Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Topological edge states in an all-dielectric terahertz photonic crystal

Open Access Open Access

Abstract

We present an analysis of the robustness of topological edge states in an all-dielectric photonic crystal slab in the terahertz (THz) frequency domain. We initially design a valley photonic crystal (VPC) exhibiting a nontrivial band topology. The excitation of the topological edge states in the structure is facilitated through a zigzag domain wall constructed by interfacing two types of VPCs with distinct band topologies. The robustness of the excited edge states is probed with respect to the magnitude and the sign of the asymmetry in terms of the hole diameters in the VPC, for different domain interfaces. Our study reveals that the topological edge states in the VPC structure are achieved only when the domain walls are formed by the larger air holes (i.e., asymmetry parameter has a positive value). In the case of the domain walls formed by relatively smaller air holes (i.e., asymmetry parameter has a negative value), the topological protection of the edge states is forbidden. For positive asymmetry, we demonstrate that the topological transport of THz becomes more robust with the increasing magnitude of asymmetry in the VPC structure. A robust propagation of topological edge states and strong confinement of electromagnetic fields within the domain wall are observed for asymmetry ranging from 28% to 42% in our structure. We have adopted a generic technique and therefore, the results of our study could be achieved at other frequency regimes by scaling the size parameters of the structure appropriately. At THz frequencies, such extensive analysis on the robustness of the topological edge states could be relevant for the realization of low-loss waveguides for 6G communication and other integrated photonic devices.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Over the recent years, topological photonic structures have fascinated the scientific community with their unique features which cannot be achieved with conventional photonic structures [15]. The most striking feature of a topological photonic structure is its ability to support electromagnetic wave propagation that is backscattering-free and robust against sharp bends and structural defects [39]. The perspective of topology was first introduced into the photonic realm by Haldane and Raghu [10,11], as an analogy of the quantum Hall effect [12]. The quantum Hall effect, which was discovered in the 1980s [12,13], demonstrates the presence of topologically protected states propagating along the perimeter of an insulating bulk material [14,15]. The first experimental realization of these topologically protected states in photonics was reported in 2009 [6], in the microwave frequency domain. In their study, the excitation of edge states was enabled by breaking the time-reversal symmetry in a 2D photonic crystal. Subsequently, topological edge states based on the quantum spin Hall effect [1619] and the quantum valley Hall effect [1922], where time-reversal symmetry is conserved, were realized in photonic structures. These successful emulations of topological edge states in photonic structures opened up new doors for designing devices with novel functionalities [2326] that can revolutionize the field of photonics [27]. Since its discovery, extensive theoretical investigations as well as experimental demonstrations, have been reported using several photonic structures based on gyromagnetic materials [6,28], bi-anisotropic metamaterials [29,30], ring resonators [31,32], Weyl semi-metals [33,34], photonic crystals [6,7,2022,35] and various other systems [3639].

Topologically protected edge states have been investigated in these systems, with the goal of realizing various applications, such as, backscattering-free waveguides [6,7], communication devices [40], splitters [19,22], lasers [41,42], slow light devices [4345], etc. Most of the investigations on topological transport of light, however, have been reported largely at the microwave [6,19,26,46] and optical frequencies [7,22,35,47,48], with very limited research at the terahertz (THz) frequencies [40,49,50]. The THz spectral band, typically ranging from $0.1$ THz to $10$ THz, is important for information transfer and communication [51,52]. The THz band, because of its massive available bandwidth and achievable data transfer up to several terabits-per-second [53], is the ideal spectral band for meeting the rising demand for high rate of data transfer [54]. However, the realization of this application is limited by loss and signal distortion suffered by conventional approaches [5557] due to back-reflection at sharp corners. Another bottleneck suffered by the THz spectral band is the lack of functional devices that could be easily integrated for on-chip applications [58,59]. In this context, topological photonic structures can be a viable solution in overcoming these limitations and provide the way forward in the development of easily integrable devices for various applications in the THz domain [40,49]. However, in spite of the considerable amount of importance, there is a significant lack of investigation of the topological phenomena in the THz domain. An extensive analysis of the topological photonic structures at the THz domain is essential in order to unveil its enormous untapped potential.

Therefore, in this article, we present an all-dielectric photonic crystal structure capable of supporting topological edge states in the THz domain. We initially engineer a valley photonic crystal (VPC) slab exhibiting a nontrivial band topology, by breaking the $C_{6}$ lattice symmetry of the structure. Motivated by the immense practical significance of the THz band in communication and information transfer, we have primarily focused our investigations on the THz frequency regime. One of the most attractive applications of THz devices is the development of low-loss waveguides for 6G communication. To realize this application using THz topological VPC structures, a clear understanding of the different structural configurations is essential. Hence, we perform a robustness analysis of the edge states as a function of asymmetry in a THz VPC structure. The novelty of our work stems from the investigation of the robustness of the edge states with respect to the magnitude and sign of the asymmetry in terms of the air hole diameters constituting the VPC structure. In the proposed VPC structure, the excitation of the THz topological edge states is facilitated by constructing a zigzag domain wall by interfacing two types of VPCs with different band topologies. Then, the robustness of the excited THz topological edge states is probed as a function of the asymmetry in the structure, for a straight and an $\Omega$-type domain wall (bending of $120^{\circ }$). A detailed analysis of the dispersion of the edge states, transmission spectra, and the electromagnetic field distributions is reported for different asymmetries in the structure. Such a comprehensive study along with a detailed analysis of the topological THz VPCs should be helpful in designing devices for 6G communication as well as various other on-chip applications in the THz domain. Our study could be extended to other frequency regimes [22,26,6063] by appropriately scaling the size parameters of the structure and by using a material having a similar dielectric constant [64]. This article is presented as follows: We first discuss the design of the THz VPC structure in section 2. In section 3, the excitation of the THz topological edge states in the VPC is discussed. Then, the effect of asymmetry on the robustness of the topological edge states is presented in section 4, followed by the conclusions in section 5.

2. Design of the terahertz valley photonic crystal

We initially investigate the bulk topology of the THz VPC structure based on a silicon-on-insulator platform. Figure 1 illustrates the design of the proposed VPC structure. The VPC design comprises of cylindrical air grooves arranged in a honeycomb lattice patterned on a Si slab of relative permittivity, $\epsilon _{r} =11.9$ with a fixed height $'h'$ atop a dielectric substrate with $\epsilon _{r} = 2.1$, having a thickness $'t'$. Figure 1(a) shows the front and lateral geometry of the proposed structure, with the red dashed line illustrating the honeycomb lattice. The unit cell of the VPC comprising of two cylindrical air grooves along with the first Brillouin zone is shown in Fig. 1(b). The parameter $'a'$ denotes the lattice constant of the unit cell while the diameters of the two cylindrical air grooves are denoted by $d_{1}$ and $d_{2}$, respectively. We also define a parameter, $\Delta d=(d_{1}-d_{2})$ representing the asymmetry of the diameter of the air grooves in our proposed THz VPC structure. Then, the proposed VPC structure could be described theoretically by an effective 2D Hamiltonian [3,7,10], which can be expressed as

$$H = {\nu}_{D} \big( {\sigma}_{x} {\delta}{k_{x}} + {\sigma}_{y} {\delta}{k_{y}} \big) + {\gamma} {\sigma}_{z}$$
where, $\nu _{D}$ represents the group velocity of the Dirac cone, $\sigma _{i}$ is the Pauli matrix, $\delta k_{x}$ and $\delta k_{y}$ are the momentum deviation measured from the Dirac points K and K$^\prime$ and $\gamma$ denotes the strength of symmetry breaking in the VPC structure. Then, the topological properties of such VPC structures can be understood from the Berry curvature and valley Chern number of the bands. The valley Chern number around the K/ K$^\prime$ point is expressed as $C_{K/K^\prime } = \frac {1}{2 \pi }\int _{\small {HBZ}} \Omega _{n}(k) \ d^2 k$ [7] where, $\Omega _{n}(k)= i \nabla _{k} < \times {u_{(n,k)}} |\nabla _{k}| {u_{(n,k)}}$ is the Berry curvature. Here, $n$ is the band index and $u_{(n,k)}$ denotes the $n^{th}$ eigenstate.

 figure: Fig. 1.

Fig. 1. (a) Schematic of the THz VPC structure with cylindrical air grooves arranged in a honeycomb lattice on a Si slab with relative permittivity, $\epsilon _{r} =11.9$ atop a substrate with $\epsilon _{r} = 2.1$, (b) The unit cell of the VPC comprising of two cylindrical air grooves having diameter $d_1$ and $d_2$, along with the first Brillouin zone, (c) The band structure of the VPC for the symmetric case of $\Delta d=0$ with fixed parameters: $a = 250$ µm, $h = 220$ µm, $d_1 = d_2 =110$ µm for TE mode. The light blue shaded region denotes the light cone of the substrate. The inset shows the pictogram of the unit cell for $\Delta d=0$.

Download Full Size | PPT Slide | PDF

In this investigation, we have fixed the geometrical parameters: $a = 250$ µm, $h= 220$ µm while the diameters of the cylindrical air grooves take values ranging from $20$ µm to $110$ µm. We have focused our analysis only on the transverse electric (TE) modes, which propagates in the x-y plane (i.e., electric field is perpendicular to the direction of propagation), and are confined in the z-direction. It is noteworthy to mention that a VPC structure with a standard honeycomb lattice with $C_{6}$ symmetry (when inversion symmetry is conserved), has a degeneracy at K (K$^\prime$) symmetry point of the first Brillouin zone in the band structure of TE modes. We have calculated the band structure of the VPC using the plane wave expansion method-based MIT photonic bands (MPB) package. The band structure for the TE modes for the symmetric case $(\Delta d=0)$, when the two cylindrical air grooves have the same diameter (i.e., $d_{1}= d_{2} = 110$ µm), is shown in Fig. 1(c). The light blue shaded region represents the light cone of the substrate. The red solid line denotes the first TE band while the blue solid line represents the second TE band of the VPC structure. It is evident from the figure that a degeneracy point between the two bands appear at $f = 0.336$ THz, at the K (K$^\prime$) symmetry point of the first Brillouin zone, which indicates the existence of a trivial band topology in the VPC structure in the presence of inversion symmetry [3,7]. This degeneracy at K (K$^\prime$) point in the band structure can be relaxed upon breaking the inversion symmetry in our proposed VPC structure. In our analysis, this symmetry breaking is realized by altering the diameter of one of the air grooves in the unit cell, while the diameter of the other groove is kept fixed. Subsequently, the $C_{6}$ symmetry of the VPC structure reduces to a $C_{3}$ symmetry, due to which the degeneracy at $f = 0.336$ THz is relaxed.

The band structure for the TE modes for different values of the asymmetry parameter $(\Delta d)$, obtained by fixing $d_{1} = 110$ µm and varying $d_{2}$ from $100$ µm to $20$ µm, is shown in Fig. 2. For $d_{1} = 110$ µm, the largest asymmetry value that can be achieved in the VPC structure without closing up the smaller air grooves is $\Delta d \sim 0.42a$. It is also worth mentioning that the largest value of the hole diameter that can be assumed without overlapping of the two grooves is $d_1 = d_2 = 144$ µm. For $d_1 (d_2) = 144$ µm, the highest asymmetry magnitude that can be achieved in the VPC structure is $\Delta d \sim 0.56 a$. In the figure, the light blue shaded region denotes the light cone of the substrate while the pink shaded region represents the band gap of the asymmetric VPC structure. The red traces represent the first TE band while the blue line signifies the second TE band of the VPC. The inset depicts the unit cell for the different values of $\Delta d$. Figure 2(a) represents the case of the lowest asymmetry $(\Delta d=0.04a)$ in our investigation with $d_{1} = 110$ µm and $d_{2} = 100$ µm. It can be seen that the degeneracy at the K (K$^\prime$) point is lifted and a bandgap ranging from $0.323$ THz to $0.328$ THz opens up around the point of degeneracy. As we reduce the value of $d_{2}$ from $100$ µm to $20$ µm in steps of $20$ µm, $\Delta d$ increases from $0.12a$ to $0.36a$ in steps of $0.08a$. Figures 2(b), 2(c) and 2(d) represent the band structures for the cases when $\Delta d=0.12a$, $0.02 a$, $0.28a$, respectively with corresponding $d_{2}$ values of $80$ µm, $60$ µm and $40$ µm. For $\Delta d=0.12a$ and $\Delta d=0.2a$, a respective band gap ranging from $0.303$ THz to $0. 315$ and $0.288$ THz to $0.308$ THz is obtained. For $\Delta d=0.28a$, a wide band gap $\sim 10\%$ ranging from $0.278$ THz to $0.304$ THz is achieved. In Fig. 2, $P_1$ and $P_2$ denotes the $K$ and $M$ points of the first and second band, respectively and the frequency difference between the points denote the band gap of the VPC. It is clearly evident (from Fig. 2) that the size of the bandgap increases with increase in $\Delta d$ of the structure. Finally, when $\Delta d$ reaches a value of $0.36a$, a wider bandgap with a range of $0.272$ THz to $0.302$ THz appears in the TE bands of the VPC structure. The Berry curvature of the proposed VPC slab is known to exhibit nonzero, identical and opposite values at the K and K$^\prime$ valleys of the first Brillouin zone. Additionally, a nonzero valley Chern number $(C_{K/K^\prime } \neq 0)$ exists at the K and K$^\prime$ valleys of the first Brillouin zone [7,22], thus, giving rise to a nontrivial topology in the VPC structure.

 figure: Fig. 2.

Fig. 2. Band structure for TE mode of the proposed VPC structure for different asymmetries. Band structures for (a) $\Delta d=0.04a$, (b) $\Delta d=0.12a$, (c) $\Delta d=0.2a$, (d) $\Delta d=0.28a$, and (e) $\Delta d=0.36a$. The inset depicts the unit cell for the different values of $\Delta d$. The light blue shaded region denotes the extended bands while the pink shaded region represents the band gap of the VPC structure. The red traces represent the first TE band while the blue line signifies the second TE band of the VPC structure. $P_1$ and $P_2$ denotes the $K$ an $M$ points of the first and second band, respectively. (f) Table indicating the range of band gap for different values of $\Delta d$.

Download Full Size | PPT Slide | PDF

3. Terahertz topological edge states

Next, we investigate the excitation of THz topological edge states in the proposed VPC with broken inversion symmetry. Following the concept of band topology, we propose a design for a VPC structure capable of supporting topological edge states in the THz frequency regime [2022]. For a VPC with a honeycomb lattice structure, different types of domain interfaces namely, bridge, zigzag, and armchair domain interfaces are possible. The bridge and the zigzag domain interface point towards the $\Gamma K$ direction, while the armchair domain interface points in the $\Gamma M$ direction. For the armchair interface, which is constructed away from the $K$ and $K'$ valleys, the robustness of the edge states is highly reduced [65]. Out of the three possible interfaces, the most robust edge states are excited for the case of the zigzag domain interface. Therefore, we shall focus our analysis only on the zigzag domain interface in these VPC structures. In order to construct a zigzag domain wall at $y = 0$ axis, we interface two VPCs (VPC$-$I and VPC$-$II) with distinct nontrivial topology. Figure 3 illustrates the scheme for the excitation of the THz topological edge states. The schematic of the VPC along with a magnified image of the zigzag domain wall (indicated by black line) is represented in Fig. 3(a). The excitation of the topological edge states in such VPCs is attributed to the valley Chern index $(C_{\Delta }= C_{K} - C_{K^{'}})$ of the structure [7,40]. It is noteworthy to mention that the signs of $C_{\Delta }$ for VPC-I and VPC-II structures for the TE modes are non-zero and opposite to each other [22] . As a consequence, the bulk-edge correspondence principle ensures the excitation of THz topological edge states in the VPC structure [66,67]. For the zigzag domain interface considered in our study, two edge modes, each locked to the anticlockwise and the clockwise phase vortices respectively at $K'$ and $K$ valleys, are excited.

 figure: Fig. 3.

Fig. 3. (a) Schematic of the topological VPC along with a magnified view, where the zigzag domain wall is denoted by the black line. The direction of incident electric field polarization is denoted by the green arrow. Dispersion of the topological edge states (b) in the absence of domain wall, and (c) in the presence of domain wall where, $k_x$ denotes the wavevector parallel to the direction of edge states propagation. (d) Transmission spectra with and without the domain wall. The red line represents the transmission in the presence of a straight domain wall while the black dashed line denotes the same for the case of no domain wall.

Download Full Size | PPT Slide | PDF

Figures 3(b), 3(c), and 3(d), respectively illustrate the dispersion of the edge states and the transmission spectrum for the VPC without/with a domain wall. The dispersion of the edge states is obtained using the MPB package. A supercell structure is considered in order to obtain the edge dispersion in the structure, which is plotted against the wavevector $(k_x)$ parallel to the direction of edge states propagation. The transmission spectra and the electromagnetic field distribution is obtained using the finite difference time domain (FDTD) method based Time Domain Solver (TDS) in CST Microwave Studio. A VPC structure having a dimension of $8.01 \times 15$ µm$^{2}$ is designed in order to study the topological edge states propagation. For the FDTD simulations, a finite mesh size of $\lambda /10$ is defined. Open boundary conditions were set on the positive and negative z-axis. Waveguide ports were used to excite the VPC structure as well as to detect electromagnetic transport. The input and output ports were placed in the x-direction, at a distance of $80$ µm away from the VPC structure. Field monitors were set at several frequencies within the bulk bandgap of the corresponding VPC structures.

The dispersion plot of the edge states for the VPC with no domain wall is shown in Fig. 3(b), while Fig. 3(c) represents the edge dispersion of the VPC in the presence of a domain wall The blue shaded region denotes the projected bulk bands of the VPC structure while the light blue shaded region signifies the light cone of the substrate. The white portion denotes the bulk bandgap of the VPC while the red solid lines represent the edge states excitation and propagation in the VPC structure. It is clearly evident from Fig. 3(b) that there is no excitation of topological edge states within the bulk bandgap of the VPC, in the absence of a domain wall. This can be understood from the transmission represented by the black dashed line in Fig. 3(d), where a transmission as low as $-30$ dB is obtained within the frequency range of $0.272$ THz to $0.302$ THz. This dip in the transmission clearly suggests the presence of a bulk bandgap, within which the transmission of electromagnetic wave is forbidden through the VPC structure. However, when a domain wall is introduced, a transmission as high as $-4$ dB is achieved within the bulk bandgap of the VPC, signifying the excitation of topological edge states in the VPC structure. The transmission spectrum in the presence of a domain wall is shown by the red solid line in Fig. 3(d). The excitation of the THz topological edge states, in the presence of a domain wall, is further elucidated with the help of the dispersion of the projected bands. From Fig. 3(c), it is observed that the edge modes (indicated by the red solid lines) are excited and appropriately guided within the bulk bandgap of the VPC structure. Hence, the proposed VPC structure is capable of supporting topological edge states in the THz frequency domain.

4. Role of asymmetry on the robustness of the edge states

The most remarkable feature of a topological photonic structure lies in its capability to support propagation of light (within the bulk bandgap of the structure) which remains robust even in the presence of disorder/defects. However, our study reveals that symmetry plays a vital role in the robustness of topological edge states in such structures. Here, we perform a comprehensive analysis on the robustness of the topological edge states with respect to the asymmetry of our VPC structure and probe the transmission as well as electromagnetic field confinement as a function of asymmetry in the structure. We examine the propagation of the topological edge states in a straight domain wall as well as an $\Omega$-type domain wall (with a $120^{\circ }$ sharp bend). We divide our analysis into two cases: $d_1 > d_2$ and $d_1 < d_2$. We first consider the case: $d_1 > d_2$, when the domain walls are made of the larger air grooves with a diameter of $110$ µm. The edge states dispersion, transmission spectra, and the electromagnetic field distributions are depicted in Fig. 4. Figure 4(i) represents the edge dispersion in the VPC and Fig. 4(ii) shows the transmission spectra (in dB) for the VPC structure with different asymmetries. The red traces represent the transmission for the straight domain wall while the blue traces denote the transmission spectra for the $\Omega$-type domain wall. Figures 4(iii) and 4(iv) represent the electromagnetic field confinement for the straight and $\Omega$-type domain walls for different values of the asymmetry parameter at appropriate frequencies. For all asymmetries, it is observed that the transmission through the straight domain wall is slightly higher than the transmission through the $\Omega$-type domain wall. For $\Delta d = 0.04a$ (lowest asymmetry), we see from Figs. 4(ii)(a), 4(iii)(a), and 4(iv)(a), that the field at $f \sim 0.3$ THz is scattered throughout the bulk of the VPC structure. This field scattering is also present for $\Delta d = 0.12a$ (see Figs. 4(ii)(b), 4(iii)(b) and 4(iv)(b)) and $\Delta d = 0.2a$ (as shown in Figs. 4(ii)(c), 4(iii)(c) and 4(iv)(c)). It is evident from these figures, that there is weak confinement of the topological edge states within the domain wall of the VPC structure, for both domain walls at $f \sim 0.298$ THz and $f =0.295$ THz for $\Delta d = 0.12a$ and $\Delta d = 0.2a$, respectively. However, for the case of $\Delta d = 0.28a$, we see a stronger field confinement at $f = 0.291$ THz for the straight (Fig. 4(iii)(d)) and the $\Omega$-type domain wall (Fig. 4(iv)(d)). A transmission as high as $\sim -5$ dB is achieved for both domain walls (Fig. 4(ii)(d)) when the asymmetry is $0.28a$. Finally, for the case when $\Delta d = 0.36a$, we achieve the maximum robust transport of THz waves (see Figs. 4(e)). For an asymmetry range where $0.36a < \Delta d \leq 0.42a$, high transmission ($\sim -4$ dB) similar to the case of $\Delta d = 0.36a$ (see Fig. 4(ii)(e)) is achieved within the bulk bandgap of the VPC structure, for both straight and the $\Omega$-type domain wall. For $\Delta d = 0.36a$, the corresponding field distributions for the straight and the $\Omega$-type domain walls at $f = 0.288$ THz are shown in Fig. 4(iii)(e) and 4(iv)(e), respectively. Furthermore, we observed that the maximum transmission amplitudes achieved for different asymmetries ranging from $0.28a$ to $0.42a$ differed by a value less than $1$ dB, respectively for both straight and $\Omega$-type domain wall.

 figure: Fig. 4.

Fig. 4. Effect of asymmetry on the robustness of the THz topological edge states. (i) The dispersion of the topological edge states for (a) $\Delta d = 0.04a$, (b) $\Delta d = 0.12a$, (c) $\Delta d = 0.2a$, (d) $\Delta d = 0.28a$, and (e) $\Delta d = 0.36a$, (ii) Transmission spectra (in dB) for the VPC structure when (a) $\Delta d = 0.04a$, (b) $\Delta d = 0.12a$, (c) $\Delta d = 0.2a$, (d) $\Delta d = 0.28a$, and (e) $\Delta d = 0.36a$, (iii) Topological edge states field confinement for a straight domain wall for (a) $\Delta d = 0.04a$, (b) $\Delta d = 0.12a$, (c) $\Delta d = 0.2a$, (d) $\Delta d = 0.28a$, and (e) $\Delta d = 0.36a$, and (iv) Topological edge states field confinement for an $\Omega$-type domain wall with (a) $\Delta d = 0.04a$, (b) $\Delta d = 0.12a$, (c) $\Delta d = 0.2a$, (d) $\Delta d = 0.28a$ and (e) $\Delta d = 0.36a$.

Download Full Size | PPT Slide | PDF

It is clear from Fig. 4., that the topological edge states are weakly confined within the domain wall for $\Delta d=0.04a$ and there is a high scattering of the edge states for the straight domain wall, which further increases when sharp bends are introduced in the domain wall. This scattering of the topological edge states can be further explained by the dispersion of the topological edge modes (Fig. 4(i)), where the red solid lines denote the edge modes within the bulk bandgap. In the figure, the blue shaded region denotes the projected bulk-bands of the VPC structure which represent the modes that are scattered throughout the VPC structure. The light blue shaded region signifies the light cone of the substrate in which no guided modes are allowed in the VPC structure. The white portion denotes the bulk bandgap of the VPC while the red solid lines represent the edge modes in the VPC structure. It is seen from Fig. 4(i)(a), that the edge modes almost overlap with the lower bulk bands of the VPC structure. Intuitively, since the edge modes overlap with the projected bulk bands of the VPC, they behave similar to the bulk modes of the VPC structure and therefore, will get scattered throughout the bulk VPC. Further, the presence of a narrow bulk bandgap for $\Delta d=0.04a$, gives rise to an intervalley scattering within the bands of the VPC structure, due to which, the edge modes are weakly guided within the domain walls. As a result, the edge modes undergo large scattering close to the domain interface of the VPC structure. However, we see an increase in the confinement as well as the robustness of the edge states with an increase in the magnitude of $\Delta d$. It is evident from Fig. 4(i), that the edge modes are pushed away from the bulk bands and towards the higher frequency with increasing $\Delta d$. This shift of the edge modes toward the middle of the bulk bandgap causes an increase in the confinement of the edge modes within the domain interface of the VPC. The robust behavior of the edge modes can be further understood from the fact that the bulk bandgap of the VPC increases with increasing $\Delta d$, reaching close to $10 \%$ for $\Delta d = 0.28a$. This large bandgap ensures complete suppression of the intervalley scattering [7,9] in the VPC structure, which further ensures a robust guiding of the topological edge states within the domain wall. Furthermore, for positive $\ \Delta d$ (i.e. the domain walls are made of larger air grooves), the separation $(g)$ between the upper and the lower air grooves forming the zigzag domain interface is smaller thus, enabling strong interaction/coupling of the vortex fields between the upper and the lower domain holes [22]. Consequently, a robust transport of the THz topological edge states is achieved even in the presence of sharp corners in the domain wall. In our analysis, a robust transport and a strong confinement of the THz topological edge states are achieved when $\Delta d$ ranges from $0.28a$ to $0.42a$.

We further extend our investigation for the case when $d_1 < d_2$ i.e., when the domain wall is formed by the relatively smaller air grooves. For this case, $d_1$ varies from $100$ µm to $20$ µm while $d_2$ is kept fixed at $110$ µm. Figure 5(i) shows the variation of the bandgap for the cases: $d_1 > d_2$, $d_1 = d_2$, and $d_1 < d_2$. It is evident from the figure that the degeneracy at $f = 0.336$ THz for $d_1=d_2$ is relaxed with the introduction of asymmetry and the bandgap not only gets wider with the increase in asymmetry, but also gets shifted towards the lower frequency. Figure 5(ii) represents the edge dispersion in the VPC and Fig. 5(iii) shows the transmission spectra for the VPC structure with different negative asymmetry parameters. The red traces represent the transmission for the straight domain wall while the blue traces denote the transmission spectra for the $\Omega$-type domain wall. Figures 5(iv) and 5(v) represent the electromagnetic field confinement for the straight and $\Omega$-type domain walls for different asymmetry. It is evident from the figure that there is a large scattering of the field throughout the bulk of the VPC for $\Delta d = - 0.04a$, $-0.12a$, and $-0.02a$. The scattering gets reduced when the magnitude of the asymmetry parameter increases. When $\Delta d = - 0.28a$, we see that the electromagnetic field gets more confined to the domain wall (see Fig. 5(iv)(d)). The scattering is suppressed and the electromagnetic field confinement is increased further for the case of $\Delta d = - 0.36a$ (Fig. 5(iv) (e)). Here, for the case of the straight domain wall, we observe a high forward transmission $\sim -5$ dB within the bulk bandgap of the VPC structure. Surprisingly, when we introduce $120 ^{\circ }$ sharp bends into the domain wall ($\Omega$-type), there is a significant drop in the transmission amplitude for all magnitude of negative asymmetry parameter. Even though a robust transport of light is expected in the VPC with broken inversion symmetry, we observe that the topological protection is lost when $\Delta d$ is $-ve$.

 figure: Fig. 5.

Fig. 5. Effect of asymmetry on the robustness of the THz topological edge states. (i) The variation of the bulk bandgap as a function of the asymmetry parameter, $\Delta d$ of the VPC structure. (ii) The dispersion of the topological edge states for (a) $\Delta d = -0.04a$, (b) $\Delta d = -0.12a$, (c) $\Delta d = -0.2a$, (d) $\Delta d = -0.28a$, and (e) $\Delta d = -0.36a$, (iii) Transmission spectra (in dB) for the VPC structure when (a) $\Delta d = -0.04a$, (b) $\Delta d = -0.12a$, (c) $\Delta d = -0.2a$, (d) $\Delta d = -0.28a$, and (e) $\Delta d = -0.36a$, (iv) Topological edge states field confinement for a straight domain wall for (a) $\Delta d = -0.04a$, (b) $\Delta d= -0.12a$, (c) $\Delta d = -0.2a$, (d) $\Delta d =- 0.28a$, and (e) $\Delta d = -0.36a$, and (v) Topological edge states field confinement for an $\Omega$-type domain wall with (a) $\Delta d = -0.04a$, (b) $\Delta d =- 0.12a$, (c) $\Delta d = -0.2a$, (d) $\Delta d = -0.28a$, and (e) $\Delta d = -0.36a$.

Download Full Size | PPT Slide | PDF

As evidenced from the figures, the topological edge states are weakly confined within the domain wall for $\Delta d=- 0.04a$, $- 0.12a$, and $-0.2a$. There is a high scattering of the edge states for the straight domain wall for the first three cases similar to that for $\Delta d = 0.04a$, $0.12a$, and $0.2a$. To understand this, we look into the dispersion of the topological edge modes (Fig. 5(ii)) of the VPC. It is evident from Fig. 5(ii)(a), that the edge modes overlap with the upper bulk bands of the studied VPC structure. As a result of this, the edge modes behave similar to the bulk modes of the VPC structure and are thus scattered throughout the VPC. Further, the presence of a narrow bulk bandgap for $\Delta d = -0.04a$ is not able to suppress the intervalley scattering within the bands of the VPC structure. As a result, the edge modes undergo large scattering close to the domain interface of the VPC structure. However, there is an increase in the confinement as well as the robustness of the edge states with an increase in the magnitude of $\Delta d$ for a straight domain wall (see Fig. 5(iv)). This could be attributed to the shifting of the edge modes away from the bulk bands and towards the middle of the bulk bandgap with the increase of $\Delta d$. As opposed to the case of positive asymmetry, the edge modes are pushed towards the lower frequency for negative asymmetry parameters. Furthermore, an increasing bandgap ensures a stronger suppression of the intervalley scattering in the VPC structure, which ultimately ensures a robust guiding of the topological edge states within the domain wall of the VPC structure. As a result, there is an increase in the transmission as well as electromagnetic field confinement as the magnitude of $\Delta d$ increase from $0.04a$ to $0.36a$ for the straight domain wall. However, if we introduce sharp bends and examine the propagation of the field in an $\Omega$-type domain wall, we see that the transport of the incident THz wave is blocked by the sharp corners of the domain wall. For negative $\Delta d$, since the domain walls are constituted by relatively smaller air holes, the separation $'g'$ increases with increasing magnitude of $\Delta d$ and the coupling of vortex fields between the upper and lower domain holes becomes weaker. Subsequently, the robustness of the excited edge states is reduced and the topological protection of the edge states is forbidden when $d_1 < d_2$.

In order to obtain more physical insights about the edge states behavior for $d_1 > d_2$ and $d_1 < d_2$, we further examine the Poynting vectors of the edge states in the VPC structure for both the straight and $\Omega$-type domain walls (Fig. 6). The Poynting vector represents the direction of the net energy flow in the VPC structure. The net flow of energy will be towards the right for the edge state locked to the $K'$ valley, while the energy flows in the leftward direction for the edge state locked to the $K$ valley [22]. Figures 6(a), 6(d) show the pictorial representation of the domain formed by larger ($\Delta d= + 0.36a$) and smaller air grooves ($\Delta d= -0.36a$), respectively for an edge state with a rightward propagation. Figures 6(b), 6(e) represent the Poynting vector for the VPC structure with a straight type domain wall, while Fig. 6(c) and 6(f) denote the same for an $\Omega$-type domain wall. When the domain wall is constituted by the larger air holes ($g = 34$ µm), we can see from Figs. 6(b) and 6(c), that the net energy flows from one domain hole to the next domain hole. An effective coupling path is established between the domain holes thus, giving rise to a unidirectional flow of energy in the VPC structure, which is free from any backscattering even in the presence of sharp bends. For this case, there is a robust transport of the incident THz light through the VPC structure. Thus, the excited edge states are topologically protected for $\Delta d= + ve$.

 figure: Fig. 6.

Fig. 6. (a) Pictorial depiction of the zigzag domain interface formed by the larger air holes ($\Delta d = +ve$). Poynting vectors of the VPC structure for (b) a straight domain wall, and (c) and an $\Omega$-type domain wall, (d) Pictorial depiction of the zigzag domain interface formed by the smaller air holes ($\Delta d = -ve$). Poynting vectors of the VPC structure for (e) a straight domain wall, and (f) and an $\Omega$-type domain wall.

Download Full Size | PPT Slide | PDF

However, when the domain wall is formed by the relatively smaller air grooves (in particular, when $\Delta d= -0.36a$), a key difference in the flow paths of energy is observed. For $\Delta d= -0.36a$ (i.e. $g = 124$ µm), there is no flow of energy between the upper and lower domain grooves. Instead, the flow of energy is concentrated within a larger area enclosed by the larger air grooves neighboring to the smaller domain holes (see Fig. 6(e)). Even though there is a unidirectional flow of energy for the straight domain, the Poynting vector is distorted around the sharp bend of the $\Omega$-type domain wall (Fig. 6(f)). The flow of energy is obstructed around the corner of the $\Omega$-type domain wall and there is no unidirectional flow of light in the VPC structure. Thus, the topological protection of the edge states in the VPC structure is lost. Although a robust edge state excitation is expected for a zigzag domain interface but our study reveals that the configuration ($d_1 < d_2$) is not favorable for the excitation and propagation of THz topological edge states. A robust transport of topological edge states is achieved only when the domain wall is formed by the larger air grooves in the proposed VPC structure. For the case: $d_1 > d_2$, high forward transmission (close to unity) with negligible backscattering is achieved even in the presence of sharp bends in the VPC structure. Further, the robustness of the excited edge states increases with the increase in the magnitude of the asymmetry parameter, wherein the maximum robustness is achieved for an asymmetry ranging from $28 \%$ to $42 \%$. Beyond this, the honeycomb lattice of the structure changes into a triangular lattice and hence, is beyond the scope of this study. This study could be significant in attaining a comprehensive insight about THz topological edge states in similar VPC structures, which could prove beneficial in designing devices for 6G communication and various other on-chip applications in the THz domain. In addition, the proposed VPC structure can easily be integrated for various on-chip applications, making it favorable for the development of various waveguide and communication technologies in the THz band.

5. Conclusion

In this work, we numerically demonstrate THz transport through topological edge states in an all-dielectric photonic crystal-based structure. We design a valley photonic crystal (VPC) structure with a nontrivial topology by breaking the $C_{6}$ lattice geometry. The excitation of the topological edge states in the VPC is then facilitated through a zigzag domain wall, formed by interfacing two types of VPCs. High forward transmission of the terahertz (THz) waves is achieved within the bulk bandgap of the VPC structure. Further, the robustness of the topological edge states is probed as a function of asymmetry in terms of the diameter of air grooves in the VPC structure, for a straight as well as an $\Omega$-type (with $120^{\circ }$ bending) domain wall. The dispersion of the topological edge states, the transmission spectra, and the electromagnetic field profiles are explored for different asymmetries introduced through the groove diameters. For $\Delta d = +ve$, it is observed that the topological transport of THz waves becomes more robust with an increase in the magnitude of asymmetry. For $\Delta d = 0.36a$, we observe a near-unity transmission of THz waves through a straight as well as an $\Omega$-type domain wall. A strong confinement of the electromagnetic field within the domain wall is also observed in the VPC for both types of domain walls. However, when the domain wall is formed by the smaller air grooves (i.e., $d_{1} < d_{2}$ case), a huge drop in the transmission is observed through the $\Omega$-type domain wall. Our study reveals that the proposed VPC supports a robust propagation of the THz waves only when the domain walls are formed by the larger air holes (i.e., $\Delta d = +ve$). This robustness of the edge states is attributed to the strong coupling of fields between the upper and lower air holes constituting the domain interface in the VPC structure. On the contrary, when the domain walls are formed by the relatively smaller air holes (i.e., $\Delta d = -ve$), there is no effective coupling between the upper and the lower holes forming the domain interface due to which, the topological protection of the edge states is forbidden in the VPC structure. For the case of the positive asymmetry, a robust transport of the THz topological edge states is achieved for an asymmetry ranging from $28 {\%}$ to $42{\%}$ in the VPC structure. Similar behavior could be achieved at other frequency regimes by appropriately scaling the size parameters of the structure and by using a material having a similar dielectric constant. Current analysis about THz topological edge states in VPC structures could be beneficial in attaining comprehensive insights toward the realization of low loss waveguides for 6G communication and other integrated photonic devices in the THz frequency domain.

Funding

Board of Research in Nuclear Sciences (58/14/32/2019-BRNS/11090); Department of Science and Technology (CRG/2019/001656).

Acknowledgments

D.R.C. acknowledges partial supports from the Department of Science and Technology (DST), Project No. CRG/2019/001656 and BRNS project, 58/14/32/2019-BRNS/11090.

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. 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, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]  

3. A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013). [CrossRef]  

4. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017). [CrossRef]  

5. B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018). [CrossRef]  

6. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [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,” Nat. Nanotechnol. 14(1), 31–34 (2019). [CrossRef]  

8. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008). [CrossRef]  

9. F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018). [CrossRef]  

10. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008). [CrossRef]  

11. 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]  

12. K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980). [CrossRef]  

13. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982). [CrossRef]  

14. L. Balents and M. P. A. Fisher, “Chiral surface states in the bulk quantum hall effect,” Phys. Rev. Lett. 76(15), 2782–2785 (1996). [CrossRef]  

15. D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008). [CrossRef]  

16. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015). [CrossRef]  

17. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004). [CrossRef]  

18. B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020). [CrossRef]  

19. Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018). [CrossRef]  

20. T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016). [CrossRef]  

21. J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017). [CrossRef]  

22. 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), 1–9 (2019). [CrossRef]  

23. M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020). [CrossRef]  

24. Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017). [CrossRef]  

25. S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform,” Opt. Mater. Express 11(2), 319–337 (2021). [CrossRef]  

26. S. Ma and S. M. Anlage, “Microwave applications of photonic topological insulators,” Appl. Phys. Lett. 116(25), 250502 (2020). [CrossRef]  

27. M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 1(10), 425-434 (2020).

28. Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017). [CrossRef]  

29. A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016). [CrossRef]  

30. Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019). [CrossRef]  

31. L. Pilozzi, D. Leykam, Z. Chen, and C. Conti, “Topological photonic crystal fibers and ring resonators,” Opt. Lett. 45(6), 1415–1418 (2020). [CrossRef]  

32. D. Leykam and L. Yuan, “Topological phases in ring resonators: recent progress and future prospects,” Nanophotonics 9(15), 4473–4487 (2020). [CrossRef]  

33. C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016). [CrossRef]  

34. M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021). [CrossRef]  

35. X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016). [CrossRef]  

36. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020). [CrossRef]  

37. Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020). [CrossRef]  

38. S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018). [CrossRef]  

39. S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018). [CrossRef]  

40. 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]  

41. Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018). [CrossRef]  

42. M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018). [CrossRef]  

43. G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021). [CrossRef]  

44. 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]  

45. J. Chen, W. Liang, and Z. Y. Li, “Broadband dispersionless topological slow light,” Opt. Lett. 45(18), 4964–4967 (2020). [CrossRef]  

46. M. Reisner, M. Bellec, U. Kuhl, and F. Mortessagne, “Microwave resonator lattices for topological photonics,” Opt. Mater. Express 11(3), 629–653 (2021). [CrossRef]  

47. Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021). [CrossRef]  

48. 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]  

49. H. Xiong, Q. Wu, Y. Lu, R. Wang, Q. Zhang, J. Qi, J. Yao, and J. Xu, “Polarization-resolved edge states in terahertz topological photonic crystal,” Opt. Express 27(16), 22819–22826 (2019). [CrossRef]  

50. B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016). [CrossRef]  

51. T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared, Millimeter, Terahertz Waves 32(2), 143–171 (2011). [CrossRef]  

52. Q. J. Gu, “THz interconnect: the last centimeter communication,” IEEE Commun. Mag. 53(4), 206–215 (2015). [CrossRef]  

53. S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013). [CrossRef]  

54. H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).

55. J. T. Lu, Y. C. Hsueh, Y. R. Huang, Y. J. Hwang, and C. K. Sun, “Bending loss of terahertz pipe waveguides,” Opt. Express 18(25), 26332 (2010). [CrossRef]  

56. F. Grillot, L. Vivien, S. Laval, and E. Cassan, “Propagation loss in single-mode ultrasmall square silicon-on-insulator optical waveguides,” J. Lightwave Technol. 24(2), 891–896 (2006). [CrossRef]  

57. S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013). [CrossRef]  

58. B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002). [CrossRef]  

59. J. A. Harrington, R. George, P. Pedersen, and E. Mueller, “Hollow polycarbonate waveguides with inner Cu coatings for delivery of terahertz radiation,” Opt. Express 12(21), 5263–5268 (2004). [CrossRef]  

60. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015). [CrossRef]  

61. S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016). [CrossRef]  

62. P. D. Anderson and G. Subramania, “Unidirectional edge states in topological honeycomb-lattice membrane photonic crystals,” Opt. Express 25(19), 23293–23301 (2017). [CrossRef]  

63. N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020). [CrossRef]  

64. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).

65. 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]  

66. M. Ezawa, “Topological Kirchhoff law and bulk-edge correspondence for valley Chern and spin-valley Chern numbers,” Phys. Rev. B 88(16), 161406 (2013). [CrossRef]  

67. M. G. Silveirinha, “Bulk-edge correspondence for topological photonic continua,” Phys. Rev. B 94(20), 205105 (2016). [CrossRef]  

References

  • View by:

  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, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
    [Crossref]
  3. A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
    [Crossref]
  4. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
    [Crossref]
  5. B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018).
    [Crossref]
  6. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
    [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,” Nat. Nanotechnol. 14(1), 31–34 (2019).
    [Crossref]
  8. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
    [Crossref]
  9. F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
    [Crossref]
  10. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
    [Crossref]
  11. 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]
  12. K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
    [Crossref]
  13. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
    [Crossref]
  14. L. Balents and M. P. A. Fisher, “Chiral surface states in the bulk quantum hall effect,” Phys. Rev. Lett. 76(15), 2782–2785 (1996).
    [Crossref]
  15. D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
    [Crossref]
  16. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
    [Crossref]
  17. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
    [Crossref]
  18. B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
    [Crossref]
  19. Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
    [Crossref]
  20. T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
    [Crossref]
  21. J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
    [Crossref]
  22. 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), 1–9 (2019).
    [Crossref]
  23. M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020).
    [Crossref]
  24. Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
    [Crossref]
  25. S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform,” Opt. Mater. Express 11(2), 319–337 (2021).
    [Crossref]
  26. S. Ma and S. M. Anlage, “Microwave applications of photonic topological insulators,” Appl. Phys. Lett. 116(25), 250502 (2020).
    [Crossref]
  27. M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 1(10), 425-434 (2020).
  28. Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
    [Crossref]
  29. A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
    [Crossref]
  30. Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
    [Crossref]
  31. L. Pilozzi, D. Leykam, Z. Chen, and C. Conti, “Topological photonic crystal fibers and ring resonators,” Opt. Lett. 45(6), 1415–1418 (2020).
    [Crossref]
  32. D. Leykam and L. Yuan, “Topological phases in ring resonators: recent progress and future prospects,” Nanophotonics 9(15), 4473–4487 (2020).
    [Crossref]
  33. C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
    [Crossref]
  34. M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021).
    [Crossref]
  35. X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
    [Crossref]
  36. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
    [Crossref]
  37. Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
    [Crossref]
  38. S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
    [Crossref]
  39. S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
    [Crossref]
  40. 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]
  41. Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
    [Crossref]
  42. M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
    [Crossref]
  43. G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
    [Crossref]
  44. 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]
  45. J. Chen, W. Liang, and Z. Y. Li, “Broadband dispersionless topological slow light,” Opt. Lett. 45(18), 4964–4967 (2020).
    [Crossref]
  46. M. Reisner, M. Bellec, U. Kuhl, and F. Mortessagne, “Microwave resonator lattices for topological photonics,” Opt. Mater. Express 11(3), 629–653 (2021).
    [Crossref]
  47. Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
    [Crossref]
  48. 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]
  49. H. Xiong, Q. Wu, Y. Lu, R. Wang, Q. Zhang, J. Qi, J. Yao, and J. Xu, “Polarization-resolved edge states in terahertz topological photonic crystal,” Opt. Express 27(16), 22819–22826 (2019).
    [Crossref]
  50. B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016).
    [Crossref]
  51. T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared, Millimeter, Terahertz Waves 32(2), 143–171 (2011).
    [Crossref]
  52. Q. J. Gu, “THz interconnect: the last centimeter communication,” IEEE Commun. Mag. 53(4), 206–215 (2015).
    [Crossref]
  53. S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
    [Crossref]
  54. H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).
  55. J. T. Lu, Y. C. Hsueh, Y. R. Huang, Y. J. Hwang, and C. K. Sun, “Bending loss of terahertz pipe waveguides,” Opt. Express 18(25), 26332 (2010).
    [Crossref]
  56. F. Grillot, L. Vivien, S. Laval, and E. Cassan, “Propagation loss in single-mode ultrasmall square silicon-on-insulator optical waveguides,” J. Lightwave Technol. 24(2), 891–896 (2006).
    [Crossref]
  57. S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
    [Crossref]
  58. B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002).
    [Crossref]
  59. J. A. Harrington, R. George, P. Pedersen, and E. Mueller, “Hollow polycarbonate waveguides with inner Cu coatings for delivery of terahertz radiation,” Opt. Express 12(21), 5263–5268 (2004).
    [Crossref]
  60. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
    [Crossref]
  61. S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
    [Crossref]
  62. P. D. Anderson and G. Subramania, “Unidirectional edge states in topological honeycomb-lattice membrane photonic crystals,” Opt. Express 25(19), 23293–23301 (2017).
    [Crossref]
  63. N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
    [Crossref]
  64. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).
  65. 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]
  66. M. Ezawa, “Topological Kirchhoff law and bulk-edge correspondence for valley Chern and spin-valley Chern numbers,” Phys. Rev. B 88(16), 161406 (2013).
    [Crossref]
  67. M. G. Silveirinha, “Bulk-edge correspondence for topological photonic continua,” Phys. Rev. B 94(20), 205105 (2016).
    [Crossref]

2021 (6)

S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform,” Opt. Mater. Express 11(2), 319–337 (2021).
[Crossref]

M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021).
[Crossref]

G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
[Crossref]

M. Reisner, M. Bellec, U. Kuhl, and F. Mortessagne, “Microwave resonator lattices for topological photonics,” Opt. Mater. Express 11(3), 629–653 (2021).
[Crossref]

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

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]

2020 (12)

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]

J. Chen, W. Liang, and Z. Y. Li, “Broadband dispersionless topological slow light,” Opt. Lett. 45(18), 4964–4967 (2020).
[Crossref]

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]

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
[Crossref]

L. Pilozzi, D. Leykam, Z. Chen, and C. Conti, “Topological photonic crystal fibers and ring resonators,” Opt. Lett. 45(6), 1415–1418 (2020).
[Crossref]

D. Leykam and L. Yuan, “Topological phases in ring resonators: recent progress and future prospects,” Nanophotonics 9(15), 4473–4487 (2020).
[Crossref]

M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020).
[Crossref]

S. Ma and S. M. Anlage, “Microwave applications of photonic topological insulators,” Appl. Phys. Lett. 116(25), 250502 (2020).
[Crossref]

M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 1(10), 425-434 (2020).

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

2019 (6)

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), 1–9 (2019).
[Crossref]

Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

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]

H. Xiong, Q. Wu, Y. Lu, R. Wang, Q. Zhang, J. Qi, J. Yao, and J. Xu, “Polarization-resolved edge states in terahertz topological photonic crystal,” Opt. Express 27(16), 22819–22826 (2019).
[Crossref]

2018 (7)

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018).
[Crossref]

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

2017 (5)

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

P. D. Anderson and G. Subramania, “Unidirectional edge states in topological honeycomb-lattice membrane photonic crystals,” Opt. Express 25(19), 23293–23301 (2017).
[Crossref]

2016 (7)

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

M. G. Silveirinha, “Bulk-edge correspondence for topological photonic continua,” Phys. Rev. B 94(20), 205105 (2016).
[Crossref]

B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016).
[Crossref]

C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
[Crossref]

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

2015 (3)

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref]

Q. J. Gu, “THz interconnect: the last centimeter communication,” IEEE Commun. Mag. 53(4), 206–215 (2015).
[Crossref]

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref]

2014 (1)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2013 (4)

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
[Crossref]

M. Ezawa, “Topological Kirchhoff law and bulk-edge correspondence for valley Chern and spin-valley Chern numbers,” Phys. Rev. B 88(16), 161406 (2013).
[Crossref]

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

2011 (1)

T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared, Millimeter, Terahertz Waves 32(2), 143–171 (2011).
[Crossref]

2010 (1)

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

2008 (4)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

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]

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

2006 (1)

2004 (2)

2002 (1)

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002).
[Crossref]

1996 (1)

L. Balents and M. P. A. Fisher, “Chiral surface states in the bulk quantum hall effect,” Phys. Rev. Lett. 76(15), 2782–2785 (1996).
[Crossref]

1982 (1)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
[Crossref]

1980 (1)

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

. Haldane, F. D. M

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]

Abbott, D.

S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
[Crossref]

Afshar, S.

S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
[Crossref]

Alouini, M. S.

H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).

Alpeggiani, F.

N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
[Crossref]

Ambacher, O.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Amin, O.

H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).

Amo, A.

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Anderson, P. D.

Anlage, S. M.

S. Ma and S. M. Anlage, “Microwave applications of photonic topological insulators,” Appl. Phys. Lett. 116(25), 250502 (2020).
[Crossref]

Antes, J.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Ao, Y.

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

Arakawa, Y.

S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform,” Opt. Mater. Express 11(2), 319–337 (2021).
[Crossref]

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

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]

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

Arora, S.

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]

Arregui, G.

G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
[Crossref]

Atakaramians, S.

S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
[Crossref]

Bahari, B.

B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016).
[Crossref]

Balents, L.

L. Balents and M. P. A. Fisher, “Chiral surface states in the bulk quantum hall effect,” Phys. Rev. Lett. 76(15), 2782–2785 (1996).
[Crossref]

Bandres, M. A.

M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 1(10), 425-434 (2020).

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Barczyk, R.

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]

Barik, S.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Bauer, T.

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]

Bellec, M.

Bliokh, K. Y.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref]

Boes, F.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Cai, T.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

Carusotto, I.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Cassan, E.

Cava, R. J.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Chan, C. T.

Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
[Crossref]

Chen, J.

Chen, X. D.

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), 1–9 (2019).
[Crossref]

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

Chen, Y. F.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018).
[Crossref]

Chen, Z.

Chen, Z. G.

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Cheng, X.

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

Chern, R. L.

Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
[Crossref]

Chong, Y.

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Christodoulides, D. N.

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Conti, C.

DeGottardi, W.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

den Nijs, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
[Crossref]

Dong, J. W.

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), 1–9 (2019).
[Crossref]

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

Dorda, G.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Elayan, H.

H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).

Ezawa, M.

M. Ezawa, “Topological Kirchhoff law and bulk-edge correspondence for valley Chern and spin-valley Chern numbers,” Phys. Rev. B 88(16), 161406 (2013).
[Crossref]

Fang, C.

C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
[Crossref]

Fei, H.

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

Ferguson, B.

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002).
[Crossref]

Filonov, D. S.

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

Fisher, M. P. A.

L. Balents and M. P. A. Fisher, “Chiral surface states in the bulk quantum hall effect,” Phys. Rev. Lett. 76(15), 2782–2785 (1996).
[Crossref]

Fleury, R.

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]

Flower, C.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

Freude, W.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Fu, L.

C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
[Crossref]

Fujita, M.

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]

Gao, F.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Garcia, P. D.

G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
[Crossref]

Genack, A. Z.

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

George, R.

Goldman, N.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Gomis-Bresco, J.

G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
[Crossref]

Gong, Q.

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

Grillot, F.

Gu, Q. J.

Q. J. Gu, “THz interconnect: the last centimeter communication,” IEEE Commun. Mag. 53(4), 206–215 (2015).
[Crossref]

Hafezi, M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Han, Y.

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

Harari, G

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Harrington, J. A.

Hasan, M. Z.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

He, X. T.

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), 1–9 (2019).
[Crossref]

Henneberger, R.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Hillerkuss, D.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Hor, Y. S.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Hsieh, D.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Hsueh, Y. C.

Hu, L.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Hu, X.

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref]

Huang, Y. R.

Hwang, Y. J.

Iwamoto, S.

S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform,” Opt. Mater. Express 11(2), 319–337 (2021).
[Crossref]

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]

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

Jacob, Z.

M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020).
[Crossref]

Jia, Z.

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

Jiang, Y.

M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021).
[Crossref]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).

Jouvaud, C.

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

Kallfass, I.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Kang, Y.

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

Kante, B.

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

Kanté, B.

B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016).
[Crossref]

Karasahin, A.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Katsumi, R.

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

Khajavikhan, M.

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Khanikaev, A. B.

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Khanikaev, A. B..

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

Kim, M.

M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020).
[Crossref]

Kivshar, Y.

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

Kivshar, Y. S.

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

Kleine-Ostmann, T.

T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared, Millimeter, Terahertz Waves 32(2), 143–171 (2011).
[Crossref]

Klitzing, K. V.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Koenig, S.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Kohmoto, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
[Crossref]

Koos, C.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Kuhl, U.

Kuipers, L.

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]

N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
[Crossref]

Kuo, C. Y.

Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
[Crossref]

Lai, K.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Laval, S.

Leuther, A.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Leuthold, J.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Leykam, D.

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

L. Pilozzi, D. Leykam, Z. Chen, and C. Conti, “Topological photonic crystal fibers and ring resonators,” Opt. Lett. 45(6), 1415–1418 (2020).
[Crossref]

D. Leykam and L. Yuan, “Topological phases in ring resonators: recent progress and future prospects,” Nanophotonics 9(15), 4473–4487 (2020).
[Crossref]

Li, C.

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

Li, M.

M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021).
[Crossref]

Li, Z. Y.

Liang, E. T.

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), 1–9 (2019).
[Crossref]

Liang, W.

Lin, H.

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

Lin, X.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Litchinitser, N. M.

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

Liu, F.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Liu, J.

C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
[Crossref]

Lopez-Diaz, D.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Lu, J. T.

Lu, L.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Lu, M. H.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018).
[Crossref]

Lu, Y.

Ma, S.

S. Ma and S. M. Anlage, “Microwave applications of photonic topological insulators,” Appl. Phys. Lett. 116(25), 250502 (2020).
[Crossref]

Ma, T.

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).

Mei, J.

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Miroshnichenko, A. E.

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

Miyake, H.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Monro, T. M.

S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
[Crossref]

Mortessagne, F.

Mousavi, S. H.

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Mueller, E.

Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref]

Nagatsuma, T.

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]

T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared, Millimeter, Terahertz Waves 32(2), 143–171 (2011).
[Crossref]

Ni, X.

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

Nightingale, M. P.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
[Crossref]

Nori, F.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref]

Notomi, M.

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

Onoda, M.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref]

Orazbayev, B.

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]

Ota, Y.

S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform,” Opt. Mater. Express 11(2), 319–337 (2021).
[Crossref]

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

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]

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

Ozawa, T.

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Palmer, R.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Parappurath, N.

N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
[Crossref]

Parto, M.

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Pedersen, P.

Pepper, M.

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Pilozzi, L.

Pitchappa, P.

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]

Price, H. M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Qi, J.

Qian, D.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Qiu, H. Y.

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), 1–9 (2019).
[Crossref]

Raghu, S.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

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]

Rechtsman, M. C.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Reisner, M.

Ren, J.

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Rho, J.

M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020).
[Crossref]

Schmogrow, R.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Schuster, D.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Segev, M.

M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 1(10), 425-434 (2020).

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Shalaev, M. I.

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

Shubair, R. M.

H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).

Shvets, G.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Bulk-edge correspondence for topological photonic continua,” Phys. Rev. B 94(20), 205105 (2016).
[Crossref]

Simon, J.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Singh, R.

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]

Slobozhanyuk, A. P.

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

Smirnova, D.

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref]

Smirnova, D. A.

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

Soljacic, M.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Song, J.

M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021).
[Crossref]

Sotomayor-Torres, C. M.

G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
[Crossref]

Su, G.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Subramania, G.

Sun, C. K.

Sun, X. C.

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Takata, K.

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

Tellez-Limon, R.

B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016).
[Crossref]

Tessmann, A.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Thouless, D. J.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
[Crossref]

Tse, W. K

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Tsukernik, A.

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

Verhagen, E.

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]

N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
[Crossref]

Vivien, L.

Waks, E.

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Walasik, W.

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

Wang, H. F.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

B. Y. Xie, H. F. Wang, X. Y. Zhu, M. H. Lu, Z. D. Wang, and Y. F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018).
[Crossref]

Wang, R.

Wang, Y.

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

Wang, Z.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Wang, Z. D.

Watanabe, K

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

Webber, J.

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]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).

Wittek, S.

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Wray, L.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Wu, L. H.

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref]

Wu, Q.

Wu, Y.

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Xia, Y.

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Xie, B.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Xie, B. Y.

Xiong, H.

Xu, J.

Xu, Y.

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

Xue, H.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Yamagami, Y.

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]

Yamaguchi, T.

Yang, Y.

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

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]

Yang, Z.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Yao, J.

Yao, S.

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

Yoshimi, H.

Yu, S. Y.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Yu, X.

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]

Yu, Y.

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Yu, Y. Z.

Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
[Crossref]

Yuan, J. J.

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), 1–9 (2019).
[Crossref]

Yuan, L.

D. Leykam and L. Yuan, “Topological phases in ring resonators: recent progress and future prospects,” Nanophotonics 9(15), 4473–4487 (2020).
[Crossref]

Zhan, P.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Zhang, B.

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]

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

Zhang, M.

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

Zhang, Q.

Zhang, X.

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Zhang, X. C.

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002).
[Crossref]

Zhang, Y.

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

Zhao, F. L.

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), 1–9 (2019).
[Crossref]

Zhao, J.

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

Zhao, Y.

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

Zhu, H.

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

Zhu, X. Y.

Zilberberg, O.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Zwick, T.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Adv. Opt. Mater. (1)

Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, “Applications of topological photonics in integrated photonic devices,” Adv. Opt. Mater. 5(18), 1700357 (2017).
[Crossref]

Adv. Opt. Photonics (1)

S. Atakaramians, S. Afshar, T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photonics 5(2), 169–215 (2013).
[Crossref]

Appl. Phys. Lett. (2)

B. Bahari, R. Tellez-Limon, and B. Kanté, “Topological terahertz circuits using semiconductors,” Appl. Phys. Lett. 109(14), 143501 (2016).
[Crossref]

S. Ma and S. M. Anlage, “Microwave applications of photonic topological insulators,” Appl. Phys. Lett. 116(25), 250502 (2020).
[Crossref]

Appl. Phys. Rev. (1)

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

Commun. Phys. (1)

Y. Ota, R. Katsumi, K Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018).
[Crossref]

IEEE Commun. Mag. (1)

Q. J. Gu, “THz interconnect: the last centimeter communication,” IEEE Commun. Mag. 53(4), 206–215 (2015).
[Crossref]

J. Infrared, Millimeter, Terahertz Waves (1)

T. Kleine-Ostmann and T. Nagatsuma, “A review on terahertz communications research,” J. Infrared, Millimeter, Terahertz Waves 32(2), 143–171 (2011).
[Crossref]

J. Lightwave Technol. (1)

Light: Sci. Appl. (2)

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]

M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2D, 3D and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020).
[Crossref]

Nanophotonics (4)

D. Leykam and L. Yuan, “Topological phases in ring resonators: recent progress and future prospects,” Nanophotonics 9(15), 4473–4487 (2020).
[Crossref]

Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020).
[Crossref]

M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 1(10), 425-434 (2020).

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]

Nat. Commun. (3)

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), 1–9 (2019).
[Crossref]

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin–valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018).
[Crossref]

Nat. Mater. (4)

A. B. Khanikaev, S. H. Mousavi, W. K Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

J. W. Dong, X. D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2017).
[Crossref]

B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002).
[Crossref]

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B.. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref]

Nat. Nanotechnol. (1)

M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019).
[Crossref]

Nat. Photonics (4)

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

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]

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7(12), 977–981 (2013).
[Crossref]

Nat. Phys. (2)

F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2018).
[Crossref]

C. Fang, L. Lu, J. Liu, and L. Fu, “Topological semimetals with helicoid surface states,” Nat. Phys. 12(10), 936–941 (2016).
[Crossref]

Nature (2)

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological Dirac insulator in a quantum spin Hall phase,” Nature 452(7190), 970–974 (2008).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref]

New J. Phys. (2)

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Opt. Commun. (1)

Y. Han, H. Fei, H. Lin, Y. Zhang, M. Zhang, and Y. Yang, “Design of broadband all-dielectric valley photonic crystals at telecommunication wavelength,” Opt. Commun. 488, 126847 (2021).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Opt. Mater. Express (2)

Phys. Rev. A (2)

Z. G. Chen, J. Mei, X. C. Sun, X. Zhang, J. Zhao, and Y. Wu, “Multiple topological phase transitions in a gyromagnetic photonic crystal,” Phys. Rev. A 95(4), 043827 (2017).
[Crossref]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Phys. Rev. B (3)

M. Li, J. Song, and Y. Jiang, “Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials,” Phys. Rev. B 103(4), 045307 (2021).
[Crossref]

M. Ezawa, “Topological Kirchhoff law and bulk-edge correspondence for valley Chern and spin-valley Chern numbers,” Phys. Rev. B 88(16), 161406 (2013).
[Crossref]

M. G. Silveirinha, “Bulk-edge correspondence for topological photonic continua,” Phys. Rev. B 94(20), 205105 (2016).
[Crossref]

Phys. Rev. Lett. (9)

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref]

G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, “Quantifying the robustness of topological slow light,” Phys. Rev. Lett. 126(2), 027403 (2021).
[Crossref]

S. Yao and Z. Wang, “Edge states and topological invariants of non-Hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018).
[Crossref]

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]

K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
[Crossref]

L. Balents and M. P. A. Fisher, “Chiral surface states in the bulk quantum hall effect,” Phys. Rev. Lett. 76(15), 2782–2785 (1996).
[Crossref]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 013905 (2008).
[Crossref]

Rev. Mod. Phys. (1)

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Sci. Adv. (1)

N. Parappurath, F. Alpeggiani, L. Kuipers, and E. Verhagen, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing,” Sci. Adv. 6(10), eaaw4137 (2020).
[Crossref]

Sci. Rep. (2)

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6(1), 22270 (2016).
[Crossref]

Y. Z. Yu, C. Y. Kuo, R. L. Chern, and C. T. Chan, “Photonic topological semimetals in bianisotropic metamaterials,” Sci. Rep. 9(1), 18312 (2019).
[Crossref]

Science (3)

S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018).
[Crossref]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref]

M. A. Bandres, S. Wittek, G Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018).
[Crossref]

Other (2)

H. Elayan, O. Amin, R. M. Shubair, and M. S. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” 2018 International Conference on Advanced Communication Technologies and Networking (CommNet)1–5 (2018).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton, 2008).

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.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of the THz VPC structure with cylindrical air grooves arranged in a honeycomb lattice on a Si slab with relative permittivity, $\epsilon _{r} =11.9$ atop a substrate with $\epsilon _{r} = 2.1$ , (b) The unit cell of the VPC comprising of two cylindrical air grooves having diameter $d_1$ and $d_2$ , along with the first Brillouin zone, (c) The band structure of the VPC for the symmetric case of $\Delta d=0$ with fixed parameters: $a = 250$ µm, $h = 220$ µm, $d_1 = d_2 =110$ µm for TE mode. The light blue shaded region denotes the light cone of the substrate. The inset shows the pictogram of the unit cell for $\Delta d=0$ .
Fig. 2.
Fig. 2. Band structure for TE mode of the proposed VPC structure for different asymmetries. Band structures for (a) $\Delta d=0.04a$ , (b) $\Delta d=0.12a$ , (c) $\Delta d=0.2a$ , (d) $\Delta d=0.28a$ , and (e) $\Delta d=0.36a$ . The inset depicts the unit cell for the different values of $\Delta d$ . The light blue shaded region denotes the extended bands while the pink shaded region represents the band gap of the VPC structure. The red traces represent the first TE band while the blue line signifies the second TE band of the VPC structure. $P_1$ and $P_2$ denotes the $K$ an $M$ points of the first and second band, respectively. (f) Table indicating the range of band gap for different values of $\Delta d$ .
Fig. 3.
Fig. 3. (a) Schematic of the topological VPC along with a magnified view, where the zigzag domain wall is denoted by the black line. The direction of incident electric field polarization is denoted by the green arrow. Dispersion of the topological edge states (b) in the absence of domain wall, and (c) in the presence of domain wall where, $k_x$ denotes the wavevector parallel to the direction of edge states propagation. (d) Transmission spectra with and without the domain wall. The red line represents the transmission in the presence of a straight domain wall while the black dashed line denotes the same for the case of no domain wall.
Fig. 4.
Fig. 4. Effect of asymmetry on the robustness of the THz topological edge states. (i) The dispersion of the topological edge states for (a) $\Delta d = 0.04a$ , (b) $\Delta d = 0.12a$ , (c) $\Delta d = 0.2a$ , (d) $\Delta d = 0.28a$ , and (e) $\Delta d = 0.36a$ , (ii) Transmission spectra (in dB) for the VPC structure when (a) $\Delta d = 0.04a$ , (b) $\Delta d = 0.12a$ , (c) $\Delta d = 0.2a$ , (d) $\Delta d = 0.28a$ , and (e) $\Delta d = 0.36a$ , (iii) Topological edge states field confinement for a straight domain wall for (a) $\Delta d = 0.04a$ , (b) $\Delta d = 0.12a$ , (c) $\Delta d = 0.2a$ , (d) $\Delta d = 0.28a$ , and (e) $\Delta d = 0.36a$ , and (iv) Topological edge states field confinement for an $\Omega$ -type domain wall with (a) $\Delta d = 0.04a$ , (b) $\Delta d = 0.12a$ , (c) $\Delta d = 0.2a$ , (d) $\Delta d = 0.28a$ and (e) $\Delta d = 0.36a$ .
Fig. 5.
Fig. 5. Effect of asymmetry on the robustness of the THz topological edge states. (i) The variation of the bulk bandgap as a function of the asymmetry parameter, $\Delta d$ of the VPC structure. (ii) The dispersion of the topological edge states for (a) $\Delta d = -0.04a$ , (b) $\Delta d = -0.12a$ , (c) $\Delta d = -0.2a$ , (d) $\Delta d = -0.28a$ , and (e) $\Delta d = -0.36a$ , (iii) Transmission spectra (in dB) for the VPC structure when (a) $\Delta d = -0.04a$ , (b) $\Delta d = -0.12a$ , (c) $\Delta d = -0.2a$ , (d) $\Delta d = -0.28a$ , and (e) $\Delta d = -0.36a$ , (iv) Topological edge states field confinement for a straight domain wall for (a) $\Delta d = -0.04a$ , (b) $\Delta d= -0.12a$ , (c) $\Delta d = -0.2a$ , (d) $\Delta d =- 0.28a$ , and (e) $\Delta d = -0.36a$ , and (v) Topological edge states field confinement for an $\Omega$ -type domain wall with (a) $\Delta d = -0.04a$ , (b) $\Delta d =- 0.12a$ , (c) $\Delta d = -0.2a$ , (d) $\Delta d = -0.28a$ , and (e) $\Delta d = -0.36a$ .
Fig. 6.
Fig. 6. (a) Pictorial depiction of the zigzag domain interface formed by the larger air holes ( $\Delta d = +ve$ ). Poynting vectors of the VPC structure for (b) a straight domain wall, and (c) and an $\Omega$ -type domain wall, (d) Pictorial depiction of the zigzag domain interface formed by the smaller air holes ( $\Delta d = -ve$ ). Poynting vectors of the VPC structure for (e) a straight domain wall, and (f) and an $\Omega$ -type domain wall.

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

H = ν D ( σ x δ k x + σ y δ k y ) + γ σ z

Metrics

Select as filters


Select Topics Cancel
© Copyright 2022 | Optica Publishing Group. All Rights Reserved