge states, which allow transmission in only one direction and are robust against local disorders and structural defects. In recent years, with the development of metamaterials, topologically protected one-way propagating edge states have been observed and experimentally demonstrated in many topological photonic systems [13–21]. Helical waveguides and arrays of coupled ring-resonators [22,23] can emulate the time reversal symmetry breaking, leading to topological modes. Topological photonic crystals (TPCs) based on bi-anisotropic materials and gyromagnetic materials could also achieve unique edge state propagation under an external magnetic field [24–30]. However, it is difficult to realize these photonic topological systems in practice, because of lacking gyromagnetic materials or requiring an external magnetic field to break the time-reversal symmetry. With the time-reversal symmetry preserved, analogues of the quantum valley Hall effect (QVHE) [31–40] or the quantum spin Hall effect (QSHE) [41–45] made of metallic or dielectric materials have been proposed, and several TPCs have been demonstrated by adjusting the structure parameters to achieve the QSHE analogues.
In this article, we present spin TPCs composed of periodically arranged metallic cross structures at terahertz frequencies. By simultaneously deforming the hexagonal honeycomb lattice and adjusting the rotation angle of the cross structures, the topological property of the structures could be changed and the operation bandwidth of the topological unidirectional transport edge states is significantly broadened. As a result, the combination of the trivial and topological areas provides an intriguing way to demonstrate broadband topologically protected unidirectionally propagating edge states around sharp bends. Due to the simultaneous tuning of the two parameters, the designed spin TPCs possess more tunability and tunable operation bandwidth, which provides a new feasible path to achieve broadband (narrowband) topological edge states in TPCs. The combination of the broadband and narrowband structures offers new application prospects for topological devices in terahertz technology, such as waveguide, beam splitter and wavelength division multiplexer.
2. Design and simulation
Using the commercial finite element method-based software COMSOL Multiphysics, we design and simulate the spin TPCs made of perfect electric conductor cross structures embedded in an air host, as displayed in Fig. 1. The lattice constant is a = 346 μm, the length of the rhombus is b = 200 μm, and the distance between the center of the lattice and cross structure is R = a/3 = 115.47 μm. The long arms of the cross structure is L1 = 0.42b = 84 μm and the short axis is L2 = 0.30b = 60 μm. The height of the cross structure is h = 110 μm, and the rotation angle between the major axis and y axis is θ = 45°.
Fig. 1. Schematic diagram of the graphene-like, broadband and narrowband unit cells, where the parameters are the lattice constant a = 346 μm, the length of the rhombus b = 200 μm, and the distance between the center of the lattice and cross structure R = a/3 = 115.47 μm. The long arm of the cross structure is L1 = 0.42b = 84 μm and the short arm is L2 = 0.30b = 60 μm. The height is h = 110 μm, and the rotation angle between the major axis and y axis is θ = 45°.
Here we consider harmonic transverse magnetic (TM) modes of the electromagnetic waves, namely the finite out-of-plane Ez and in-plane Hx and Hy components with the others being zero. Based on Maxwell’s equations, the master equation for the TM modes in the spin TPCs can be expressed by the following equation [45–47]
3. Result and discussion
3.1 Photonic bands
To start with, we have obtained the dispersion relation of the spin TPCs by employing the finite element method to solve Eq (1). The band structures of the rhombic primitive cell and hexagonal honeycomb lattice in which θ = 45° and a/R = 3 are displayed in Figs. 2(a) and 2(b), respectively. We can see that the two-fold degenerate eigenfrequencies change steadily and cross, and a Dirac dispersion is formed at the KII point. Due to the zone folding mechanism, the two-fold degenerate Dirac dispersions at the KII point are folded to a double Dirac dispersion at the ГI point in the hexagonal honeycomb lattice.
Fig. 2. (a) Band structure of rhombic primitive cell in which θ = 45° and a/R = 3. (b) Band structure of hexagonal honeycomb lattice in which θ = 45° and a/R = 3, and a double Dirac cone is seen at Г point.
In order to open the topological band gap in the vicinity of the Dirac point, we can deform the hexagonal honeycomb lattice a/R or vary the rotation angle θ. The eigenfrequencies of the dipole and quadrupole modes of the hexagonal honeycomb lattice with the change of distance a/R at the Brillouin zone center Γ1 are shown in Fig. 3(a), where there is a Dirac point for a/R = 3. By expanding or shrinking the distance a/R, the Dirac point is opened and band inversion occurs between the dipole mode p state and quadrupole mode d state. The greater the distance, the wider the photonic bandgap. When the distance is reduced to a/R = 3.30, a topological band gap opens from 0.508-0.527 THz. Clearly, the p-type state is located at the lower side of the bandgap, while the d-type state at the upper side. Band inversion happens when the distance a/R is expanded to 2.80, which also brings about a similar topological bandgap within the range of 0.506-0.525 THz. When we vary the rotation angle θ, the corresponding eigenfrequencies of the hexagonal honeycomb lattice are illustrated in Fig. 3(b). Both the structures with θ = 0° and 90° have a band gap emerging within the range of 0.511-0.521 THz, where the bands are inverted, too.
Fig. 3. (a) Eigenfrequencies of dipole and quadrupole modes of hexagonal honeycomb lattice with change of distance a/R at Brillouin zone center Γ1. (b) Eigenfrequencies of dipole and quadrupole modes of hexagonal honeycomb lattice with change of rotation angle θ at Brillouin zone center Γ1.
By simultaneously deforming the hexagonal honeycomb lattice a/R and adjusting the rotation angle θ, we can acquire two broadband structures and two narrowband structures. The broadband structures correspond to a/R = 2.84, θ = 0° and a/R = 3.28, θ = 90°, and their opened bandgap is 0.505-0.537 THz. The narrowband structures correspond to a/R = 2.84, θ = 90° and a/R = 3.28, θ = 0°, and their bandgap range is 0.510-0.518 THz. Because the broadband and narrowband structures are based on the QSHE, below we take the broadband structures as an example to demonstrate this fact. When simultaneously shrinking the cross structures to a/R = 3.28 and adjusting the rotation angle θ to 90°, we observe from Fig. 4(a) that the double Dirac dispersion splits into accidental doubly degenerate states. Identically, the p-type states are located at the lower side of the bandgap, while the d-type states at the upper side, which is consistent with the classical photonic theory. However, when cross structures are expanded to a/R = 2.84 and the rotation angle θ is varied to 0°, band inversion will occur, and the d-type states rise above the p-type states, as illustrated in Fig. 4(b). Figure 4(c) displays the simulated electric field Ez distributions and the process of topological phase transition between the p± and d± states. The bands ${p_ \pm } = ({p_x} \pm i{p_y})/\sqrt 2$ and ${d_ \pm } = ({d_{{x^2} - {y^2}}} \pm i{d_{xy}})/\sqrt 2$ correspond to pseudo spin-up (spin-down) states in photonic structures.
Fig. 4. (a) and (b) Dispersion relations for broadband structures with a/R = 3.28, θ = 90° and a/R = 2.84, θ = 0°, respectively. (c) Simulated field intensity field distributions of dipole modes px(py) and quadrupole modes dxy(dx2-y2) at Г point. Ez field modes of px(py) and dxy(dx2-y2) correspond to different bands, which are separated by a complete bulk band gap.
To further reveal the topology, we apply the $\vec{k} \cdot \vec{p}$ perturbation method and tight-banding approximation to calculate the effective Hamiltonian and the spin Chern numbers. Based on the four basis eigenmodes [ px, py, dxy, dx2-y2] around the Γ point, the effective Hamiltonian $H(\vec{k})$ could be described as [15]
3.2 Bands of domain-wall interfaces
We could explain the edge states at the interface between the trivial and topological regions by overlapping the band gaps. Here, we investigate a ribbon-shaped supercell which consists of the trivial and topological unit cells, as shown in Fig. 5(a). The bandgaps of the trivial and topological structures have an overlapping frequency region. In Figs. 5(b) and 5(c), we show the projected band diagram along the ГK direction of the ribbon-shaped supercell, which results in a complete bulk bandgap within the range of 0.505-0.532 THz. Clearly, two additional topological edge modes are localized within the bulk bandgap, which are represented by the degenerated red curves. These topologically protected edge states have opposite group velocities at the same frequency, which indicates that there exist back propagation and pseudo spin orbit coupling between the edge states.
Fig. 5. (a) Ribbon-shaped supercell of designed broadband structures composed of trivial and topological unit cells. (b) Projected band diagram and (c) its local magnification displayed for supercell with domain walls in vicinity of Γ point. Shaded areas represent bulk states, and red curves represent two topological edge states with opposite group velocities at the same frequency.
3.3 Topological edge states
To demonstrate the unidirectional propagation of the topological interface modes, we provide a spin TPC with a straight interface constructed from the trivial and topological photonic structures. The spin TPC is excited with a chiral source that includes six electric dipoles with ±π/3 phase difference between the adjacent dipole points (Fig. 6(a)). The chiral source can generate a left-circularly polarized (LCP) wave ${S_ + } = {H_0}{e^{i\omega t}}(\hat{x} - i\hat{y})$ or right-circularly polarized (RCP) wave ${S_ - } = {H_0}{e^{i\omega t}}(\hat{x} + i\hat{y})$, where H0 is the amplitude and ω is the angular frequency, as shown in Figs. 6(b) and 6(c), respectively. The source is located around the interface center, which matches well with the profile of the mode propagating along the x direction. A topologically protected leftward or rightward one-way transmission of the edge states at 0.52 THz can be created by the chiral source, which can selectively excite an LCP or RCP phase vortex, as displayed in Figs. 6(d) and 6(e), respectively.
Fig. 6. (a) Schematic of point-like chiral source made by a six-antenna array with ±π/3 phase delay between adjacent elements. (b) Electric field distribution stimulated by chiral source with anticlockwise phase delay. (c) Electric field distribution stimulated by chiral source with clockwise phase delay. (d) Leftward unidirectional electromagnetic wave propagation at 0.52 THz excited by LCP chiral source along interface. (e) Rightward unidirectional electromagnetic wave propagation at 0.52 THz excited by RCP chiral source along interface.
In order to further demonstrate the robustness of the broadband spin TPCs, we have first designed a topological straight waveguide with a finite lattice constructed by the trivial and topological structures at terahertz frequencies. A chiral source is situated at the left side of the waveguide to excite an RCP phase vortex. Topological edge states appear owing to the analog of the QSHE, and propagate along the domain wall interface without backscattering (Fig. 7(a)). Then we have designed another zigzag waveguide with sharp bends, where the edge states could be well transmitted along the sharp bends without notable scattering losses (Fig. 7(b)). Figures 7(c) and (d) illustrates that the simulated transmission at the position S3 (S1) is almost identical to that at S4 (S2), indicating that the pseudospin polarization-dependent edge state propagates without noticeable backscattering from the sharp bends, too.
Fig. 7. Unidirectional propagation of pseudospin-dependent edge states localized at interface. (a) Simulated EZ field intensity distribution of topological edge states at 0.52 THz propagating along straight interface. (b) Simulated EZ field intensity distribution of topological edge states at 0.52 THz propagating along zigzag interface with sharp bends. In both (a) and (b), a chiral source located at left side of the waveguide could generate an RCP phase vortex. (c) and (d) Simulated transmission spectra (in dB) at S1– S4 for straight and zigzag configurations, respectively.
4. Conclusion
We propose a feasible approach to achieve broadband or narrowband spin TPCs with more tunability using an array of regular metallic cross structures in the terahertz region. The topological property and bandgap are engineered by simultaneously deforming the honeycomb lattice and adjusting the rotation angle of the cross structures, and topologically protected unidirectional transmission of the edge states along the domain wall between the trivial and topological parts of the TPCs is achieved. The proposed spin TPCs can work in a broad or narrow terahertz frequency range. Terahertz technology is key to the development of information technology and 6G mobile communications, so basic terahertz devices, such as waveguides, beam splitters and wavelength division multiplexers are indispensable, and the combination of the broadband and narrowband structures can help achieve these devices. The proposed scheme offers new application prospects and may play an important role in terahertz communications.
Funding
National Key Research and Development Program of China (2017YFA0701004); National Natural Science Foundation of China (61775159, 61875150, 61805129, 61605143, 61735012).
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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