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Abstract
We have proposed a tunable topological slow-light in a photonic crystal (PC) waveguide with unified magnetic field. The waveguide is constructed by bringing close two gyromagnetic photonic crystals (GPCs) with different structural parameters and introducing a row of Al2O3 rods as the coupling layer. The two GPCs are applied with a unified external static magnetic field (ESMF) instead of two opposite ESMFs. Such waveguide supports a slow-light state originated from the coupling effect of two one-way edge states on both sides of the waveguide. By simply changing the strength of ESMF, one can achieve a tunable slow-light state with large normalized delay-bandwidth product (NDBP) (0.36< NDBP <0.84). Based on these excellent properties, we further design an optical delayer with a compact structure and expansibility simultaneously. This unique topological slow-light state with simple unified magnetic condition, high maneuverability and strong immunity to defects holds promise for many fields such as signal processing, optical modulation, and the design of various slow-light devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Slow-light with a remarkably low group velocity (vg) has attracted a lot of attention in the past decades for its promising properties in enhancing light–matter interaction strength, such as time-domain processing of optical signal and spatial compression of optical energy [1,2] etc. Therefore, slow-light can significantly reduce the energy and optical path required to realize the same linear and nonlinear effects as photons fully interact with matter compared to the light at conventional speeds [3,4]. In addition, the flat dispersion curve means the photon momentum of slow-light is more sensitive to the frequency change. These characteristics allow us to design ultracompact, low-power, and high-sensitive optical devices base on slow-light, such as optical buffer [5,6], optical delay [7,8], and optical switch [9,10]. The typical technologies for slow-light generation include Bose-Einstein condensation [11,12], electromagnetically induce transparency [13], stimulated Brillouin amplification [14], simulated Raman scattering [15,16], optical fiber [16–18], and so on. However, these traditional technologies usually require complex devices with strict operating conditions.
Photonic crystal (PC) has been proved to be an excellent platform for realizing slow-light, which possesses many advantages such as room-temperature operation, large bandwidth, compatibility with on-optical-chip integration and high manipulation of dispersion relationship [19–24]. Up to now, various PC waveguides with different structures (such as PC coupled cavity [25], traditional ordinary PC [26,27], chirped PC [28,29], valley PC [30,31] and magnetic-fluid-infiltrated PC [32]) have been used to create slow-light successfully. With the flourishing development of topological photonics in recent years, researchers have found that gyromagnetic photonic crystal (GPC) is an excellent system to generate one-way state because of the external static magnetic field (ESMF) and the consequent broken time-reversal symmetry [33–35]. Subsequently, many works reported a rich wealth of interesting photonic behaviors based on GPC, including chiral one-way edge state [35,36], anti-chiral one-way edge state [37], one-way bulk state [38,39], and especially slow-light transmission [19,20].
There are mainly three types of slow-light effects based on GPC systems [40]. The first type is nontopological slow-light which usually realized in a waveguide consisting of two identical GPCs applied with a single magnetic bias [41–43]. The generation of this type of the slow-light can be attributed to the strong coupling of two anti-propagating one-way edge states. It is not topologically protected and suffers from the scattering caused by disorders and defects. The second type is conventional topological slow-light with narrow effective bandwidth [44–47]. In these systems, the one-way edge states generally traverse the bulk bandgap within a single Brillouin zone. Furthermore, it can be engineered to realize topological slow-light over a narrow frequency window by tuning the geometrical parameters of the edge rods [46,47]. The third type is broadband topological slow-light which was proposed by Guglielmon and Rechtsman based on the Haldane model of a honeycomb GPCs [48]. The dispersion curve of one-way edge states can wind many times around the Brillouin zone. Such broadband topological slow-light is realized theoretically by modifying the nearest-neighbor and next-nearest-neighbor couplings in a topological insulator lattice. Recently, some works have translated this idea into a realistic photonic structure through periodically loading designed resonators at the edge termination of a photonic topological insulator [40,49–51]. Inspired by the above researches, we find that introducing a tunable coupling mechanism of one-way edge states in GPC system is a good way to generate topological slow-light. What’s more, a tunable topological slow-light can be realized by using the similar response characteristics of GPCs with the same structure to the change of ESMF.
In this work, we propose a physical mechanism to realize tunable topological slow-light state in a one-way waveguide constructed by bringing close two GPCs with appropriate structural parameters and introducing a row of Al2O3 PC in between as the coupling layer. Both a slow-light state and a fast-light state are supported in this waveguide under a unified ESMF. Especially, the slow-light state originates from the strong coupling effect of two one-way sub edge modes on both sides of the waveguide. The group velocity and operation bandwidth of the slow-light state can be tuned by simply changing the strength of the ESMF. What’s more, an optical time delayer with a compact structure and expansibility is designed. These results have potential in various fields such as signal processing, optical modulation, and the design of various slow-light devices.
2. Structural design and band structure analysis
Let us start with two isolated triangular GPCs composed of cylindrical yttrium iron garnet (YIG) rods with different radii. Both of two GPCs have the same lattice constant a = 30 mm under the same ESMFs of H0 = + 1800 Oe. The YIG rods of supercells in Figs. 1(a1) and 1(a2) have radii of r1 = 0.1a and r2 = 0.12a, respectively. The relative permittivity of YIG is ε1 = 15. Two metallic claddings (denoted by yellow color) are placed close to the edge of each GPC to form two isolated waveguides whose width are wd1 = wd2 = a. When no ESMF is applied, the magnetic permeability of YIG rods is µ = 1. However, when an ESMF H0 is applied to the YIG rods along the + z direction, the gyromagnetic anisotropy is strongly induced to break time-reversal symmetry and the magnetic permeability of fully magnetized YIG rods becomes a tensor as follows,
Fig. 1. (a1), (a2) Two GPC supercells with different YIG rods radii (r1 = 0.1a, r2 = 0.12a) which are used to calculate the projected band structures of TM mode. (b1), (b2) The projected band structures of TM mode for both GPCs shown in (a1) and (a2) respectively. (c1), (c2) The Ez field distributions of M1 and M2 modes for both GPC waveguides when a line source denoted by the magenta stars is excited at ωwork = 0.6(2πc/a).
The projected band structure of transverse magnetic (TM) mode with electric field parallel to z axis for the two isolated triangular GPCs are calculated with the aid of the commercial finite-element analysis package COMSOL Multiphysics, as shown in Figs. 1(b1) and 1(b2) respectively. In addition, floquet periodic boundary and scattering boundary condition conditions were used to simulate the eigenmode-field distribution. Obviously, there exist two counter-propagating one-way edge states (i.e., the red and blue curves) in the second and third band gaps respectively for both GPCs in Figs. 1(a1) and 1(a2). The properties of one-way edge states can be characterized by Chern number Cn and the gap Chern number Cgap which is calculated by summing the Cn of all the bands below the band-gap (${C_{gap}} = \sum {{C_n}} $) [52]. The Chern number of the nth band of a 2D periodic photonic crystal is an integer defined by [35]:
Based on these two isolated co-propagating one-way edge states at the presence of the same ESMF, we further construct a line defect waveguide by moving two GPCs in Figs. 1(a1) and 1(a2) close to each other head by head (the metallic slabs are removed). Besides, one Al2O3 layer with a relative permittivity of ε3 = 8.9 is added in the middle of the waveguide to provide the coupling effect between two topologically YIG GPCs, as shown in Fig. 2(a). The radius of Al2O3 rod and width of the waveguide play key roles in the coupling effect. After careful calculations, we choose wd3 = 1.35a and r3 = 0.25a for our further study.
Fig. 2. (a) The supercell includes YIG rods with different radius (r1, r2) on both sides and a layer of Al2O3 rods with radius r3 in the middle. (b) The projected band structure. A purple dashed line represents a typical frequency ω=0.606(2πc/a) intersects with two bands at M3 and M4 points respectively. (c) The intensity and phase distributions of Ez fields of M3 mode and M4 mode.
There are two one-way waveguide states (marked by red and blue) in the band gap [ranging from 0.596(2πc/a) to 0.622(2πc/a)] as shown in Fig. 2(b). More importantly, the red band curve descends from kx = -0.11(2π/a), later flattens out in the Brillouin zone range of 0.04(2π/a) < kx < 0.23(2π/a), and continues to decline with increasing kx. The changing trend of the curve meaning that its slope is always negative. As a result, a one-way slow-light state is achieved. To deeply explore the characteristics of these waveguide modes, we choose a typical frequency [ωs = 0.606(2πc/a)] in the bandgap which intersects with two one-way waveguide bands at points M3 and M4, respectively. Figure 2(c) shows the eigen field of |E| and the corresponding phase distributions of the M3 and M4 modes, respectively. One can find that the slow-light state (i.e., the M3 mode) is an even-like mode with incomplete symmetry, while the M4 mode shows the characteristics of odd-like mode but with asymmetrical distributions of |E| and phase on both sides of the waveguide. It is remarkable that these two one-way modes (i.e., the M3 and M4 modes) emerge at the simple condition of unified ESMF, in contrast to those edge modes in conventional GPC waveguides which must be applied with two reverse ESMFs strictly.
3. Group velocity and group velocity dispersion characteristics
We further explore the propagation characteristics of the M3 and M4 modes. We mainly focus on three important parameters, i.e., group velocity vg, group index ng and group velocity dispersion (GVD), which are often used to evaluate the performance of slow-light modes. The calculation formula of vg, ng and GVD are as follows [54],
As shown in Fig. 3(a1), for the M3 mode, vg is always negative or equal to zero. The negative group velocity indicates that M3 mode is transmitted to the left in the waveguide. When ${k_x}$ ncreases from 0.09(2πc/a) to 0.24(2πc/a), vg first ascends from -0.013c to 0 and then descends slowly to -0.012c. The M3 mode has near-zero group velocity (vg ≈ 0) in the flat region near ks = 0.15(2π/a), indicating that the M3 mode is a one-way slow-light mode. Figure 3(b1) shows the GVD of the M3 mode. It can be observed that the GVD tends to infinity when the frequency closes to ωs = 0.606(2πc/a) (corresponding to vg = 0). This feature can be explained by formula (5) that GVD is inversely proportional to vg3. Therefore, if vg tends to zero, the GVD will tend to infinity. Obviously, the slow-light mode with lower vg (also meaning high ng) tends to coincide with larger GVD. Thus, the signal pulse will spread after transmitting a certain distance in waveguide with excessive GVD (such as points ω3 and ω4 denoted in Fig. 3(b1)), resulting in the overlap of two adjacent optical signal pulses. However, interestingly, the GVD tends to zero sharply if the frequency is away from ωs, meaning that there are states near ωs with very small vg (but not zero) and acceptable GVD simultaneously, such as ω2 = 0.6049(2πc/a) with vg2 = -0.012c, ng2 = -83.33, GVD2 = -1.4 × 10−18s2/mm and = 0.6067(2πc/a) with vg5 = -0.027c, ng5 = -37.04, GVD5 = 1.1 × 10−18s2/mm. The states away from ωs have near zero GVD, such as ω1 = 0.6032(2πc/a) and ω6 = 0.6087(2πc/a), but possess too high vg simultaneously.
Fig. 3. Group velocity (vg) and group velocity dispersion (GVD) curve. (a1) (a2) vg curves of the M3 and M4 modes, respectively. the purple dash line in (a1) denotes vg = 0. (b1) (b2) GVD curves of the M3 and M4 modes, respectively. All insets in Fig. 3 are partial enlarged view.
On contrary, the case of the M4 mode is quite distinctive from that of the M3 mode. The vg of M4 mode is always less than -0.05c as shown in Fig. 3(a2). Therefore, we call M4 mode the fast-light state in contrast to the M3 mode. Besides, the M4 mode with three zero GVD points (ω7, ω8 and ω9) shows a completely different transmission mode. In the follow sections, we mainly focus on the M3 slow-light mode.
4. Tunable one-way slow-light state
We proceed to study the tunable characteristics of the one-way slow-light state. The diagram of the slow-light states (M3 mode) with different ESMFs is shown in Fig. 4(a). As H decreases from 1800 Oe to 1600 Oe, the curve shifts to a lower frequency region. Besides, the middle near-linear part of the curve (denoted by colored line) gradually slopes to right scarcely, indicating that it still has excellent slow-light characteristics. This image clearly reveals that the slow-light mode can be magnetically tuned within a considerable frequency range. We further study the influence of ESMF on the group velocity and GVD of the M3 mode. Figures 4(b) and 4(c) illustrate that the vg gradually becomes larger with decreasing GVD as H decreases from 1800 Oe to 1600 Oe.
Fig. 4. (a) The M3 mode with H ranging from 1600 Oe to 1800 Oe. The colored lines represent the near-linear parts which are used to calculate the NDBP. (b) and (c) are vg and GVD curves of the near-linear part when H varies from 1600 Oe to 1800 Oe, respectively.
The characteristics of the M3 and M4 modes have shown that a state with low vg tends to match large GVD. The highly dispersion would introduce a strong dependence on wavelength, and thus it is easy to cause pulse broadening and distortion of optical signal. Therefore, it is prominent and critical to take measures to balance vg and GVD. To evaluate the slow-light states more comprehensively, we adopt another important parameter called normalized delay-bandwidth product (NDBP) [27–29], which is more reasonable for evaluating and comparing the performance of slow-light devices working in different frequency range. NDBP is given by [54],
Table 1. The NDBP calculation data with H ranging from 1600 Oe to 1800 Oe.
5. Numerical simulations of the slow-light state
In order to further explore the transmission behaviors of the M3 mode, we construct a waveguide channel consisting of 30 supercells in x direction with a waveguide width of 1.35a. Two in-phase 2D line-sources oscillating at ωs = 0.606(2πc/a) are symmetrically placed on at the both sides of the middle of waveguide at x = 20.5a, as denoted by the purple stars in Fig. 5(a). A left-propagating one-way edge state is excited [Fig. 5(a)]. The amplitude of electric field (|E|) along the centers of the rods of Al2O3 PC layer is calculated and shown in Fig. 5(b). It clearly shows the topology property of the one-way mode from the view of numerical terms. On the one hand, when $x$ anging from 0 to 22a, the |E| always fluctuates near 2.5 × 106 V/m, indicating that the light field is stably transmitted to the left. On the other hand, the amplitude of electric field decays rapidly to zero when x is greater than 22a, indicating that light is not allowed to transmit to the right.
Fig. 5. Ez field, the amplitude of electric field (|E|) and Poynting vector distributions. (a) Ez field distribution in the waveguide of the M3 mode. Two magenta stars denote line sources. (b) |E| of the M3 mode along purple dashed cut line 1 in (a). (c) The Poynting vector distribution.
To clearer understand the physical picture of the slow-light, the Poynting vector distribution is calculated and presented in Fig. 5(c). It illustrates a complicated internal energy flux circulation that exhibits an eight-shaped loop (denoted by blue arrows) generated by strong coupling effect of two sub-waveguide one-way modes. The energy concentrated at Al2O3 rods manifests that there exists the strong energy exchange between the two sub-waveguides through the Al2O3 rod coupling layer. Therefore, the physical picture of the slow-light mode (M3 mode) can be summarized as: Two left-transmitting edge modes of the upper and lower sub-waveguides are generated from the broken time-reversal symmetry because of the positive ESMF. The M3 mode emerges as a result of the strong coupling effect between the two one-way edge modes when the coupling layer is introduced into the middle of the waveguide. The energy flows in the form of an eight-shaped loop since it is strongly localized in the marginal YIG rods and center Al2O3 rods. Finally, the energy flux flows very slowly between adjacent eight-shaped loops, leading to the slow-light phenomenon.
6. Robustness analysis of the slow-light mode
To demonstrate the robustness of the M3 mode, a rectangular perfect electric conduct (PEC) obstacle with size of 1.5a × 0.1a is inserted vertically in the center of the waveguide. The distributions of Ez and |E| are shown in Figs. 6(a) and 6(b). They clearly show that the EM wave gets around the PEC and continuously propagates leftwards with almost no backscattering [Figs. 6(a) and 6(b)]. The physical picture of such strong robustness is further verified by the Poynting vector distributions and longitudinal electric field intensity distributions shown in Figs. 6(c) and 6(d) respectively. The energy flux bypasses the both ends of the PEC and then recovers its original distributions to continuously propagate along the waveguide, which is verified by |E| distributions along cut lines 2 and 3 in Figs. 6(a).
Fig. 6. Ez field, the amplitude of electric field (|E|), Poynting vector distributions and transmission. (a) The Ez field distribution of M3 mode with a metallic rectangular obstacle which marked in black color. (b) (d) |E| along cut line 1, cut line 2 and cut line 3 in (a) respectively. (c) The Poynting vector distribution around the metallic obstacle. (e) The transmission of M3 mode without and with the metallic obstacle respectively.
In Fig. 6(e), the transmission spectra from right to left in the bandgap are both nearly 100% for the cases without or with the PEC obstacle, which also proves the robustness of the M3 mode. Despite a PEC cannot block the transmission of one-way states, it can cause a phase delay of the one-way modes, which can be found by comparing the minimum or maximum amplitude distribution in Figs. 5(a) and 6(a).
7. Design of optical delayer
Based on the excellent characteristic of the M3 mode, we design an optical delayer which is shown in Fig. 7(a). The length of the delayer is 38a in the x direction and 18a in the y direction. Three 120° bends are used to construct a 180° deflection of the parallel waveguide in order to achieve large optical signal delay with a compact structure. In addition, we increase the density of the Al2O3 rods to ensure the coupling strength of the GPCs on both sides of waveguide because of the larger waveguide width at the bend. The distance between the three parallel waveguides is $2\sqrt 3 a$, which is sufficient to ensure optical signals transmit in parallel waveguides without signal crosstalk.
Fig. 7. Schematic showing of optical delayer and Ez field distribution. (a) Structure of the proposed delayer where two in-phase line sources are excited in the waveguide. The structural parameters and ESMF are the same as Fig. 3. (b) Ez field distribution.
Figure 7(b) displays the schematic diagram of optical signal transmitting in the optical delayer. For the sake of without loss of generality, we set two in-phase line sources oscillating at ω = 0.600(2πc/a) as denoted in Fig. 7(a) under the ESMF H = + 1750 Oe as an example. The EM wave transmits leftwards firstly in the top parallel waveguide. When the optical signal arrives at the curved waveguide, it is effectively coupled to the middle parallel waveguide and propagates forwards due to the higher density of the Al2O3 rods. This transmission behavior raises a possibility that the optical signal transmits layer by layer in parallel waveguides and finally reaches the left side. Moreover, the topological property of the slow-light state ensures that the optical signal can only be transmitted from the right side to the left side. The idea of introducing the curved waveguide not only reduces the length of the delayer, but also enhances the expansibility of the optical delayer, such as increasing the number of parallel waveguides to improve the delay time and forming a Y-shaped waveguide, etc. The time delay through the optical delayer can be calculated by the following formula,
where L is the effective length of the waveguides through which the optical signal is transmitted in the delayer, vg is the group velocity of optical signal. After simple qualitative analysis, the delay time of this optical signal transmitting in the horizontal part of the three parallel waveguides (approximately 80a long) is about 1.04µs while it will take 8 ns for the optical signal to propagate the same distance in vacuum.Conclusions
In conclusions, we first study the properties of two topological one-way edge states in two isolated triangular PCs. Then we construct a waveguide by bringing close two GPCs with different radii of YIG rods which are applied with a unified ESMF and introducing a row of Al2O3 rods as the coupling layer. Such waveguide supports a slow-light state (M3 mode) and a fast-light state (M4 mode) originated from the coupling effect of two one-way edge states on both sides of. By simply changing the strength of ESMF, we achieve a tunable slow-light state with large NDBP. Furthermore, we design an optical delayer with a compact structure and expansibility simultaneously. This unique topological slow-light state with simple unified magnetic condition, high maneuverability and strong robustness against defects holds promise for many fields such as signal processing, optical modulation, and the design of various slow-light devices.
Funding
National Natural Science Foundation of China (12074127); Natural Science Foundation of Guangdong Province (2023A1515010951); National Key Research and Development Program of China (2018YFA 0306200); Special Project for Research and Development in Key areas of Guangdong Province (2020B010190001); Guangdong Province Introduction of Innovative R&D Team (2016ZT06C594).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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.
References
1. Y. Zhao, H. W. Zhao, X. Y. Zhang, B. Yuan, and S. Zhang, “New mechanisms of slow light and their applications,” Opt. Laser Technol. 41(5), 517–525 (2009). [CrossRef]
2. T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2(8), 448–450 (2008). [CrossRef]
3. C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, and L. O’Faolain, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Select. Topics Quantum Electron. 16(1), 344–356 (2010). [CrossRef]
4. N. A. Mortensen and S. Xiao, “Slow-light enhancement of Beer-Lambert-Bouguer absorption,” Appl. Phys. Lett. 90(14), 374 (2007). [CrossRef]
5. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, “Optimization of the storage capacity of slow light photonic crystal waveguides,” Opt. Lett. 37(22), 4585–4587 (2012). [CrossRef]
6. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: Comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005). [CrossRef]
7. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71(2), 023801 (2005). [CrossRef]
8. R. Almhmadi, “Slow Light Mode at the Degenerate Band Edge for True Time Delay Applications,” IEEE Photonics Technol. Lett. 34(12), 657–660 (2022). [CrossRef]
9. L. Torrijos-Moran, A. Brimont, A. Griol, P. Sanchis, and J. Garcia-Ruperez, “Ultra-Compact Optical Switches Using Slow Light Bimodal Silicon Waveguides,” J. Lightwave Technol. 39(11), 3495–3501 (2021). [CrossRef]
10. C. Husko, T. D. Vo, B. Corcoran, J. Li, T. F. Krauss, and B. J. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19(21), 20681–20690 (2011). [CrossRef]
11. Z. Dutton, “Observation of Quantum Shock Waves Created with Ultra Compressed Slow Light Pulses in a Bose-Einstein Condensate,” Science 293(5530), 663–668 (2001). [CrossRef]
12. F. Grusdt and M. Fleischhauer, “Tunable Polarons of Slow-Light Polaritons in a Two-Dimensional Bose-Einstein Condensate,” Phys. Rev. Lett. 116(5), 053602 (2016). [CrossRef]
13. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
14. Z. Zhu, A. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS Slow Light in an Optical Fiber,” J. Lightwave Technol. 25(1), 201–206 (2007). [CrossRef]
15. G. Qin, R. Jose, and Y. Ohishi, “Stimulated Raman scattering in tellurite glasses as a potential system for slow light generation,” J. Appl. Phys. 101(9), 497–529 (2007). [CrossRef]
16. O. D. Herrera, L. Schneebeli, K. Kieu, R. A. Norwood, and N. Peyghambarian, “Slow light based on stimulated Raman scattering in an integrated liquid-core optical fiber filled with CS2,” Opt. Express 21(7), 8821–8830 (2013). [CrossRef]
17. M. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-Bandwidth Brillouin Slow Light in Optical Fibers,” Opt. Express 14(4), 1395–1400 (2006). [CrossRef]
18. K. Totsuka and M. Tomita, “Dynamics of fast and slow pulse propagation through a microsphere-optical-fiber system,” Phys. Rev. E 75(1), 016610 (2007). [CrossRef]
19. C. Bohley, V. Jandieri, B. Schwager, R. Khomeriki, D. Schulz, D. Erni, D. H. Werner, and J. Berakdar, “Thickness-dependent slow light gap solitons in three-dimensional coupled photonic crystal waveguides,” Opt. Lett. 47(11), 2794–2797 (2022). [CrossRef]
20. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008). [CrossRef]
21. Y. Lu, Q. Liu, Y. Chen, Z. Chen, D.-W. Zhang, H. Deng, S.-L. Zhu, P. Han, N.-H. Shen, T. Koschny, and C. M. Soukoulis, “Topological Transition Enabled by Surface Modification of Photonic Crystals,” ACS Photonics 8(5), 1385–1392 (2021). [CrossRef]
22. G. J. Tang, X. T. He, F. L. Shi, J. W. Liu, X. D. Chen, and J. W. Dong, “Topological Photonic Crystals: Physics, Designs, and Applications,” Laser Photonics Rev. 16(4), 2100300 (2022). [CrossRef]
23. T. Yu, R. Luo, T. B. A. Wang, D. J. Zhang, W. X. Liu, T. B. Yu, and Q. H. Liao, “Enhancement of Casimir Friction between Graphene-Covered Topological Insulator,” Nanomaterials 12(7), 1148 (2022). [CrossRef]
24. B. Toshihiko, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
25. M. Svaluto Moreolo, V. Morra, and G. Cincotti, “Design of photonic crystal delay lines based on enhanced coupled-cavity waveguides,” J. Opt. A: Pure Appl. Opt. 10(6), 064002 (2008). [CrossRef]
26. N. Le Thomas, H. Zhang, J. Jágerská, V. Zabelin, R. Houdré, I. Sagnes, and A. Talneau, “Light transport regimes in slow light photonic crystal waveguides,” Phys. Rev. B 80(12), 125332 (2009). [CrossRef]
27. M. Patterson, S. Hughes, S. Combrie, N. Tran, A. D. Rossi, R. Gabet, and Y. Jaoueen, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102(25), 253903 (2009). [CrossRef]
28. J. Hou, H. Wu, D. S. Citrin, W. Mo, D. Gao, and Z. Zhou, “Wideband slow light in chirped slot photonic-crystal coupled waveguides,” Opt. Express 18(10), 10567–10580 (2010). [CrossRef]
29. D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13(23), 9398–9408 (2005). [CrossRef]
30. H. Zhang, L. Qian, C. Wang, C. Y. Ji, Y. Liu, J. Chen, and C. Lu, “Topological rainbow based on graded topological photonic crystals,” Opt. Lett. 46(6), 1237–1240 (2021). [CrossRef]
31. H. Yoshimi, T. Yamaguchi, Y. Ota, Y. Arakawa, and S. Iwamoto, “Slow light waveguides in topological valley photonic crystals,” Opt. Lett. 45(9), 2648–2651 (2020). [CrossRef]
32. S. Pu, S. Dong, and J. Huang, “Tunable slow light based on magnetic-fluid-infiltrated photonic crystal waveguides,” J. Opt. 16(4), 045102 (2014). [CrossRef]
33. F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008). [CrossRef]
34. S. Raghu and F. Haldane, “Analogs of quantum Hall effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008). [CrossRef]
35. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef]
36. S. Liu, W. Lu, Z. Lin, and S. T. Chui, “Molding reflection from metamaterials based on magnetic surface plasmons,” Phys. Rev. B 84(4), 045425 (2011). [CrossRef]
37. J. Chen, W. Liang, and Z.-Y. Li, “Antichiral one-way edge states in a gyromagnetic photonic crystal,” Phys. Rev. B 101(21), 214102 (2020). [CrossRef]
38. J. Chen and Z. Y. Li, “Prediction and Observation of Robust One-Way Bulk States in a Gyromagnetic Photonic Crystal,” Phys. Rev. Lett. 128(25), 257401 (2022). [CrossRef]
39. M. Wang, R. Y. Zhang, L. Zhang, D. Wang, Q. Guo, Z. Q. Zhang, and C. T. Chan, “Topological One-Way Large-Area Waveguide States in Magnetic Photonic Crystals,” Phys. Rev. Lett. 126(6), 067401 (2021). [CrossRef]
40. F. Chen, Haoran Xue, Y. Pan, M. Wang, Y. Hu, L. Zhang, Q. Chen, S. Han, G.-g. Liu, Z. Gao, P. Zhou, W. Yin, H. Chen, B. Zhang, and Y. Yang, “Demonstration of broadband topological slow light,” arXiv, arXiv:2208.07228 (2022).
41. J. Chen, W. Liang, and Z.-Y. Li, “Switchable slow light rainbow trapping and releasing in strongly coupling topological photonic systems,” Photonics Res. 7(9), 1075 (2019). [CrossRef]
42. Y.-t. Fang, H.-Q. He, J.-x. Hu, L.-k. Chen, and Z. Wen, “Flat and self-trapping photonic bands through coupling of two unidirectional edge modes,” Phys. Rev. A 91(3), 033827 (2015). [CrossRef]
43. J. Chen, W. Liang, and Z.-Y. Li, “Strong coupling of topological edge states enabling group-dispersionless slow light in magneto-optical photonic crystals,” Phys. Rev. B 99(1), 014103 (2019). [CrossRef]
44. J. Chen, W. Liang, and Z. Y. Li, “Broadband dispersionless topological slow light,” Opt. Lett. 45(18), 4964–4967 (2020). [CrossRef]
45. Y. Liu and C. Jiang, “Dispersionless one-way slow wave with large delay bandwidth product at the edge of gyromagnetic photonic crystal,” Int. J. Mod. Phys. B 34(10), 2050086 (2020). [CrossRef]
46. Z. Lan, J. W. You, and N. C. Panoiu, “Nonlinear one-way edge-mode interactions for frequency mixing in topological photonic crystals,” Phys. Rev. B 101(15), 155422 (2020). [CrossRef]
47. Y. Yang, Y. Poo, R.-x. Wu, Y. Gu, and P. Chen, “Experimental demonstration of one-way slow wave in waveguide involving gyromagnetic photonic crystals,” Appl. Phys. Lett. 102(23), 231113 (2013). [CrossRef]
48. J. Guglielmon and M. C. Rechtsman, “Broadband Topological Slow Light through Higher Momentum-Space Winding,” Phys. Rev. Lett. 122(15), 153904 (2019). [CrossRef]
49. S. A. Mann and A. Alu, “Broadband Topological Slow Light through Brillouin Zone Winding,” Phys. Rev. Lett. 127(12), 123601 (2021). [CrossRef]
50. S. A. Mann, D. L. Sounas, and A. Alù, “Broadband delay lines and nonreciprocal resonances in unidirectional waveguides,” Phys. Rev. B 100(2), 020303 (2019). [CrossRef]
51. L. Yu, H. Xue, and B. Zhang, “Topological slow light via coupling chiral edge modes with flatbands,” Appl. Phys. Lett. 118(7), 071102 (2021). [CrossRef]
52. S. A. Skirlo, L. Lu, Q. Yan, Y. Igarashi, J. Joannopoulos, and M. Soljacic, “Experimental Observation of Large Chern numbers in Photonic Crystals,” Phys. Rev. Lett. 115(25), 253901 (2015). [CrossRef]
53. R. Zhao, G. D. Xie, M. L. N. Chen, Z. Lan, Z. Huang, and W. E. I. Sha, “First-principle calculation of Chern number in gyrotropic photonic crystals,” Opt. Express 28(4), 4638–4649 (2020). [CrossRef]
54. Y. Zhao, Y.-N. Zhang, Q. Wang, and H. Hu, “Review on the Optimization Methods of Slow Light in Photonic Crystal Waveguide,” IEEE Trans. Nanotechnol. 14(3), 407–426 (2015). [CrossRef]
