Abstract
Photonic topological insulators (PTIs) bring markedly new opportunities to photonic devices with low dissipation and directional transmission of signal over a wide wavelength range due to the broadband topological protection. However, the maximum gap/mid-gap ratio of PTIs is below 10% and hardly further improved duo to the lack of new bandwidth enhancement mechanism. In this paper, a PTI with the gap/mid-gap ratio of 16.25% is proposed. The designed PTI has a honeycomb lattice structure with triangular air holes, and such a wide bandwidth is obtained by optimizing the refractive-index profile of the primitive cell for increasing the energy proportion in the geometric perturbation region. The PTI shows a large topological nontrivial gap (the gap/mid-gap ratio 33.4%) with the bandwidth approaching its theoretical limit. The edge states propagate smoothly around sharp bends within 1430–1683 nm. Due to topological protection, the bandwidth only decreases 1.38% to 1450–1683 nm under 1%-random-bias disorders. The proposed PTI has a potential application in future high-capacity and nonlinear topological photonic devices.
1 Introduction
Photonic topological insulators (PTIs) [1], [2], [3], [4], [5], which utilize photonic crystals rather than electronic lattices [6], [7] to engineer the topological band structure with emergence of topologically protected surface or edge states, have attracted great attentions. Unlike traditional topologically trivial waveguide, the forward and backward edge states of the PTIs have the energy flow vortices along two opposite directions [8]. This effectively suppresses the coupling between these two states, enabling the edges one-way propagation without undergoing any backscattering even in the presence of structural defects that preserve symmetry. These exotic properties of the PTIs bring markedly new opportunities to photonic devices with low dissipation and directional transmission of signal. To date, two dimensional PTIs have been reported in various photonic structures, e. g., the coupled resonator arrays [9], [10], metacrystals superlattices [11], [12], photonic crystals with C6v-symmetric unit cells [13], [14], [15], and valley photonic crystals (VPC) [16], [17], [18], [19]. Furthermore, various novel photonic devices, that are robust to their structural disorders based on these topological structures, have been experimentally demonstrated, including the directional optical waveguide without backscattering [14], the robust photonic delay line [9], [16], the topological power splitter [14], and the topological photonic router [18].
However, the limited bandwidth of the PTIs makes the topological photonic devices typically operating within a relatively narrow wavelength range, which hinders their potential applications in high-capacity photonic systems. Specifically, the PTIs based on the coupled resonator arrays [10], metacrystals superlattices [12], photonic crystals with C6v-symmetric unit cells [15], and VPCs [19] are reported to have the maximum bandwidths with the gap/mid-gap ratios (γ) of 0.02%, 10%, 6%, and 10%, respectively. Moreover, their bandwidths are hardly further improved. The bandwidth of the coupled resonator structures is inherently limited by the narrow resonant transmission band of high-Q resonators. While, for the metacrystals superlattices, there exists a big challenge to open a large topological nontrivial gap but simultaneously match the permittivity and the permeability of meta-atoms required for realizing the pseudo-spin states. In terms of the photonic crystal with C6v-symmetric unit cells, their bandwidths are limited because the C6v-symmetric clusters can be only shrunk (or expanded) within a relatively small extent in order to keep the topological nontrivial gap open. In stark contrast to the above schemes, the valley photonic crystal is usually considered to be a good candidate for achieving a wide bandwidth. The valley photonic crystal [16], [17], [18], [19] only requires the units with the C3 symmetry, and can be realized by various materials, flexible element types, and the simpler band structure. The studies have been shown that the topological nontrivial gap of the VPC is proportional to the strength of perturbation that reshapes circle into C3 symmetric geometry [16]. Furthermore, in order to increase the perturbation strength, two methods are usually considered: (1) to increase the refractive-index contrast between the photonic crystal and background; (2) to increase the geometric perturbation area from circle to C3 symmetric shape. By carefully designing to maximize both the refractive-index contrast and geometric perturbation area, the maximum bandwidth with
In this Letter, we report a broadband PTI by increasing the energy proportion in the geometric perturbation region, and discuss its theoretical limit bandwidth. Enlightened by the strong evanescent field of nanofiber waveguide [20], we design the primitive cell with resonating rods surrounding hole instead of antiresonant air cavities surrounding rod. The bound modes of the resonating rods can efficiently spread into holes when reducing the size of rods. Therefore, when considering a honeycomb lattice with triangular air holes, its transverse magnetic (TM) modes have more energy contribution to the geometric perturbation region than that of a honeycomb lattice with triangular solid rods, and its topological nontrivial gap is opened with
2 Broadband PTI optimization
Using the finite element method (outlined in the Appendix), we consider a generic photonic honeycomb lattice constructed by the circular air holes on the Si slab (the refractive index RI = 3.4757). The lattice constant is a, and the side length of the regular hexagon is
The topological nontrivial gap of the VPC with triangular air holes can be increased by optimizing the size and the shape of the triangular air hole in unit cell. The triangular air hole has the round corner with radius
In order to explain that the gap/mid-gap ratio of the VPC with air hole arrays can be much larger than the VPC with solid scatter arrays, we consider the following low-energy Hamiltonian at Dirac points [16]:
where vD is the Dirac velocity,
where V is the geometric perturbation area in the primitive cell when the circle air hole is deformed into the triangular air hole. It is composed of two parts: the side area V1 and the angle area V2, as shown in Figure 2a.
However, for the VPC with the solid scatters (see Figure 2c), the unit cell consists of the triangular solid scatter surrounded by six air resonators (the antiresonant reflecting optical waveguides). Each TE mode of the array is the antiresonant mode of three air cavities. The antiresonant mode is formed by the reflection of the solid scatter, and is hardly penetrated into the solid scatter even when the size of air resonantor is reduced. Therefore, only the side area V1 has the electric field of eR(K). Then the perturbation strength is
According to Eqs. (3) and (4), the air hole array has larger perturbation strength than the solid scatter array, because it can support much more electric field distribution in the geometric perturbation area. Therefore, the air hole array has larger gap/mid gap ratio. Particularly noteworthy here is the above results are only suit for the dielectric VPCs. For the metallic VPCs, the net perturbation strength can not be expressed by (3) because the mode fields are always localized in air and hard to leak into the metal.
Besides the topological nontrivial gap, the bandwidth of the VPC is also inherently limited by the group velocity of edge states vedge. Here, the bandwidth of the VPC is defined as the frequency range for light smoothly propagating along the zigzag path with negligible backscattering, which is the overlapped band dispersions of edge states at two different interfaces (see Figure 4). Due to the crossover bands structure with near linear dispersion around K/K′ valley, the maximum bandwidth of the VPC with a large energy gap has a theoretical limit
where 2|KM| is the distance between the high symmetry K and M points in the first Brillouin zone of photonic graphene, and
3 Wideband topological edge states
The interface between two valley photonic crystals with different Chern numbers can support the topologically protected edge states. The valley Chern number of bulk bands at K/K′ valley is
To demonstrate the topological robustness of the wideband edge modes, we simulate (see Appendix) one-way edge mode propagation along a zigzag path with 120° bends (see Figure 4a, d). As shown in Figure 4b, e, the edge modes for the interfaces IFS12 and IFS21 can smoothly propagate along the zigzag path. Figure 4c, f show the transmission spectra of the edge state propagating along the zigzag paths and straight paths. For the mid-gap modes from 1430 to 1683 nm, the edge states can smoothly pass through the zigzag path with negligible backscattering. Therefore, the designed PTI has a gap/mid-gap ratio of 16.25%. The bandwidth is slightly smaller than the results of band structure because the edge states near kx = 0 are more easily reflected. It is interesting that the low-loss window of IFS12 zigzag path is the overlapped operating band of edge states at IFS12 and IFS21 straight paths. The reason may be that the sharp corner part of IFS12 zigzag path owes the same structure as IFS21 straight path (highlighted by the red circles in the insets of Figure 4a, d), which inhibits the light beyond the bandwidth of IFS21 straight path.
The bandwith of the designed PTI can be immune to the disorder. In Figure 5, we consider four kinds of structural defects: the offset, the rotation, the scaling, and the deformation of the triangular air hole, respectively. All cells of IFS21 zigzag path have the same kind of defect with the different random value. The coupling effect between K valley and K′ valley will increase with increasing the size of defect and the robustness of the edge states is broken. Due to the larger group velocity vedge, the edge states near
4 Conclusion
In conclusion, we demonstrate a broadband PTI achieved by a honeycomb lattice with triangular air holes. The triangular air hole configuration distributes more energy in the geometric perturbation region and opened the topological nontrivial gap up to 33.4%. We also show that the theoretical limit bandwidth of edge states without backscattering is limited by both the energy gap and the group velocity of edge states. The topological nontrivial gap size of the designed VPC is larger than its bandwidth and the edge state bands of both interfaces do not extend across the whole gap. Based on the designed VPCs, broadband transmission of the edge states without backscattering is achieved within 1430–1683 nm
Note added—After the submission of this paper, we found the independent discovery of the VPC with the quasi-hexagonal holes array, which was used for the realization of the electrically pumped topological laser [23].
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: U1609219
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: U1766220
Funding source: National Key R&D Program of China
Award Identifier / Grant number: 2017YFB0405500
Funding source: Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program
Award Identifier / Grant number: 2017BT01X137
Funding source: China Postdoctoral Science Foundation
Award Identifier / Grant number: 2019M652872
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: 2018ZD01
Acknowledgments
The authors gratefully acknowledge financial support from National Natural Science Foundation of China (U1609219, U1766220), the National Key R&D Program of China (2017YFB0405500), the Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X137), Project funded by China Postdoctoral Science Foundation (2019M652872), and the Fundamental Research Funds for the Central Universities (2018ZD01).
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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Appendix Numerical simulation
The band structures are calculated by the COMSOL RF module. The model equation is
The electric field distributions of the edge state traveling along interface are calculated by the Lumerical FDTD Solutions. The model equation is the Maxwell equations
© 2020 Zhishen Zhang et al., published by De Gruyter, Berlin/Boston
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