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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 [1–5]. 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 [3–9]. 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 [16–19] and the quantum valley Hall effect [19–22], 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 [23–26] 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 [6,7,20–22,35] and various other systems [36–39].
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 [43–45], 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 [55–57] 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 [22,26,60–63] 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
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 $\Delta d=0$.
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.
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 $\Delta d$.
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 [20–22]. 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.
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.
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.
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) $\Delta d = 0.36a$.
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 $-ve$.
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) $\Delta d = -0.36a$.
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 $\Delta d= + ve$.
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.
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.
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