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Abstract
Photonic heterostructure has recently become a promising platform to study topological photonics with the introduction of mode width degree of freedom (DOF). However, there is still a lack of comprehensive analysis on the coupling of dipole emitters in photonic heterostructures, which constrains the development of on-chip quantum optics based on chiral dipole sources. We systematically analyze the unidirectional coupling mechanism between dipole emitters and valley photonic heterostructure waveguides (VPHWs). With the eigenmode calculations and full-wave simulations, the Stokes parameters are obtained to compare the coupling performance of two types of valley-interface VPHWs. Simulation results show that compared to the zigzag interface with inversion symmetry, the strategy of bearded interface with glide symmetry is easier to realize high-efficiency coupling. By adjusting the position and chirality of dipole emitters in VPHWs, the transmission of light reverses with guided modes coupled to different directions. Furthermore, a topological beam modulator is realized based on VPHWs, which maintains the robustness to large-area potential barriers and sharp corners. Our work supplies a powerful guide for chiral light-matter interaction, which is expected to be applied to increasingly compact and efficient on-chip optical platforms in the future.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Topological photonics is a flourishing flied that receives significant attention in photonics for its immunity against backscattering and robustness to defects. [1–4]. As a remarkable branch of topological photonics, valley photonics [5–7] is characterized by the broken valley degeneracy and the resultant quantum valley-Hall effect. Serving as a convenient approach, quantum valley-Hall effect can be realized by valley photonic crystals (VPCs) with broken inversion symmetry and behaves as a pair of valley edge states [8,9] with opposite pseudospin at non-trivial domain walls. With the topologically protection, valley edge state is immune to intervalley scattering, spurring significant potentials in advanced optoelectronic devices, such as topological lasers [10–12], logic gates [13,14], coupled resonators [15,16], and on-chip optical platforms [17–19].
Generally, the valley edge states only exist at the domain walls of two topological phases with different valley invariants, which is unfavorable for the modulation of transmission width in topological waveguides. Recently, this problem is well addressed by the proposed heterostructure [20–25] with additional incorporation of Dirac units. This strategy of heterostructure provides high flexibility to engineer the flow of light by introducing the width DOF and derives multiple applications in topological photonics, such as beam modulators [26–28], channel intersections [29,30], high-capacity topological cavity [31]. However, the unidirectional coupling (UDC) mechanism of chiral dipoles in VPHWs, an important issue to maximize the transmission efficiency, has still not been systematically analyzed, which limits their further applications in efficient optical transmission platforms and flexible light manipulation.
In this paper, we take a systematic investigation of the UDC mechanism in VPHWs with two different valley-interfaces [32–34], which complements the interaction system between chiral dipole emitters and photonic heterostructures. By means of eigenmode solver and full-wave calculations, the directionality and coupling efficiency are obtained to quantify the UDC performance in VPHWs. Simulation results show that bearded-stack VPHW is accessible for high-efficiency coupling than that of zigzag-stack VPHW. The Stokes parameters in the eigenmodes further indicate that switching the chirality and position of the circularly polarized dipoles can reverses the transmission direction, which presents potential applications on photonic routing and optical switch. In addition, a topological beam modulator is proposed, which is not only capable to compress or expand the beam width, but also robust to sharp corners and large-area potentials. These results advance the current knowledge on the control of dipole emission, paving the way for the design of optoelectronic devices, intense light-matter interaction, and on-chip communication.
2. Design of photonic heterostructure
We initially investigate a schematic of terahertz VPC array based on a silicon (of relative permittivity 11.7) slab, as illustrated in Fig. 1(a). The unit cell follows the honeycomb lattice (lattice constant a = 50 µm), consisting of two equilateral hexagonal airholes with side length l1 and l2, respectively. Focusing on the transverse-electric (TE) modes in the VPC, we use a geometric parameter r (the average value of l1 and l2) and a detuning parameter δ to characterize the evolution of spatial inversion symmetry. With the existence of inversion symmetry (e.g., r = 0.3a, δ = 0), the VPC exhibits C6 symmetry in Fig. 1(b), resulting in a pair of degenerate Dirac cones at K (K′) point. As the inversion symmetry (e.g., r = 0.3a, δ = 0.15a) is broken, the VPC symmetry is reduced to C3 that lift the degeneracy at K (K′) point with the emergence of a bandgap (The numerical simulations in this work are carried out using the COMSOL Multiphysics software). It is obvious that the gap of band edges at K point can be controlled by the detuning parameter δ, as shown in Fig. 1(c). We sweep δ from -0.25a to 0.25a (with r = 0.3a), taking blue and pink lines to depict right circular polarized (RCP) and the left circular polarized (LCP) modes, respectively. When δ = 0 (defined as B-state), there is no band gap at K point; when δ > 0 (defined as A-state), RCP power flow is exhibited in the lower band, with LCP power flow in the upper band; when δ < 0 (defined as C-state), the scenario is reversed for both bands. Figure 1(d) gives the Poynting power flow and the z-component of magnetic field (Hz) of the eigenmodes at K point for both bands in two specific VPCs (VPC-A: r = 0.3a, δ = 0.15a; VPC-C: r = 0.3a, δ = -0.15a), which are labeled as green crosses in Fig. 1(c). With time-reversal symmetry, these phenomena are quite the opposite at K’ point, as detailed in Supplement 1. In fact, the non-trivial properties can be described numerically by valley Chern number to explain the phase transition. For VPC-A, by integrating the Berry curvature around two valleys in the first band, we obtain two half-integer valley Chen numbers, i.e., CK = 1/2 and CK′ = -1/2. Taking valley Chern index CV = CK-CK′ [6,17] to describe the topological property in the VPC systems, we can get CV = 1 in the first band of VPC-A, while CV = -1 for VPC-C. According to the bulk-edge correspondence, a pair of twist states locked to different valleys appear at the domain wall formed by VPC-A and VPC-C.
Fig. 1. (a) The schematic picture of the VPC structure, described by geometric parameter r and detuning parameter δ. The red hexagonal dashed line or green rhombic dashed line shows the unit cell. (b) Band diagrams of the VPC with C6 symmetry (r = 0.3a, δ = 0, marked by black dashed line) versus C3 symmetry (r = 0.3a, δ = 0.15a, marked by red solid line). (c) Band edges at K point as a function of the detuning parameter δ, where the band edges of VPC-A (r = 0.3a, δ = 0.15a) and VPC-C (r = 0.3a, δ = -0.15a) are marked by the green crosses. The pink (blue) dashed line indicates LCP (RCP) phase modes, respectively. The blue and pink shadowed regions are gap regions of band edges for VPCs in the A-state (δ > 0) and C-state (δ < 0), respectively. (d) Mode profiles at K point for the former two bands of VPC-A and VPC-C. The color scale shows the distributions of Hz. The green and black arrows indicate the Poynting power flow.
To extend the valley twist states to a larger area, we insert VPC-B (r = 0.35a, δ = 0) featuring Dirac cones in the A|C structure to form a valley photonic heterostructure waveguide (VPHW) A|Bx|C with mode width DOF, where x is the number of layers of VPC-B, referred to Fig. 2(a). The projected band structures along kx of A|Bx|C (bearded interface) is depicted in Fig. 2(b) with x = 2, 6, 10, 14, where gapless bands (green solid lines) traverse the entire bulk bandgap, defined as the topological valley-locked waveguide states (TVWSs) [20]. The TVWSs maintain the topological property of momentum-valley locked, that is, opposite group velocities at different valleys. The area highlighted in blue indicates a propagating frequency window only TVWSs exist, which narrows with enlarging x, accompanied by the appearance of several gapped non-topological higher-order waveguide states. It can be inferred that when x is large extremely, the frequency window finally turns off with the band structures recovered to the bulk band of VPC-B. The electric field intensity distributions of TVWSs at K point for the whole VPWHs and along y-axis of VPWHs are illustrated in Figs. 2(c) and 2(d), respectively. It is evident that energy is centralized in domain B, proving the feasibility of large-area transmission in VPHWs. Furthermore, bearded-stack VPHW exhibit a larger topological frequency window compared to the zigzag-stack one (as detailed in Supplement 1), which means a broader bandwidth to manipulate and better performance for photonic devices.
Fig. 2. (a) Schematic of the bearded-stack photonic heterostructure A|Bx|C, with x = 2, 6, 10, and 14. VPC-A, VPC-B, and VPC-C are marked in blue, green, and red, respectively. (b) The band structures of A|Bx|C supercells. The area highlighted in blue indicates the frequency window where only the TVWSs (green solid line) exist. (c) Electric field intensity distributions at K point for the TVWSs of A|Bx|C supercells, where the frequency is 1.406THz, 1.391THz, 1.387THz and 1.385THz, respectively. (d) Simulated normalized electric field intensity across the length of A|Bx|C supercells with x = 2, 6, 10, and 14 in black solid, blue dashed, green solid, and pink dashed lines, respectively.
3. Analysis of unidirectional coupling in VPHWs
Coupling dipole emitters into optical structures such as high Q-factor micro-nano cavities, photonic crystal waveguides, can enhance the photon emission rates via the Purcell effect [35–39]. Considering a photonic crystal waveguide where waveguide modes transmit along x-axis, the Green’s function ${\boldsymbol G}({\boldsymbol r},{{\boldsymbol r}_0})$ characterizes the electric-field response at position r excited by a dipole emitter at r0 [36]. For our VPHW, Green’s function includes two parts, guided modes ${{\boldsymbol G}^{GD}}$ as a primary part and the other modes ${{\boldsymbol G}^O}$ as the secondary part. In detail, for the guided modes ${{\boldsymbol G}^{GD}}$, the Green’s function is expressed as [35,40,41]
The extreme case D = 1 (-1) represents ideal rightward (leftward) UDC. For our VPWH with topological TE modes, the in-plane electric field can be written as ${{\boldsymbol e}_k}({\boldsymbol r}) = {e_x}{\boldsymbol x} + {e_y}{\boldsymbol y}$. Then, Eq. (2) is simplified as the form of Stokes parameters [43,44]
Based on the above analysis and derivation, we investigate the rightward coupling behavior of chiral dipole emitters in valley-interface VPHWs. As shown in Figs. 3(a) and 3(b), two typical types of interfaces along ΓK direction are discussed, that is, bearded interface with glide symmetry and zigzag interface with inversion symmetry. First, we get the directionality D by eigenmode calculations, with kx = 0.78π/a, f1 = 1.326 THz in the bearded-stack VPHW and kx = 0.5π/a, f2 = 1.288 THz in the zigzag-stack VPHW, as illustrated in Fig. 3(c). Next, Fig. 3(d) shows the directionality D’ obtained by full-wave simulation, where a chiral point source with dipole moment ${{\boldsymbol d}_ + } = ({{\boldsymbol x} + i{\boldsymbol y}} )/\sqrt 2 $ is placed to excite the rightward modes in two VPHWs. To quantify the UDC efficiency, we use the proximity of directionality in eigenmode calculation and full-wave simulation, i.e., $\beta = 1 - |{D - D^{\prime}} |$. . Figure 3(e) depicts the UDC efficiency β of two waveguides calculated by sampling 10 points in the bearded-stack VPHW and zigzag-stack VPHW, which are labeled in Fig. 3(d) using yellow triangles and circles, respectively. From the results, the bearded-stack VPHW behaving a high UDC efficiency above 97%, which is superior to that of zigzag interface. We speculate that this might be due to the intense transverse localization of valley twist states in VPHWs caused by the glide symmetry [45,46], which is more tolerant to the offset of dipole position to receive higher UDC efficiency. Moreover, the glide plane, leading to a compact structure, is friendly to VPC packing and, thus, easier to realize the coupling of chiral dipole emitters with edge modes. Additionally, it has been shown [47] that the transmission bandwidth, wave velocity, intrinsic losses, and robustness in bearded-stack VPC waveguides are functions of the glide parameter, which provides a new DOF to optimize device performance and functionality.
Fig. 3. (a) Schematic view of bearded-stack VPHW with glide symmetry. (b) Schematic view of zigzag-stack VPHW with y-axis inversion symmetry. VPC-A, VPC-B, and VPC-C are marked in blue, green, and red, respectively. (c) Stokes parameter S3/S0 profiles retrieved from eigenmode calculations, with kx = 0.78π/a, f1 = 1.326 THz in the left panel and kx = 0.5π/a, f2 = 1.288 THz in the right panel. (d) D’ obtained by full-wave simulations. The frequency of d+ dipole is 1.326 THz and 1.288 THz in two types of VPHWs, respectively. (e) The calculated coupling efficiency β of 10 samples in two types of VPHWs, which are marked in Fig. 3(d) with yellow triangles and yellow circles, respectively.
4. Appilications of TVWSs
As observed in Fig. 3, there is a remarkble agreement between D (obtained by eigenmode calculation) and D’ (obtained by full-wave simulations), which suggests the feasiblity to manipulate the valley twist states by adjusting the position of dipole emitters. To demonstrate it, we use a d+ dipole emitter to selectively excite the rightward or leftward modes in the VPHWs with the layer x = 2. Figures 4(a) and 4(d) shows the positions of dipole emitters in the VPHW with bearded interface and zigzag interface respectively, where the color scale donates the directionality D retrieved by eigenmode calculation. As can be inferred from the electric field distributions in Figs. 4(b) and 4(c), the excitation source will couples guided modes to different directions if its position changes. In particular, when the d+ dipole emitter is placed at positions where D close to 1 and -1, it exhibits opposite and good unidirectional coupling, e.g., A and C in Fig. 4(a), whereas in the case where D is close to zero, it displays bidirectional property, e.g., B in Fig. 4(a). Note that the directionality of coupled modes reverses when the chirality of the dipole flips, as detailed in Supplement 1. In addition, the VPHW with bearded interface manifests a stronger confinement of electric intensity, leading to a narrower beam width which has great application prospect in compact on-chip integration platforms.
Fig. 4. (a) The positions of d+ dipole emitter in bearded-stack VPHW, labeled as A, B, and C. The color scale shows the directionality D retrieved by eigenmode calculation. (b) Distributions of electric field intensity in bearded-stack VPHW, with the location of d+ dipole emitter changing from A to C. (c) Distributions of electric field intensity in zigzag-stack VPHW, with the location of d+ dipole emitter changing from D to F. (d) The positions of d+ dipole emitter in zigzag-stack VPHW, labeled as D, E, and F. The color scale shows the directionality D retrieved by eigenmode calculation.
With an additional mode width DOF, it is easy to realize beam expanding or concentrating by combining several A|Bx|C waveguides featuring different x [26,29,31]. For instance, Fig. 5(a) shows a scheme of topological beam modulator with bearded interface composed of A|B2|C and A|B14|C heterostructures. We analysis the robustness of the modulator by introducing defects (VPC-B': r = 0.35a, δ = 0, labeled in orange) with width L to domain B (VPC-B: r = 0.3a, δ = 0, labeled in green). As is obvious from the electric-field profiles at 1.32 THz illustrated in Fig. 5(b), the beam width is expanded then concentrated without the influence of sharp corners and defects. Figure 5(c) provides the transmission rate as a function of frequency, which is received by integrating the energy flux density at the left and right terminals, and the area highlighted in grey shows a high transmission frequency range from 1.24 to 1. 39 THz. With the inversion symmetry, VPC-B (VPC-B’) features a Dirac frequency of 1.315 THz (1.38 THz), as shown in Fig. 5(d). In the Hamiltonian theory, the frequency shift in photonics can be analogized to the potential shift [48] and can therefore be explained by Klein tunneling [49]. As schematically shown in Fig. 5(e), we refer to the domain B’ as a potential barrier which features width L and height V = 0.065 THz. Klein tunneling shows that a massless Dirac particle under normal incidence can traverse the potential barrier perfectly regardless of its width and height [50], and the high transmission frequency range in Fig. 5(c) demonstrate it. Three conservation laws [48–51] are manifested in the Klein tunneling here: Firstly, due to the spatial translation symmetry parallel to the interfaces, the Dirac particles preserve transverse momentum, i.e., ky = 0. Secondly, the loss of material is negligible, indicating the conservation of energy. Thirdly, two linear dispersions with opposite pseudospins are depicted in Fig. 5(e), demonstrating the valley pseudospin conservation. Collectively, these findings indicate the robustness of the beam modulator to sharp corners and large area potential barriers, which is significant for signal processing and on-chip communications.
Fig. 5. (a) Schematic diagram of the beam modulator consisiting of A|B2|C and A|B14|C, where defects B’ (r = 0.35a, δ = 0, marked in orange) are introduced to domain B (r = 0.3a, δ = 0, marked in green). (b) Electric-field profiles with defect at 1.32 THz. (c) Calculated transmission rate with defect (pink solid line) and without defect (black solid line). The black dashed lines mark the Dirac frequencies of VPC-B and VPC-B’. The frequency range of high transmission efficiency is shaded in grey. (d) Band diagrams of VPC-B (VPC-B’), which Dirac frequency is 1.315 THz (1.38 THz). The potential barriers formed by two VPCs are depicted by black dashed lines. The frequency range from 1.24 to 1.39 THz (shaded in grey) shows high-efficiency transmission. (e) Top panel: Schematic diagram of Klein tunneling in optics, where a potential barrier features height V and width L. Two linear dispersions with opposite pseudospins are depicted by blue and red lines. The orange arrows show the rightward group velocity excited by d+ dipole emitters. Bottom panel: the sandwich-like structure constructed by VPC-B and VPC-B’.
5. Conclusions
In summary, we have theoretically analyzed and simulatively verified the coupling of dipole emitters with the valley kink states in VPHWs, which provides a guidance for high-capacity optical communication. In detail, the VPHW is realized by a trivial VPC sandwiched between two topologically distinct VPCs and still supports valley locked, gapless guide modes. To study the coupling performance in VPHWs, two types of valley-interfaces are considered, i.e., bearded interface and zigzag interface. The results demonstrate that bearded-stack VPHW with glide symmetry owns a larger bandgap as well as a narrower beam width, and manifests higher UDC efficiency. It is further shown that switching the chirality and position of dipole emitters can couple the optical modes to different valleys, offering a novel scheme to manipulate the direction of light transmission. Finally, the guided modes can transverse the large potential barriers with trivial loss in the proposed topological beam modulator, which simulatively proves the Klein tunneling. Our work carries essential influence for the evolution of the interaction of quantum dots with optical structures and the design of on-chip integrated platforms.
Funding
National Key Research and Development Program of China (Grant No. 2022YFA1203500); National Natural Science Foundation of China (NSAF, No. U2230114).
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.
Supplemental document
See Supplement 1 for supporting content.
References
1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
2. T. Ozawa, H. M. Price, A. Amo, et al., “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]
3. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017). [CrossRef]
4. Z. Wang, Y. Chong, J. D. Joannopoulos, et al., “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef]
5. H. Xue, Y. Yang, and B. Zhang, “Topological valley photonics: Physics and device applications,” Adv. Photonics Res. 2(8), 2100013 (2021). [CrossRef]
6. J. W. Liu, F. L. Shi, X. T. He, et al., “Valley photonic crystals,” Adv. Phys. X 6(1), 1905546 (2021). [CrossRef]
7. J. W. Dong, X. D. Chen, H. Zhu, et al., “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017). [CrossRef]
8. X. Wu, Y. Meng, J. Tian, et al., “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017). [CrossRef]
9. Y. Kang, X. Ni, X. Cheng, et al., “Pseudo-spin-valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018). [CrossRef]
10. S. Han, Y. Chua, Y. Zeng, et al., “Photonic Majorana quantum cascade laser with polarization-winding emission,” Nat. Commun. 14(1), 707 (2023). [CrossRef]
11. Y. Zeng, U. Chattopadhyay, B. Zhu, et al., “Electrically pumped topological laser with valley edge modes,” Nature 578(7794), 246–250 (2020). [CrossRef]
12. D. Smirnova, A. Tripathi, S. Kruk, et al., “Room-temperature lasing from nanophotonic topological cavities,” Light: Sci. Appl. 9(1), 127 (2020). [CrossRef]
13. F. Zhang, L. He, H. Zhang, et al., “Experimental Realization of Topologically-Protected All-Optical Logic Gates Based on Silicon Photonic Crystal Slabs,” Laser Photonics Rev. 17(8), 2200329 (2023). [CrossRef]
14. L. He, W. Zhang, and X. Zhang, “Topological all-optical logic gates based on two-dimensional photonic crystals,” Opt. Express 27(18), 25841–25859 (2019). [CrossRef]
15. M. J. Mehrabad, A. P. Foster, R. Dost, et al., “Chiral topological photonics with an embedded quantum emitter,” Optica 7(12), 1690–1696 (2020). [CrossRef]
16. Y. Yang and Z. H. Hang, “Topological whispering gallery modes in two-dimensional photonic crystal cavities,” Opt. Express 26(16), 21235–21241 (2018). [CrossRef]
17. X. He, E. Liang, J. Yuan, et al., “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019). [CrossRef]
18. Y. Yang, Y. Yamagami, X. Yu, et al., “Terahertz topological photonics for on-chip communication,” Nat. Photonics 14(7), 446–451 (2020). [CrossRef]
19. M. I. Shalaev, W. Walasik, A. Tsukernik, et al., “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019). [CrossRef]
20. M. Wang, W. Zhou, L. Bi, et al., “Valley-locked waveguide transport in acoustic heterostructures,” Nat. Commun. 11(1), 3000 (2020). [CrossRef]
21. J. Wang, Z. Zhang, S. Yu, et al., “Extended topological valley-locked surface acoustic waves,” Nat. Commun. 13(1), 1324 (2022). [CrossRef]
22. S. Liu, W. Deng, X. Huang, et al., “Robust acoustic waveguide transport in heterostructures based on acoustic topological insulators,” Phys. Rev. Appl. 18(3), 034066 (2022). [CrossRef]
23. Z. Lan, M. L. N. Chen, J. W. You, et al., “Large-area quantum-spin-Hall waveguide states in a three-layer topological photonic crystal heterostructure,” Phys. Rev. A 107(4), L041501 (2023). [CrossRef]
24. A. Vakulenko, S. Kiriushechkina, D. Smirnova, et al., “Adiabatic topological photonic interfaces,” Nat. Commun. 14(1), 4629 (2023). [CrossRef]
25. S. Li, M. L. N. Chen, Z. Lan, et al., “Coexistence of large-area topological pseudospin and valley states in a tri-band heterostructure system,” Opt. Lett. 48(17), 4693–4696 (2023). [CrossRef]
26. M. Wang, R. Zhang, L. Zhang, et al., “Topological one-way large-area waveguide states in magnetic photonic crystals,” Phys. Rev. Lett. 126(6), 067401 (2021). [CrossRef]
27. X. Yu, J. Chen, Z. Li, et al., “Topological large-area one-way transmission in pseudospin-field-dependent waveguides using magneto-optical photonic crystals,” Photonics Res. 11(6), 1105–1112 (2023). [CrossRef]
28. Q. Zhang, X. Xing, D. Zou, et al., “Robust topological valley-locked waveguide transport in photonic heterostructures,” Results Phys. 54, 107066 (2023). [CrossRef]
29. S. Yan, J. Yang, S. Shi, et al., “Transport of a topologically protected photonic waveguide on-chip,” Photonics Res. 11(6), 1021–1028 (2023). [CrossRef]
30. Z. Hu, M. Qin, H. Lingjuan, et al., “Manipulating the optical beam width in topological pseudospin-dependent waveguides using all-dielectric photonic crystals,” Opt. Lett. 47(20), 5377–5380 (2022). [CrossRef]
31. Q. Chen, L. Zhang, F. Chen, et al., “Photonic topological valley-locked waveguides,” ACS Photonics 8(5), 1400–1406 (2021). [CrossRef]
32. W. Ruan, X. He, F. Zhao, et al., “Analysis of unidirectional coupling in topological valley photonic crystal waveguides,” J. Lightwave Technol. 39(4), 889–895 (2021). [CrossRef]
33. L. Wang, R. Zhang, M. Xiao, et al., “The existence of topological edge states in honeycomb plasmonic lattices,” New J. Phys. 18(10), 103029 (2016). [CrossRef]
34. Z. Li, Z. He, L. Zhuo, et al., “Investigation of three topological edge states in honeycomb lattices based on graphene plasmonic crystal,” J. Phys. D: Appl. Phys. 55(27), 275102 (2022). [CrossRef]
35. P. Yao, V.S.C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photonics Rev. 4(4), 499–516 (2010). [CrossRef]
36. I. Söllner, S. Mahmoodian, S. L. Hansen, et al., “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10(9), 775–778 (2015). [CrossRef]
37. V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007). [CrossRef]
38. H. Siampour, C. O’Rourke, A. J. Brash, et al., “Observation of large spontaneous emission rate enhancement of quantum dots in a broken-symmetry slow-light waveguide,” NPJ Quantum Inf. 9(1), 15 (2023). [CrossRef]
39. K. Kuruma, H. Yoshimi, Y. Ota, et al., “Topologically-Protected Single-Photon Sources with Topological Slow Light Photonic Crystal Waveguides,” Laser Photonics Rev. 16(8), 2200077 (2022). [CrossRef]
40. A. B. Young, A. C. T. Thijssen, D. M. Beggs, et al., “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115(15), 153901 (2015). [CrossRef]
41. G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90(20), 205406 (2014). [CrossRef]
42. B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6(1), 6695 (2015). [CrossRef]
43. Z. Zhang, Z. Lan, Y. Xie, et al., “Bound topological edge state in the continuum for all-dielectric photonic crystals,” Phys. Rev. Appl. 16(6), 064036 (2021). [CrossRef]
44. Z. Lan, J. W. You, Q. Ren, et al., “Second-harmonic generation via double topological valley-Hall kink modes in all-dielectric photonic crystals,” Phys. Rev. A 103(4), L041502 (2021). [CrossRef]
45. S. Ok, L. Muechler, D. D. Sante, et al., “Custodial glide symmetry of quantum spin Hall edge modes in monolayer WTe2,” Phys. Rev. B 99(12), 121105 (2019). [CrossRef]
46. M. Bieniek, J. I. Väyrynen, G. Li, et al., “Theory of glide symmetry protected helical edge states in a WTe2 monolayer,” Phys. Rev. B 107(19), 195105 (2023). [CrossRef]
47. X. Zhou, J. Tong, J. Chang, et al., “Gliding photonic topological edge waves,” J. Phys. D: Appl. Phys. 56(47), 475303 (2023). [CrossRef]
48. W. Li, Q. Chen, Y. Sun, et al., “Topologically Enabled On-Chip THz Taper-Free Waveguides,” Adv. Opt. Mater. 11(21), 2300764 (2023). [CrossRef]
49. O. Klein, “Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac,” Z. Phys. 53, 157–165 (1929). [CrossRef]
50. X. Jiang, C. Shi, Z. Li, et al., “Direct observation of Klein tunneling in phononic crystals,” Science 370(6523), 1447–1450 (2020). [CrossRef]
51. P. E. Allain and J. N. Fuchs, “Klein tunneling in graphene: optics with massless electrons,” Eur. Phys. J. B 83(3), 301–317 (2011). [CrossRef]
