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Abstract
Finite-size effect plays a significant role in topology photonics not to mention in reality all experimental setups are in finite-size. A photonic bandgap is opened in the topological edge state dispersion if a topological photonic crystal with finite width is considered, and the bandgap size relies on the finite-size effect. Pseudospin-preserving and pseudospin-flipping processes can be realized when a selectively switch of the pseudospin of edge states are customized by our designs. Our microwave experiments also successfully demonstrate pseudospin switch-on and -off behaviors in a finite-width photonic crystal. By combining photonic crystals with finite widths, a multi-tunneling proposal of topological photonic crystals can also be achieved. Our study of the finite-size effect will provide new approaches and thoughts to improve the development of topological photonic devices in the future.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In analogy to topological insulators in condensed matter physics [1–4], photonic and acoustic topological materials possess unique properties, in which backscattering is suppressed and interface or edge states are immune to defects and disorders [5–23]. Photonic topological insulators composed of photonic crystals (PCs), periodic distributions of dielectric materials, provide a new platform for the light propagation and manipulation. Though the topology arouses from infinitely periodic structures, by considering the finite-size effect, novel degrees of freedom are introduced to tune the topology properties, not to mention that in reality all experimental samples are in finite-size. Thus, it is very important to investigate the transport and coupling of topological interface states in finite structures. Two-dimensional finite topological PCs, which are finite in both x and y directions, have discrete spectra of edge states [24], and have been used in the construction of topological whispering gallery modes [25,26] and lasers [27–29].
For a finite-stripped topological insulator, which is only finite in one direction, edge states along the two opposite sides can couple with each other to generate a band gap in the spectrum due to the finite -size effect [30–38]. Even in a three-dimensional topological insulator, a surface state can also tunnel from one to another surface owing to the finite-size effect [31]. Moreover, compared with an infinite geometry, edge states in a finite strip is reported to have stronger robustness [32]. Enormous efforts have been devoted to explore the spin-preserving and spin-flipping tunneling processes between topological edge states in finite-width nanoribbons [33–35] and acoustic crystals [36], which could give rise to designs of topological transistors in device applications [37]. To our knowledge, the tunneling and pseudospin switching of topological edge states have not yet been experimentally demonstrated in topological PCs.
In this work, we investigate the finite-width effect of a two-dimensional topological PC ribbon and the evolution of its band diagram, which bears similarity to the finite-size effect of a topological insulator ribbon in condensed matter physics. The pseudospin of topological edge states in finite-width structures can be selectively switched. Microwave experiments are performed to observe the tunneling behavior of topological interface states. A design is also proposed to realize multi-tunneling by manipulating the structure widths between topological and trivial PCs.
2. Pseudospin switching
The artificial honeycomb lattice is considered to explore the behavior of pseudospin switching in a photonic topological insulator. Figure 1(a) shows the schematic of a PC supercell, composed of hexagonal unit cells with a lattice constant $a = 17\sqrt 3 \,mm$. We use alumina to construct the dielectric cylinders with a relative permittivity $\varepsilon = 7.5$, and diameters $d = 6\,mm$. The intra couple distance ${h_1}$ and inter couple distance ${h_2}$, defining as the distances between neighboring cylinders, satisfy the relationship $2{h_1} + {h_2} = a$. When ${h_1} > {h_2}$ and ${h_1} < {h_2}$, the PCs are topological (yellow circles) and trivial (blue circles), respectively [17]. The two distinct PCs are assembled along the zigzag interface (marked by a dashed line). The transverse-magnetic (TM, with ${E_z}$ polarization) photonic band diagram of topological PC (${h_1} = 10.8{\kern 1pt} \,mm$) unit cell is shown in Fig. 1(b), as calculated using COMSOL Multiphysics software based on the finite-element method. Though when ${h_1} = {h_2}$, a degenerate Dirac cone appears at Γ point of the first Brillouin zone (the inset in Fig. 1(b)), with ${h_1} \ne {h_2}$, a complete band gap emerges in topological PCs. The trivial PC (${h_1} = 8.8{\kern 1pt} \,mm$) has the common band gap but inverted eigenmodes with those of the topological PC. We calculate the projected bands of PC supercells with periodic boundaries along the x and y directions, in which different widths W of the gapped trivial PCs between topological PCs is used. As displayed in the left panel of Fig. 1(c), two sets of edge state bands are nearly degenerate, located separately along the two interfaces between topological and trivial PCs, while the width $w = 6a$. At this width, we may conclude that the trivial PC will be of bulk effect. However, when we further decrease the trivial PC width, the two sets of edge states tend to couple together and a band gap appears in the spectrum owing to the finite-size effect, as shown in the right panel of Fig. 1(c). The width of PC definitely affects the gap size of edge state dispersion (see Fig. 1(d)). The gap size decreases dramatically and eventually vanishes when the width increases to 6a. A similar phenomenon can also be observed when a finite-width topological PC is gapped between two trivial PCs, where the finite thickness alters the band gap size in the edge state dispersions.
Fig. 1. (a) The schematic of a PC supercell, consisting of a finite-width trivial PC (blue circles) gapped between two topological PCs (yellow circles). The lattice constant of the hexagonal unit cell is $a = 17\sqrt 3 \,mm$. (b) The band diagram of the topological PC. Inset: the first Brillouin zone. (c) Edge state dispersions are degenerate and open a gap for the trivial PC at $w = 6a$ (left panel) and $w = 2a$ (right panel), respectively. Projected bulk bands are painted in gray. In frequency region III (in yellow), the edge state preserves pseudospin while transporting. In region II (in blue), the pseudospin of edge states flips. In region I (in red), the edge state occurs perfect reflection in band gap. (d) The bandgap between the two sets of edge state bands as a function of the trivial PC width.
We also analyze the transport properties of edge states in finite-size PC structures. A H-shape trivial PC structure surrounded by topological PC is studied, as depicted in Fig. 2(a). The widths of the left and right ribbons of trivial PCs are set as w0, which are chosen as 6a to ensure gapless edge state dispersion. The width of middle constriction of trivial PC is $w = 2a$, which corresponds to a gapped band diagram (see Fig. 1(c)). A length $L = 23a/\sqrt 3 $ of the trivial PC is first used in simulations. An exciting source is placed at the input port P1, so that the pseudospin up edge state (red arrows) is excited to propagate rightward. Three output ports are defined as P2-P4. Although topological edge state localizes at edges, it decays exponentially into bulk PCs. In the finite-width topological PC, edge sates along the two opposite interfaces can interfere through the finite-size bulk PC, leading to a tunneling effect from the top channel to the bottom one. Interestingly, at different frequency ranges, totally different coupling and propagation properties are achieved. For the lowest and highest band of the edge state dispersion (region III in Fig. 1(c)), the edge state coupling is very weak: edge states within these frequencies tend to propagate along its original interface. As illustrated in the simulated ${E_z}$ field distribution in Fig. 2(b) at 6.48 GHz, pseudospin-up edge state, for example, preserves its initial spin while traveling through the middle constriction with narrower width. For frequencies within frequency region II of the dispersion, pseudospin flipping will occur when the edge state propagates through the middle constriction. The pseudospin up edge state will tunnel to the bottom channel and exits at port 4 with pseudospin flipping (see the simulation in Fig. 2(c) at 6.42 GHz). Please be noted that only pseudospin down state can propagate rightward through the bottom channel of ${w_0} = 6a$ ribbon. Furthermore, as a band gap opens in frequency region I due to the $w = 2a$ middle constriction, the gapless edge state originally exists on ${w_0} = 6a$ ribbon will be blocked. As the same pseudospin is forbidden to propagate along the same channel backwardly, a nearly total reflection occurs by tunneling to the opposite channel with the same pseudospin, shown in Fig. 2(d) at 6.31 GHz. We have witnessed that transmitted electromagnetic energy redistributes among the three output ports at different frequencies, and thus we also calculate the transmitted energy proportion to individual ports at different frequencies, as shown in Fig. 2(e). The energy is evaluated through line integral of electric field intensity along white dashed lines in simulations (see Fig. 2(b)). The finite-size tunneling effect also explains that in frequency region III, electromagnetic energy is still received at port 2. The constriction of trivial PC structure induces inevitable scattering tunneling to port 2. With time reversal symmetry preserved here, it is no wonder that the similar phenomena of pseudospin switching can still be observed when an exciting source is applied at the port 3. In Fig. 2(e), different frequency ranges corresponding to different pseudospin manipulation functions are also painted for better eye guidance purpose, which are total reflection (in red), pseudospin flipping (in blue) and pseudospin preserving (in yellow), respectively. By controlling widths and frequencies of topological PC structures, a spin filter device is thus constructed.
Fig. 2. (a) The schematic of topological spin filter. The widths of trivial PC are $w = 2a$ in the middle constriction and ${w_0} = 6a$ in left and right ribbons on both sides, respectively. The length of trivial PC constriction is $L = 23a/\sqrt 3 $ in simulations. Red (green) arrows indicate spin up (down) edge states existing in the top and bottom channels between topological and trivial PCs. (b) The simulated ${E_z}$ field intensity distribution of the pseudospin preserving at frequency 6.48 GHz. White star: an exciting source; White dashed lines: line integral of electric field intensity at three output ports. (c) Pseudospin flipping at 6.42 GHz. (d) The nearly total reflection of the pseudospin-up edge state at 6.31 GHz. (e) The transmitted energy distributions at three output ports. Red, black and blue lines correspond to ports 2, 3 and 4, respectively. Vertical arrows indicate the frequencies for panels (b)-(d).
To fully understand the pseudospin flipping properties, we carry out a further study by calculating the transmitted energy distribution at different lengths of the middle constriction in simulations. As shown in Fig. 3(a) at 6.42 GHz, the transmitted energies to output ports P3 and P4 are as functions of the length, which indicates the behavior of pseudospin preserving and flipping can be controlled by choosing a proper length. The pseudospin flipping process at $L = 20a/\sqrt 3 $ is displayed in Fig. 3(b), where the detailed in-plane magnetic field distributions (${H_x}$ and ${H_y}$) at different channels are shown in the inset. Within frequency region II where pseudospin flipping occurs, when the excited spin-up edge state (marked by a red arrow) propagates to the middle narrower constriction, the spin flipping occurs and its mode changes to pseudospin-down edge state (green arrow) propagating along the top channel firstly. Due to the finite-size effect, the spin-down edge state will tunnel to the bottom channel during propagation and change back to the pseudospin up state. By meeting the boundary between constriction and thicker ribbon, the spin flipping occurs once again and thus the pseudospin-down state exits at Port 4. Two types of pseudospin flipping process occur here: it will occur when propagating to the boundary between bulk trivial PC ribbon and the constriction with narrower width; it will also occur when propagating through the constriction. Thus, it is totally understandable that when considering a longer constriction, for instance with $L = 47a/\sqrt 3 $ as shown in Fig. 3(c), the pseudospin flipping occurs twice during the propagating along the constriction, and most electromagnetic energy exits at Port 3 instead. The length is critical to select a port to exit. If the electromagnetic field is mostly confined on the top/bottom channel when reaching the boundary between the constriction and bulk ribbon, it tends to exit at Port 3/Port 4 accordingly. Moreover, as electromagnetic field oscillation (also with pseudospin oscillation) during propagation, at a certain length, it will exit at both Ports 3 and 4. Displayed in Fig. 3(d) with $L = 35a/\sqrt 3 $, as both pseudospin up and down states exist in the constriction, nearly equal portion of energy will be distributed to Ports 3 and 4. The energy distribution will oscillate with different lengths of the constriction. It is totally understandable that after a third spin flipping during the propagating along the constriction, electromagnetic energy will again exit at Port 4, as shown in Fig. 3(e) with $L = 59a/\sqrt 3 $. Except for the ribbon width, the length of middle trivial constriction is another adjustable parameter to manipulate PC structures, so that we have one more degree of freedom for the designs of spin flipping devices.
Fig. 3. (a) Transmitted energies of ports P3 and P4 at different lengths of the middle constriction at 6.42 GHz. Vertical arrows indicate different lengths for panel (b)-(e). (b)-(e) Simulated ${E_z}$ electric fields at $L = 20a/\sqrt 3 $, $47a/\sqrt 3 $, $35a/\sqrt 3 $ and $59a/\sqrt 3 $, respectively. The red and green arrows indicate pseudospin-up and -down edge states, obtaining from ${H_x}$ and ${H_y}$ magnetic field distributions (the insets).
3. Experimental verification
To demonstrate the pseudospin switching in finite-width topological PCs, we also perform microwave experiments. The experimental setup of parallel-plate waveguide is shown in Fig. 4(a). In this configuration, alumina cylinders are assembled on the bottom metallic plate of waveguide. All other geometrical parameters are the same as in Fig. 1. An emitting antenna at the star position is used to excite edge states, while a probe antenna inserted through the upper metallic plate (not shown) measuring the ${E_z}$ electric field (z direction is along the cylinder axis). The height of alumina cylinders is 8 mm. In order to facilitate the movement of the translational stage, a thin air separation between cylinders and top metallic plate is used and thus experimental frequencies are higher than in simulations [16]. To avoid unexpected scattering from the boundary of the sample, the sample is surrounded by absorbing materials. Figure 4(b) displays the measured electric field distribution in experiments at 6.92 GHz. The pseudospin up edge state (red arrow) is excited but gradually tunnels to the bottom channel through the trivial PC constriction. Its pseudospin also flips to pseudospin down edge state (green arrow) and exits at port 4. Pseudospin switch-on is thus achieved. Please be noted that the scattering field measured at the bottom channel of left ribbon of trivial PC is only due to the exciting source. At 7.0 GHz frequency in Fig. 4(c), the experimental results prove that the pseudospin-up edge state is blocked by trivial PC constriction, and reflects to the bottom channel with the same pseudospin. Our electric field distributions measured in experiments also agree well with simulation results. The measured transmissions at output ports 2-4 (dashed lines in Fig. 4(b)) are displayed in Fig. 4(d). Ignoring the scattering of the exciting source at Port 2 (red circles), we can clearly see the transmission to Port 4 (blue triangles) is greater than that to Port 3 (black squares) in two blue shade regions, corresponding to pseudospin flipping. Nearly total reflection occurs within the red shaded frequency range where a band gap exists for the trivial PC constriction with narrower width.
Fig. 4. (a) Photograph of experimental setup. (b) Experimentally measured ${E_z}$ electric field distribution corresponding to pseudospin flipping at 6.92 GHz. The location of the exciting source is marked by a star. (c) The edge state is nearly totally reflected to the opposite channel at 7.0 GHz. (d) Integrations of magnitude of measured electric fields in experiments at three output ports corresponding to dashed lines in (b). Two vertical arrows indicate frequencies for panels (b)-(d).
4. Multi-tunneling
A complete analogy between electronic and photonic crystals can thus be concluded for that they both exhibit band gaps. Similar to photonic quantum well consisting of a quantum well structure [39,40], we can design a multi-tunneling device by utilizing finite-width PCs. The left and right panels in Fig. 5 shows the schematic of multi-tunneling configuration composed of PCs and the simulated electric field distribution at 6.42 GHz frequency. The excited pseudospin-up edge state transports in the vertical channel between infinite-width topological and trivial PCs. Then the edge state experiences tunneling twice in the two finite-width PCs, and the pseudospin also flips twice shown in red and green arrows. By tuning the PC width and length, we can control output at different ports. Thus, the finite-width topological and trivial PCs can be engineered to design some devices, such as topological optical switch, spin filter and logical gate.
Fig. 5. Left panel: The schematic of multi-tunneling consisting of infinite- and finite- width topological and trivial PCs. Right panel: The simulated ${E_z}$ electric field of a multi-tunneling device at 6.42 GHz. The excited edge state tunnels to opposite channels twice during the propagation.
5. Conclusion
In summary, we utilize the finite-size effect in finite-width topological PCs to manipulate the band dispersion of edge states. The pseudospin preserving and flipping of edge states can be realized by constructing topological and trivial PCs. Two mechanisms leading to pseudospin flipping are discovered which may help the design of spin filter devices. Our experimental results provide the direct observation of switch-on and off states. The pseudospin switching behavior offers us an efficient method to control the topological edge state transport, which paves a way to construct optical devices.
Funding
Qinglan Project of Jiangsu Province of China; Priority Academic Program Development of Jiangsu Higher Education Institutions; Natural Science Foundation of Jiangsu Province (BK20200630); National Natural Science Foundation of China (11874274, 12004425).
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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