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Abstract
Robust and efficient light coupling into and out of quantum valley-Hall (QVH) topological interfaces within near-infrared frequencies is demanded in order to be integrated into practical two-dimensional (2D) optical chips. Here, we numerically demonstrate efficient light coupling between a QVH interface and a pair of input/output silicon photonic waveguides in the presence of photonic crystal line defects. When the topological QVH interface is directly end-butt coupled to the silicon waveguides, the input-to-output transmission efficiency is lower than 50% and the exterior boundaries associated with a QVH interface also cause inevitable back-reflections and high-order scatterings, further reducing the transmission efficiency. The transmission efficiency is substantially increased to 95.8% (94.3%) when photonic crystal line defects are introduced between the bridge (zigzag) QVH interface and the waveguides. The buffering line defect mode, with an effective group refractive index between the interface state and the waveguide mode will ease their mode profile conversion. The design we present here brings no fabrication complexity and may be used as a guide for future implementation of on-chip 2D topological photonics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recent advances in silicon-on-insulator technologies have led to the fast development of integrated photonic devices based on two-dimensional topological photonic crystal (PhC) platforms because of their excellent confinement and manipulation of light [1–3]. Guaranteed by bulk-edge correspondence, topological interface states can be induced when assembling two PhC slabs with different topology properties at proximity. Within the photonic bandgap of both PhC slabs, unidirectional light propagation along interfaces can be maintained without backscattering. The interface states are also not susceptible to local disorders even without breaking time-reversal symmetry or under time-domain modulations [4]. Borrowing idea from valley-Hall effect in graphene [5], by breaking the Dirac point at Brillouin zone boundary, photonic quantum valley-Hall (QVH) interface states have attracted intensive research interest because of its simple geometrical designs. Following the demonstrations of robust topological waveguiding of valley-Hall states on the CMOS-compatible platforms [6–10], the exploration of high-quality cavities-based topological lasing [11–15], optical routing [16–19], filtering [20,21], and light slowing [22,23] have been enabled in designing on-chip topological photonic integrated circuits [24,25]. Moreover, fundamental phenomena such as chiral topological photonic interfaces engineering intense light-matter interactions have also been explored by integrating QVH interfaces with semiconductor quantum dots [26,27]. In order to integrate to photonic chips, achieving efficient light transfer from optical waveguide structures to these topology-based photonic devices is essential. End-butt coupling enables easy transfer of light from optical waveguides to conventional PhC slabs since the conventional and PhC waveguide modes are linearly polarized. Photonic systems utilizing QVH interface states do not benefit from end-butt coupling since it is intrinsically an optical vortex carrying orbital angular momentum (OAM). As a result, such systems are prone to experiencing high coupling losses because of the gigantic difference in mode profiles between the conventional optical waveguide and the QVH interface state. To address this problem, Ma et al. proposed using a low-effective-index subwavelength structure as a buffer to suppress unwanted diffraction between the silicon waveguide and the bridge QVH interface, achieving an input-to-output coupling efficiency of at least 40% within the bandgap [25]. He et al. proposed using a subwavelength microdisk to add a phase vortex to the silicon waveguide during its coupling to the QVH interface state [16]. By coupling out via another silicon waveguide that connects to a grating coupler, a measured transmission efficiency exceeding 50% was achieved. In spite of that, fabricating a microdisk greatly multiplies the complexity and footprint of the system.
Here, we discuss light coupling between the QVH interface and the silicon waveguide in two different ways. As schematically shown in Fig. 1(a) (1b), the QVH interface is end-butt coupled by a pair of waveguides (with two PhC line defects in-between). Two different QVH interfaces are considered, namely the bridge and zigzag interfaces [28]. If the waveguides are symmetrically positioned with regard to the bridge interface, the interface state coupled from a fundamental waveguide mode will couple to evanescent waves or high-order modes, instead of coupling back to the fundamental waveguide mode. The coupling position of the silicon waveguides is determined by the eigen-mode symmetry of the topological interface state in order to achieve the largest mode overlap between the topological interface state and fundamental waveguide mode. As depicted in Fig. 1(c), the waveguides are positioned deviated from the bridge interface because the interface eigen-mode is asymmetric with regard to the interface (see Fig. 2(d)). When it comes to the coupling to the zigzag interface, the waveguides are positioned aligned with the interface as the interface eigen-mode has a symmetric mode pattern the same as the fundamental waveguide mode (see Fig. 2(e)). A schematic of the line defect positioned in-between the waveguide and the topological interface is shown in Fig. 1(e). The line defect is constructed by removing one-row triangular air holes that are aligned with the coupled waveguide. By doing so, the largest mode overlap among fundamental waveguide mode, line defect mode, and topological interface state can be achieved. Here, the QVH PhC slabs are constructed by drilling triangular holes of different size inside a silicon slab, adopting a lattice constant $a$ = 423 nm [7]. See Supplement 1 for more simulation details.
Fig. 1. Schematic of the (a) wg1-interface-wg2 end-butt coupled and (b) wg1-defect-interface-defect-wg2 coupled systems. The QVH topological interface is a domain wall between two valley PhCs (VPhC1 and VPhC2) with opposite Valley Chern numbers. The silicon waveguides are positioned (c) deviated from the bridge interface but (d) aligned with the zigzag interface. The PhC line defects connected to the waveguides are shown in (e).
Fig. 2. Projected band structures of the (a) bridge QVH interface, (b) zigzag QVH interface, and (c) PhC line defect. Interface states and projected bulk states are represented as solid lines and dark gray shades, respectively. Light lines are represented as gray lines. Out-of-plane magnetic field distributions of the QVH interface states and the line defect mode are shown in (d, e) and (f), respectively. In (d, e), frequency-domain simulated field distributions are attached on the right side.
2. Results and discussions
Two-dimensional finite-element eigen-frequency simulations are firstly performed to determine the projected band structures of the bridge and zigzag QVH interfaces (see Figs. 2(a) and 2(b) respectively) with transverse electric field polarization (TE). The QVH interface states appear close to the boundaries of the first Brillioun zone and below the light line, indicating that they are inherently bound to the interface without radiation [29]. A PhC line defect is formed by removing a single line of triangular air holes of the zigzag QVH interface. Its projected band structure in Fig. 2(c) indicates that only the fundamental TE mode with the largest group velocity spans the entire topologically trivial bandgap [30]. Figures 2(d)–2(f) show the out-of-plane magnetic field distributions of the QVH interface states and the TE defect mode, respectively. The interface state mode profile is asymmetric (symmetric) with regard to the bridge (zigzag) interface along x-axis.
Frequency-domain simulations are also performed to evaluate the coupling performance of the waveguide-interface coupled systems as described in Figs. 1(a) and 1(b). Boundary mode analysis with numerical ports is used to compute the wave number and shape of the propagating mode in silicon waveguides. The port excitation at wg1 is applied using the fundamental TE eigenmode. In the simulations, monitors are set to determine the transmitted time-average power through points at wg1 (Pwg1), the middle point of QVH interface (Pedge), and at wg2 (Pwg2). The wg1-to-interface, interface-to-wg2, and wg1-to-wg2 transmission efficiencies, by comparing the transmitted energy ratio at corresponding points, are calculated as Pedge/Pwg1, Pwg2/Pedge, and Pwg2/Pwg1, respectively. The transverse dimension of the waveguide (Dwg) is also a key factor to transmission efficiency. We firstly simulate the wg1-interface-wg2 end-butt system with bridge interface and the three transmission efficiency contours depending on frequency f (THz) and Dwg are shown in Fig. 3. The direct end-butt coupling from a QVH interface state to waveguide is low. According to Fig. 3(a), the end-butt in-coupling from wg1 to bridge interface can only reach a maximum Pedge/Pwg1 of 40.2% at [f (THz), Dwg (nm)] [157.34 (THz), 400 (nm)]. To explain the low coupling efficiency, we plot the E-field norm distributions for the coupled system as shown in Fig. 4. The standing-wave pattern generated in wg1 indicates that the low in-coupling efficiency is due to the strong back-reflection occurring when mode conversion is inevitable from waveguide to topological interface. This mode conversion occurs again from topological interface to waveguide (see Fig. 3(b)), and as a result, the total wg1-to-wg2 transmission efficiency Pwg2/Pwg1 is further reduced to less than 20% for Dwg = 400 nm (see Fig. 3(c)). Here, our primary concern is to couple interface state to small waveguides, maintaining single-mode transmission with low loss. In Fig. 4(d), the interface state out-couples to evanescent wave with energy reaching beyond the waveguide core and dissipates along the exterior boundaries of the QVH PhC. In addition, the color bar clearly shows that both parts suffer energy losses of the same magnitude.
Fig. 3. Frequency-domain calculated transmission efficiency (a) Pedge/Pwg1, (b) Pwg2/Pedge, and (c) Pwg2/Pwg1 for the bridge wg1-interface-wg2 end-butt coupled system.
Fig. 4. (a) Frequency-domain simulated E-field norm distributions of the coupling in bridge wg1-interface-wg2 end-butt coupled system. (b-d) Zoom-in views of the field distributions.
We continue our wg1-interface-wg2 end-butt coupling study with zigzag interface (Figs. 5 and 6). As shown in Figs. 5(a) and 5(b), a maximal in-coupling efficiency of 34.8% and no more than 60% out-coupling efficiency are obtained for Dwg = 400 nm. Likewise in Fig. 5(c), the wg1-to-wg2 total transmission efficiency is less than 30%. Zoom-in views of the E-field norm distributions in Fig. 6 clearly illustrate the back-reflection into wg1, and the evanescent component of the out-coupled mode in wg2. Moreover, the interface state strongly dissipates along the exterior boundaries at both sides of the QVH PhC rather than remaining contained with the interface. Another fact shown in Figs. 3(c) and 5(c) is that transmission efficiencies Pwg2/Pwg1 approaching 30% can be achieved for both small and large waveguides at certain frequencies. However, this is due to an undesired out-coupling of interface state to high-order modes when Dwg exceeds 600 nm.
Fig. 5. Frequency-domain calculated transmission efficiency (a) Pedge/Pwg1, (b) Pwg2/Pedge, and (c) Pwg2/Pwg1 for the zigzag wg1-interface-wg2 end-butt coupled system.
Fig. 6. (a) Frequency-domain simulated E-field norm distributions of the coupling in zigzag wg1-interface-wg2 end-butt coupled system. (b-d) Zoom-in views of the field distributions.
It can be seen from the above results that during the end-butt coupling, the mode mismatch between the linearly polarized waveguide mode and the topological interface state can cause significant mode distortion and energy leakage into the background. Here we also show the low-efficiency in/out-coupling can be understood by the group refractive index difference. The group refractive indices of the interface states can be obtained from their calculated dispersions (Figs. 2(a) and 2(b)), as
Fig. 7. Group refractive indices ${n_\textrm{g}}$ v.s. effective refractive indices ${n_{\textrm{eff}}}$ spectrum. The ${n_\textrm{g}}$ is calculated using Eq. (2) for the interface state and the defect mode. The ${n_{\textrm{eff}}}$ increases as Dwg varies from 100 nm (black squares) to 600 nm (magenta triangles).
Additionally, the existence of multiple scattering channels during the end-butt coupling process can also contribute to mode distortion. Therefore, the method of increasing waveguide dimension plays a limited role in improving the coupling efficiency, but rather creates a higher-order mode that makes the coupled system unsuitable for single-mode applications. The above problems can be solved when line defects are present. The green squares shown in Fig. 7 indicate a further narrowed gap in group index by making this gap less abrupt. The line defects contribute to mode evolution during in-coupling and out-coupling since they share the same mode profile as the waveguide, as shown in Figs. 8(c) and 8(e). The coupled bridge (zigzag) interface state in Fig. 8(d) (Fig. 9(d)) does not bear mode distortion, because high-order modes are prevented from interfering with the coupling by the line defects. It relies on the line defect’s complete bandgap that blocks unnecessary scattering channels besides the specific coupling with interface states. According to the contour plots in Figs. 8(b) and 9(b), the wg1-to-wg2 total transmission efficiency Pwg2/Pwg1 reaches 95.8% (94.3%) for the bridge (zigzag) interface state coupling at [f (THz), Dwg (nm)] [165.13 (THz), 400 (nm)]. The two contours differ from those shown in Figs. 3(c) and 5(c) in the following aspects. A transmission spectral window spanning the complete frequency bandgap occurs when coupling to either the bridge or the zigzag interface in the presence of the line defects. The spectral window can be maintained across a broad waveguide size range, basically from 300 nm to 1000 nm, which indicates that the waveguide size is no longer a limiting factor. The line defects can be used in a variety of applications involving large-size silicon waveguides.
Fig. 8. Frequency-domain simulated (a) E-field norm distributions (b) transmission efficiency Pwg2/Pwg1 for the bridge wg1-defect-interface-defect-wg2 coupled system. (c-e) Zoom-in views of the field distributions. The silicon waveguide dimension is Dwg = 400 nm.
Fig. 9. Frequency-domain simulated (a) E-field norm distributions (b) transmission efficiency Pwg2/Pwg1 for the zigzag wg1-defect-interface-defect-wg2 coupled system. (c-e) Zoom-in views of the field distributions. The silicon waveguide dimension is Dwg = 400 nm.
The following is a brief summary that explains how introducing line defects improves the mode coupling. Firstly, the line defect mode has a ${n_\textrm{g}}$ between the waveguide mode and the interface state, thereby effectively reducing the difference between their indices. Secondly, the mode profile of defect mode bears similarity to both waveguide and topological interface, which facilitates the mode evolution during in-coupling and out-coupling processes. In the simulations above, the line defects have a length of 30$a$ that is sufficient to facilitate mode evolution. Our simulation results also show that as long as the line defect's length is greater than 10$a$ a high transmission efficiency (> 90%) can be maintained (see Fig. S3 in Supplement 1).
Figures 10(a)–10(d) illustrate the distributions of energy flow in the end-butt coupled system, showing back-reflections and mode distortions. The in-coupling and out-coupling have distinctly different energy flow densities and intensities. Contrary to this, in Figs. 10(e)–10 l, the energy flow from waveguide to interface is clear and concentrated when line defects are present, and vice versa. In this way, the suppression of unnecessary scattering channels and protection of mode profile have been validated further.
Fig. 10. Energy flow of the (a-d) wg1-interface-wg2 end-butt and (e-i) wg1-defect-interface-defect-wg2 coupled system, indicated by orange arrows.
We also carry out frequency-domain simulations to demonstrate the unidirectionality of the topological interface state and its robustness against sharp corners. A Z-shaped bridge interface is considered as shown in Fig. 11 where both defect mode and waveguide mode are considered as well. As shown in the overview of the E-field norm distribution in Fig. 11(a), whose zoom-in views can be found in Figs. 11(b)–11(d), a calculated transmission efficiency Pwg2/Pwg1 of 94.2% is obtained at f = 165.13 THz, which is only slightly different to a ratio of 95.8% when the Z-shaped path is absent from the transmission path as shown in Fig. 8.
Fig. 11. Simulated E-field norm distributions for the bridge wg1-defect-interface-defect-wg2 coupled system involving a Z-shaped path with two sharp corners (orange arrows). (b-d) Zoom-in views of the field distributions. The silicon waveguide dimension is Dwg = 400 nm.
3. Conclusions
In this article, we present a way to promote transmission performance of the two-dimensional QVH topological interface integrated with conventional silicon waveguides. The following conclusions can be drawn after sufficient simulations. Mode matching between the waveguide and QVH interface is essential to reduce back-reflections and mode distortions. This can be achieved by maximizing their field overlap according to the symmetry of the eigen-field distributions. Furthermore, a PhC line defect is positioned between the input/output waveguide and the interface to facilitate phase matching by shrinking the abrupt gap between their group indices. Another effective promotion to the mutual transition between the two types of modes (i.e., linearly polarized, and optical vortex carrying OAM) is the suppression of additional scattering channels enabled by the complete photonic bandgap of the line defect. In this way, the 2D QVH topological interface-waveguide coupled system presents a broad frequency band with high input-to-output total transmission efficiency (above 90%) and operates in single mode despite the large waveguides. Our simulation work has the potential to guide the future realization of 2D topological photonic systems. For example, QVH interface-cavity resonant systems can be built by directly integrating topological cavities into the interface. Combining valley degrees of freedom with existing photonic configurations can produce enhanced performance for optical filtering, switching, and multiplexing.
Funding
National Natural Science Foundation of China (11874274, 11974003); Priority Academic Program Development of Jiangsu Higher Education Institutions; the Science and Technology Project of Shenzhen (GJHZ20180928160407303); NWO Zwaartekracht program on Integrated Nanophotonics; ZJU-TU/e IDEAS project; Open Fund of the State Key Laboratory of Optoelectronic Materials and Technologies (Sun Yat-sen University); Sichuan Science and Technology Program (2020YFH0108); China Postdoctoral Science Foundation (2020M682863); State Key Laboratory of Information Photonics and Optical Communications (IPOC2020ZT01).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding author on reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nature Photon 11(12), 763–773 (2017). [CrossRef]
2. A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater 12(3), 233–239 (2013). [CrossRef]
3. M. S. Rider, S. J. Palmer, S. R. Pocock, X. Xiao, P. Arroyo Huidobro, and V. Giannini, “A perspective on topological nanophotonics: Current status and future challenges,” J. Appl. Phys. 125(12), 120901 (2019). [CrossRef]
4. J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nature Mater 16(3), 298–302 (2017). [CrossRef]
5. J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat Rev Mater 1(11), 16055 (2016). [CrossRef]
6. Q. Chen, L. Zhang, M. He, Z. Wang, X. Lin, F. Gao, Y. Yang, B. Zhang, and H. Chen, “Valley-Hall Photonic Topological Insulators with Dual-Band Kink States,” Adv. Opt. Mater. 7(15), 1900036 (2019). [CrossRef]
7. M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nature Nanotech 14(1), 31–34 (2019). [CrossRef]
8. Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci Rep 8(1), 1588 (2018). [CrossRef]
9. L. Ye, Y. Yang, Z. H. Hang, C. Qiu, and Z. Liu, “Observation of valley-selective microwave transport in photonic crystals,” Appl. Phys. Lett. 111(25), 251107 (2017). [CrossRef]
10. X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat Commun 8(1), 1304 (2017). [CrossRef]
11. D. Smirnova, A. Tripathi, S. Kruk, M.-S. Hwang, H.-R. Kim, H.-G. Park, and Y. Kivshar, “Room-temperature lasing from nanophotonic topological cavities,” Light Sci Appl 9(1), 127 (2020). [CrossRef]
12. M. Yuan, T. Xu, and Z. Hang, “Construction of Optical Topological Cavities using Photonic Crystals,” Light Sci Appl 9(1), 127 (2020). [CrossRef]
13. H. Zhong, Y. Li, D. Song, Y. V. Kartashov, Y. Zhang, Y. Zhang, and Z. Chen, “Topological Valley Hall Edge State Lasing,” Laser Photonics Rev. 14(7), 2000001 (2020). [CrossRef]
14. Y. Gong, S. Wong, A. J. Bennett, D. L. Huffaker, and S. S. Oh, “Topological Insulator Laser Using Valley-Hall Photonic Crystals,” ACS Photonics 7(8), 2089–2097 (2020). [CrossRef]
15. W. Noh, H. Nasari, H.-M. Kim, Q. Le-Van, Z. Jia, C.-H. Huang, B. Kanté, B. Kanté, and B. Kanté, “Experimental demonstration of single-mode topological valley-Hall lasing at telecommunication wavelength controlled by the degree of asymmetry,” Opt. Lett. 45(15), 4108–4111 (2020). [CrossRef]
16. X.-T. He, E.-T. Liang, J.-J. Yuan, H.-Y. Qiu, X.-D. Chen, F.-L. Zhao, and J.-W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat Commun 10(1), 872 (2019). [CrossRef]
17. G.-C. Wei, Z.-Z. Liu, D.-S. Zhang, and J.-J. Xiao, “Frequency dependent wave routing based on dual-band valley-Hall topological photonic crystal,” New J. Phys. 23(2), 023029 (2021). [CrossRef]
18. M. Makwana, R. Craster, and S. Guenneau, “Topological beam-splitting in photonic crystals,” Opt. Express 27(11), 16088 (2019). [CrossRef]
19. Y. Wang, W. Zhang, and X. Zhang, “Tunable topological valley transport in two-dimensional photonic crystals,” New J. Phys. 21(9), 093020 (2019). [CrossRef]
20. L. Gu, Q. Yuan, Q. Zhao, Y. Ji, Z. Liu, L. Fang, X. Gan, and J. Zhao, “A Topological Photonic Ring-Resonator for On-Chip Channel Filters,” J. Lightwave Technol. 39(15), 5069–5073 (2021). [CrossRef]
21. C.-Y. Ji, Y. Zhang, B. Zou, and Y. Yao, “Robust Fano resonance in the photonic valley Hall states,” Phys. Rev. A 103(2), 023512 (2021). [CrossRef]
22. H. Yoshimi, T. Yamaguchi, Y. Ota, Y. Arakawa, and S. Iwamoto, “Slow light waveguides in topological valley photonic crystals,” Opt. Lett. 45(9), 2648 (2020). [CrossRef]
23. X. Xie, S. Yan, J. Dang, J. Yang, S. Xiao, Y. Wang, S. Shi, L. Yang, D. Dai, Y. Yuan, N. Luo, T. Cui, G. Chi, Z. Zuo, B.-B. Li, C. Wang, and X. Xu, “Topological Cavity Based on Slow-Light Topological Edge Mode for Broadband Purcell Enhancement,” Phys. Rev. Applied 16(1), 014036 (2021). [CrossRef]
24. Y. Yang, Y. Yamagami, X. Yu, P. Pitchappa, J. Webber, B. Zhang, M. Fujita, T. Nagatsuma, and R. Singh, “Terahertz topological photonics for on-chip communication,” Nat. Photonics 14(7), 446–451 (2020). [CrossRef]
25. J. Ma, X. Xi, and X. Sun, “Topological Photonic Integrated Circuits Based on Valley Kink States,” Laser Photonics Rev. 13(12), 1900087 (2019). [CrossRef]
26. M. Jalali Mehrabad, A. P. Foster, R. Dost, E. Clarke, P. K. Patil, A. M. Fox, M. S. Skolnick, and L. R. Wilson, “Chiral topological photonics with an embedded quantum emitter,” Optica 7(12), 1690 (2020). [CrossRef]
27. T. Yamaguchi, Y. Ota, R. Katsumi, K. Watanabe, S. Ishida, A. Osada, Y. Arakawa, and S. Iwamoto, “GaAs valley photonic crystal waveguide with light-emitting InAs quantum dots,” Appl. Phys. Express 12(6), 062005 (2019). [CrossRef]
28. B. Orazbayev and R. Fleury, “Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides,” Nanophotonics 8(8), 1433–1441 (2019). [CrossRef]
29. S. Arora, T. Bauer, R. Barczyk, E. Verhagen, and L. Kuipers, “Direct quantification of topological protection in symmetry-protected photonic edge states at telecom wavelengths,” Light Sci Appl 10(1), 9 (2021). [CrossRef]
30. M. L. N. Chen, L. jun Jiang, Z. Lan, and W. E. I. Sha, “Pseudospin-Polarized Topological Line Defects in Dielectric Photonic Crystals,” IEEE Trans. Antennas Propag. 68(1), 609–613 (2020). [CrossRef]
31. M. J. A. De Dood, E. Snoeks, A. Moroz, and A. Polman, “Design and optimization of 2D photonic crystal waveguides based on silicon,” Opt. Quantum Electron. 34(1/3), 145–159 (2002). [CrossRef]
32. G. Sun and A. G. Kirk, “On the relationship between Bloch modes and phase-related refractive index of photonic crystals,” Opt. Express 15(20), 13149 (2007). [CrossRef]
33. M. Ghebrebrhan, M. Ibanescu, S. G. Johnson, M. Soljaćić, and J. D. Joannopoulos, “Distinguishing zero-group-velocity modes in photonic crystals,” Phys. Rev. A 76(6), 063810 (2007). [CrossRef]
