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Abstract
We propose a heterogeneous structure, which are composed of two valley photonic crystals (VPCs) with opposite valley Chern numbers and air channel. With the increasing width of the air channel, valley-locked waveguide modes are found in topological bandgap by analyzing energy bands. Finite element method (FEM) simulation results show that the fundamental and high order modes are valley-locked, propagating unidirectionally under the excitation of chiral source, and possess higher flux compared to the valley-locked topological edge state in the domain wall. Besides, the immunity to backscattering in bend and couplers, and the robustness to random disorders are discussed in detail. We also investigate the one-way multimode interference (MMI) effect based on valley-locked waveguide modes, and design topological beam splitters. Our study provides a novel idea for topological transport with high flux, and more freedom to design valley-locked waveguide devices, including bends, couplers and splitters.
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1. Introduction
Topological phase has become a very important area of research in condensed matter physics [1–4], which also has been introduced into photonic [5–9] and acoustic [10–14] systems. Topological states were firstly observed in photonic crystals (PCs) with broken time-reversal symmetry by applying magnetic field [4], and then were also investigated in systems with broken spatial symmetry [15–17]. Edge states with topological protection are well confined along domain wall, with the immunity to backward scattering and robustness [18–20], which provide a method to resist the inevitable disorder and defects in the manufacturing process. Therefore, topological edge states can be applied to topological transport in photonic integrated circuits [21–25], including topological photonic routing [26], topological slow light [27] and the ultracompact thermo-optic switches [28].
Based on the bulk-edge correspondence, the valley-locked edge states are achieved in 2-D valley photonic crystals (VPCs) with opposite valley Chern numbers [29–31]. The propagation directions of valley edge states, which pass through sharp bends without backscattering [32–35], are determined by the excitation of chiral source. Besides, topological VPCs waveguides work at room temperature with large bandwidth and low dispersion, which can find applications in phototunable, robust on-chip topological terahertz devices [36–38], topological interconnect–cavity systems [39] and electrically pumped topological laser [40]. Traditional studies of VPCs focus on the topological transport effects of single domain wall. However, the energy flux of the valley-locked edge states in the domain wall structures is very low, which is difficult to be integrated with conventional waveguides. Recently, researchers proposed a sandwiched structure composed of VPCs and Dirac crystal and found the fundamental mode is valley-locked with high-flux [34].
In this work, we propose a valley topological waveguide which are composed of dielectric cylinder arrays with air channel. Besides the fundamental mode, our waveguides support high-order valley-locked waveguide modes. The simulation is conducted finite element method (FEM), which shows that the proposed waveguide modes possess high flux, providing a possible way for integrating with conventional waveguide devices. The topological characteristics, including momentum-valley locking, immunity to backscattering, and the robustness to random defects are verified. Furthermore, the unidirectional multimode interference (MMI) effects and self-imaging phenomena are also found, which originate from the valley-locked waveguide modes. Most studies of VPCs focus on the topological transport of single domain wall. The width of the transport channels formed by single domain wall is narrow, and the energy transport capacity is limited [28–31]. Our heterogeneous structure, composed of two VPCs and air channel, provides a possible method of achieving high energy flux. Therefore, our proposed heterostructure paves a new way for constructing high-flux and valley-locked waveguides, such as topological bends, couplers and splitters, enriching the researches of VPC.
2. Sandwiched heterostructure and band analysis
Figure 1(a) shows the 2-D schematic of the dielectric cylinder array (gray) in the air background. The green diamond dashed line indicates a unit cell. Each cell consists of two infinitely high dielectric cylinders A and B with relative permittivity ε (= 11.7). The lattice constant is a (= 10 mm). Here, we only consider the transverse magnetic (TM) mode. The radii of cylinders A and B are denoted as rA and rB, respectively. When rA = rB = a/2.3, the PC is named by PC0, whose photonic band is shown in the first left panel of Fig. 1(b). It can be seen that the degeneracy occurs at point K due to the symmetry of C6. Changing rA (or rB) can break the C6 symmetry and PC is reduced to C3 symmetry while keeping other parameters constant. When rA = a/2(a/4) and rB = a/4(a/2), PC is named by PC1 (PC2), whose photonic band is plotted in the middle (right) panels of Fig. 1(b). It is observed that the degenerate point at K is opened. The band gaps of PC1 and PC2 are both from 7.386 to 9.543 GHz. For a valley topological bandgap, the pseudo-spin direction of the state at K(K’) above the bandgap is opposite to that below the bandgap. The green and yellow dots of the K points in Fig. 1(b) represents the pseudo-spin-up and pseudo-spin-down states, respectively. The phase distribution of the Ez field at point K is obtained by calculation as shown in Fig. 1(c). Although both PC1 and PC2 retain C3 symmetry, the direction of the phase evolution at the center is opposite. The phase of the eigenmode of the first band at K point rotates clockwise in PC1 (yellow arrow), while that in PC2 rotates along counterclockwise direction (green arrow). Thus, topological property of PC1 is opposite to that of PC2.
Fig. 1. (a) Structure of 2-D VPC composed of dielectric cylinders in air background. (b) The band structure of PC0 (rA = rB = a/2.3), PC1 (rA = a/2, rB = a/4), PC2 (rA = a/4, rB = a/2). (c) Phase profiles of Ez at K point.
According to the k•p perturbation method [41,42], the effective Hamiltonian around the K/K’ valley is related to:
According to the bulk-edge correspondence, the interface between the PCs with opposite valley Chern numbers support the valley-locked topological edge state (VL-TES). However, the strong confinement of VL-TES along interface limits the flux. In this work, an air channel will be introduced between two PCs with opposite topological properties. The sandwiched structure of PC1/air/PC2 is formed, which can support valley-locked waveguide modes with high flux. The supercell of the proposed sandwiched structure in this paper is shown in Fig. 2(a), and the PC1 (PC2) is indicated by red (blue) cylinder array. We define the distance between PC1 and PC2 as H, and the distance between the adjacent large cylinders in PC1 (PC2) along the y direction as h (=${{\sqrt 3 a} / 2}$). When H is ${{\sqrt 3 a} / 3}$, the PC1/air/PC2 sandwiched structure is reduced to the domain wall of PC1/PC2. Figure 2(b) plots the projected band of PC1/air/PC2, when H is chosen as H1(=${{\sqrt 3 a} / 3}$+h), H2(=${{\sqrt 3 a} / 3}$+2 h), and H3(=${{\sqrt 3 a} / 3}$+3 h). Here, the grey region represents the bulk bands, and the red (blue) lines represent the 0th (1st) waveguide mode in the sandwiched structure of PC1/air/PC2. With increasing H, the dispersion of the 0th waveguide mode red shifts and the 1st waveguide mode inters the valley bandgap from the bulk band. Besides, the solid lines, the dashed lines, and the dotted line represents dispersions of waveguide modes when H is chosen as H1, H2, and H3, respectively. Here, we firstly discuss the 0th waveguide mode, which shows valley-locked characteristics. The Ez field distributions of the 0th waveguide mode at E1, E2, and E3 of Fig. 2(b) when the frequency is 8.92 GHz are shown in Fig. 2(c), respectively. E1, E2, and E3 points are around the K valley. It can be found that the Ez fields are concentrated at the air region, which are well confined along PC1/air/PC2 waveguide. Besides, the distribution of the Ez field shows symmetrical pattern. Figure 2(d) shows that there are rotational changes of phase at the interfaces between the VPC and air, and the rotational change direction of phase for the upper interface is opposite to that of lower interface. Thus, PC1/air/PC2 structure provides a possible way of achieving the valley-locked states based on the 0th waveguide mode in the sandwiched structure of PC1/air/PC2.
Fig. 2. (a) Schematic of PC1/air/PC2 supercell (indicated by green rectangle). (b) Energy band structure of supercell, when the width of the air layer is H1, H2 and H3 respectively. (c) Electric field distribution of the 0th waveguide mode at the three points E1, E2 and E3. (d) The phase distribution of Ez at the three points E1, E2 and E3.
3. Valley-locked and backscattering immune characteristics of waveguide states
To verify the valley-locked characteristics of the fundamental (0th) waveguide mode, we proposed the PC1/air/PC2 straight waveguide, and performed a 2-D FEM simulation using COMSOL software. The 0th waveguide mode with symmetrical pattern can be excited by setting up a pair of excitation source with reversed orbital angular momentum (OAM), which is indicated by yellow stars in Figs. 3(a), (d) and (g) when H equals to H1, H2, and H3, respectively. Each excitation consists of four-line sources in square array for achieving the circularly polarization. Here, we simulated the |Ez| field distributions when the frequency is 8.92 GHz, corresponding to E1, E2, and E3 of Fig. 2(b). When one source array with clockwise phase distribution is put at upper position of PC1/air/PC2 waveguide and the other source array with counterclockwise phase distribution is put at below position, which are used for mode matching around the K’ valley. Figures 3(b), (e) and (h) show the |Ez| field distributions, which demonstrates that the 0th waveguide mode unidirectionally propagates along the left direction with valley-locked characteristics. On the contrary, we choose another pair of excitation source array, including one source array with counterclockwise phase distribution put at upper position of PC1/air/PC2 waveguide and the other source array with clockwise phase distribution put at below position, for mode matching around the K valley. The valley-locked waveguide mode can be excited and propagates unidirectionally along the right side, and the |Ez| field distributions are shown as Figs. 3(c), (f) and (i). Thus, it is verified that the 0th waveguide mode of PC1/air/PC2 is valley-locked, and the propagation direction can by controlled by changing the chirality of the source.
Fig. 3. Waveguide structure and simulated |Ez| field distribution at f = 8.92 GHz for the fundamental (0th) waveguide mode of Fig. 2. A pair of chiral sources are set at the yellow stars. (a)-(c) The width of the air layer is H1. (d)-(f) The width of the air layer is H2. (g)-(i) The width of the air layer is H3.
Next, we designed waveguide couplers and bends based on PC1/air/PC2 structure to explore the backscattering immune properties of the 0th waveguide mode in Fig. 4. Figures 4(a) and (c) show a waveguide coupler, which achieves connecting two PC1/air/PC2 structures width different H. Here, H is chosen as H1(=${{\sqrt 3 a} / 3}$+h) and H3(=${{\sqrt 3 a} / 3}$+3 h), respectively. Based on the results of Fig. 2, the working frequency is chosen as 8.92 GHz for ensuring the 0th waveguide mode around the K valley being supported in two PC1/air/PC2 structures with width H1 and H3. Figure 4(b) shows the |Ez| field distribution when light is input from the left. The 0th valley-locked waveguide mode around K valley is excited and coupled from narrow to wide PC1/air/PC2 structure without backscattering, and transmission efficiency is almost equal to 100%. On the contrary, Fig. 4(d) shows the |Ez| field distribution when light is input from the right. The 0th valley-locked waveguide mode around K’ valley is excited, and coupled from wide to narrow PC1/air/PC2 structure without backscattering. Figures 4(e) and (g) shows a zigzag waveguide based on PC1/air/PC2 structure with H1(=${{\sqrt 3 a} / 3}$+h) and H2(=${{\sqrt 3 a} / 3}$+2 h), respectively. The |Ez| field distributions are shown in Figs. 4(f) and (g) when the working frequency is chosen as 8.84 GHz. Obviously, the energy flux propagates through the bens without backscattering. The transmittance of 8.0–9.5 GHz for these waveguides is shown in Figs. 4(i) and (j). Figure 4(i) plots the transmission of the 0th waveguide mode in waveguide coupler. Both transmission directions (black for left input and red for right input) show 100%-transmittance around the K(K’) valley in Fig. 4(i). Figure 4(j) shows the transmittance of the 0th waveguide mode in zigzag waveguide. The black (red) line also exhibits 100%-transmittance around the K(K’) valley for air width H = H1(H2). Besides, the two designs also show higher flux, compared with the single domain wall structure. Thus, based on valley-locked waveguide mode in PC1/air/PC2, various waveguide structures with backscattering immunity and high flux can be designed.
Fig. 4. (a) and (c) waveguide coupler based on PC1/air/PC2 for the 0th waveguide mode. The width of the narrow (broad) waveguide section is H1 (H3). (b) and (d) Simulated |Ez| field distribution of (a) and (c) when f = 8.92 GHz, respectively. (e) Structure of zigzag waveguide with width H1. (f) Simulated |Ez| field distribution of (e) when f = 8.84 GHz. (g) Structure of zigzag waveguide with width H2. (f) Simulated |Ez| field distribution of (g) when f = 8.84 GHz. (i) Transmittance of waveguide coupler for both input directions. The black (red) line represents left (right) input. (j) Transmittance of zigzag waveguide. The black (red) line represents air width H = H1(H2).
In a further way, we explore the valley topology of the higher order modes in PC1/air/PC2 waveguide. Figure 5(a) shows the projected band of PC1/air/PC2 with H increasing from H4 (=${{\sqrt 3 a} / 3}$ + 4 h) to H6 (= ${{\sqrt 3 a} / 3}$ + 6 h), and other parameters are the same with those of Fig. 2. Obviously, the bands of waveguide modes in bandgap red shift and more high order waveguide modes enter from bulk band to valley bandgap with increasing H. Here, the solid lines, the dashed lines, and the dotted line represents dispersions of waveguide modes when H is chosen as H4, H5, and H6. The red (blue, green and purple) lines represent the 0th (1st, 2nd and 3rd) waveguide modes. Here, we take the 1st waveguide mode for example. The Ez field distributions of the 1st waveguide mode at E4, E5, and E6 of Fig. 5(a), when the working frequency is 8.92 GHz, are shown in Fig. 5(b), respectively. Here, E4, E5, and E6 points are around the K valley. It can be found that Ez fields are concentrated at the air region, and show anti-symmetrical pattern. Figure 5(c) shows the phase distribution, and there are phase vortexes at the interface between VPCs and air, which indicates it is possible to achieve valley-locked characteristic for the higher waveguide modes.
Fig. 5. (a) Energy band structure of supercell, when the width of the air channel is H4, H5 and H6 respectively. (b) Ez field distribution of the first waveguide mode at the three points E4, E5 and E6. (c) The phase distribution of Ez at the three points E4, E5 and E6
We discuss the valley-locked and unidirectional transmission characteristics of higher order waveguide modes (taking the 1st waveguide mode for example) in the straight PC1/air/PC2 waveguide with H increasing from H4 (=${{\sqrt 3 a} / 3}$ + 4 h) to H6 (= ${{\sqrt 3 a} / 3}$ + 6 h). The yellow stars represent a pair of chiral sources with opposite OAM, just like the same way of Fig. 3, and the working frequency is chosen as 8.92 GHz. In Figs. 6(b), (e) and (h), when one source array with clockwise phase rotation is put at upper position of PC1/air/PC2 waveguide and the other source array with counterclockwise phase rotation is put at below position, the 1st waveguide mode can be exited due to mode matching around the K’ valley. The 1st waveguide mode can propagate along the left direction unidirectionally with valley-locked characteristic. In Figs. 6(c), (f) and (i), when one source array with counterclockwise phase rotation is put at upper position of PC1/air/PC2 waveguide and the other source array with clockwise phase rotation is put at below position, the 1st waveguide mode around the K valley can be exited, and propagate along the right direction unidirectionally. Thus, PC1/air/PC2 structure provides a possible way of achieving the high-order valley-locked modes, which can greatly enhance the flux in a further way, compared with the valley-locked edge state supported by the single domain wall between PCs with opposite valley Chern numbers.
Fig. 6. Waveguide structure and simulated |Ez| field distribution of 1st waveguide mode at f = 8.92 GHz. The chiral sources with opposite OAM are set at the yellow stars. (a)-(c) The width of the air layer is H4. (d)-(f) The width of the air layer is H5. (g)-(i) The width of the air layer is H6.
4. Robustness and multimode interference
The valley-locked waveguide modes also possess immunity to defects. To verify the robustness, we built heterogeneous structures containing different defects in Figs. 7(a) and (c). We choose the same frequencies (f = 8.92 GHz) and source settings as in Figs. 3(d) and 6(a). As shown in Figs. 7(a) and (c), the gray color represents the fluctuation of the radius and position of the column. The position of the cylinder varies from [0.1a, 0.2a] in any direction, and the radius of the cylinder varies from [0.8r, 1.2r] (r represents the original radius). The |Ez| field distributions of the 0th and 1st waveguide modes around K valley are shown in Figs. 7(b) and (d), respectively. The energy fluxes can be calculated by U = 1/2 ∫l Re (E × H*)·dl. We measured the energy fluxes U1 and U2 as the two ports of the waveguide. Then we defined the normalized transmission directionality (NTD) as |U1 - U2| / (U1 + U2). The calculated NTD are 0.994 for Fig. 7(b) and 0.986 for Fig. 7(d). Thus, valley-locked and unidirectional propagation of waveguide modes are unchanged, which demonstrates robustness to defects and proves the valley-locked waveguide modes are highly resistant to common defects.
Fig. 7. (a) and (c) Schematic diagram of the waveguide with perturbation introduced. The width of the air layer is H2 and H4. The gray color represents the fluctuation of the radius and position of the column. (b) |Ez| field mode distribution of (a) at f = 8.92 GHz with the 0th mode excited. (d) |Ez| field pattern distribution of (c) at f = 8.92 GHz with the 1st mode excited.
Our heterogeneous structure support multiple waveguide modes with topological properties, which provides a possible way to achieve MMI with valley-locked characteristic. When H = H6(= ${{\sqrt 3 a} / 3}$ + 6 h), the projected band of PC1/air/PC2 is shown in Fig. 8(a). The grey region is bulk band. The red (blue, green, and purple) lines represent the 0th (1st, 2nd, and 3rd) waveguide mode. The right part of Fig. 8(a) also depicts the Ez and phase distribution of Ez for the 2nd waveguide mode when the frequency is 9.01 GHz, which is corresponding to orange dots in left part of Fig. 8(a) around K valley. Phase vortexes along the interfaces between VPC and air indicate valley-locked characteristic. Besides, the valley-locked and backscattering immunity of the 0th modes have been discussed. Therefore, we discuss MMI based on the 0th and 2nd waveguide modes in PC1/air/PC2 structure with H (=H6). According to the self-imaging principle in MMI [43], the interference pattern can be characterized by the beat length Lπ.
Here, βv is the propagation constant of v order waveguide mode. We construct a waveguide coupler to achieve valley-locked MMI as shown in Fig. 8(b). The H of the left narrow waveguide is H2, and that of the wide waveguide on the right is H6. The yellow stars represent a pair of chiral sources with opposite OAM, just like the sources of Fig. 3(d), which can excite the 0th valley-locked waveguide mode. The 0th waveguide mode around K valley propagates unidirectionally to the narrow waveguide exit, which can act as the input image. And then the 0th and 2nd waveguide modes will be excited due to mode symmetry in PC1/air/PC2 structure with H (=H6). The two valley-locked modes propagate along one way, and unidirectional MMI happens. The |Ez| field distributions when the frequency are 9.01 and 9.20 GHz are shown in Figs. 8(c) and (d). Base on self-imaging principle of Eq. (2), the anti-images and positive images of the input field unidirectionally propagate and alternate. The positive image of the input field will appear in position 6nLπ/8 (n is positive integer) and the anti-image will appear in position (6n-3) Lπ/8 (n is positive integer). Thus, the period L of the MMI is 3Lπ/4. For the MMI based on 0th and 2th waveguide modes, Lπ (= 8π/ (3(β0 − β2))) can be calculated to be Lπ (9.01 GHz) = 13.68a and Lπ (9.20 GHz) = 15.33a, based on Eq. (2). β0 and β2 can be extracted from Fig. 8(a). So, the theoretical values 3Lπ/4 of MMI period are 10.26a and 11.5a, which coincide with the simulation results L (=10.18a and 11.45a) in Figs. 8(c) and (d), respectively.
Fig. 8. (a) Energy band structure with width of H6. Ez field distribution and phase distribution of Ez at the air layer width of H6. (b) Waveguide coupler structure that supports MMI effects. (c) Simulated |Ez| field distribution at f = 9.01 GHz and f = 9.20 GHz. (e) Structure of beam splitter. (f) and (g) Simulated |Ez| field distribution of beam splitter at f = 9.07 GHz and f = 9.31 GHz.
To further demonstrate the application of MMI in PC1/air/PC2, we design a splitter as shown in Fig. 8(e). The width of the incident waveguide is H2, and that of the multimode region is H6. The widths of output arms are H1, H2 and H1. The length of the MMI zone is 42a. Base on Eq. (2), Lπ of MMI based on 0th and 2th waveguide modes can be calculated to be Lπ (9.07 GHz) = 14.10a and Lπ (9.31 GHz) = 16.12a. So, the theoretical values 3Lπ/4 of period are 10.57a and 12.09a, which coincide with the simulation results L (=10.51a and 12.02a) in Figs. 8(f) and (g). In Fig. 8(f), the positive image of input field with the frequency f = 9.07 GHz propagate after the distance of 4 period, and go through along the horizontal arm with width H2. In Fig. 8(g), the anti-image of input field with the frequency f = 9.31 GHz propagate after the distance of 3 period, and go through along the other two arms with width H1. Thus, the unidirectional splitter based on valley-locked MMI is achieved.
We have theoretically verified and numerically simulated the feasibility of the VPC/air/VPC waveguide structure, and then we built a coupler and splitter based on this sandwiched structure. However, there are some difficulties in the practical fabrication of tiny structures, especially at high frequencies. One solution is to employ a top-down nanofabrication process, defining VPC patterns on SOI wafers with electron beam lithography. Afterwards, the VPC structure is etched on top silicon layer of the SOI wafer by using inductively coupled plasma (ICP) etching step. Finally, the sample need to be ultrasonicated, cut up and polished to achieve efficient coupling [26,44].
5. Conclusions
In summary, we propose a sandwiched structure of VPC/air/VPC, which supports valley-locked waveguide modes. Besides the fundamental mode, high-order valley topological waveguide modes inter the topological band gap from bulk bands with increasing the width of air channel. FEM simulation results demonstrate that the waveguide modes propagate unidirectionally under the excitation of chiral source with high flux. The characteristic of the immunity to backscattering are verified in couplers and bends based on heterostructure. After introducing random disorders, the waveguide modes display the robust transport of the valley topology. Finally, the unidirectional self-imaging phenomena originating from MMI of the valley-locked waveguide modes are found, which paves a new way of designing topological splitters. Our work demonstrates that it is possible to manipulate the topological transport phenomena of waveguide modes by introducing the air channel. Exploiting the diversity of waveguide modes allows richer topological transport effects, which contributes to the integration with conventional PC waveguides where energy is concentrated in the air channel. Thus, the heterostructure based on VPCs find new freedom for constructing topological photonic integrated devices.
Funding
Postgraduate Research and Practice Innovation Program of Jiangsu Province (SJCX22_1103); National Natural Science Foundation of China (114041434).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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